The Hunting of the Hopf Ring
Andrew Stacey Sarah Whitehouse
November 23, 2007
Abstract
We provide a new algebraic description of the structure on the set of
all unstable cohomology operations for a suitable generalised cohomology
theory, E*(). Our description is as a graded and completed version of a
TallWraith monoid. The E*cohomology of a space X is a module for this
TallWraith monoid. We also show that the corresponding Hopf ring of
unstable cooperations is a module for the TallWraith monoid of unstable
operations. Further examples are provided by considering operations from
one theory to another.
____________________________
2000 Mathematics Subject Classification. Primary: 55S25; Secondary: 55N20, *
*16W99
The authors acknowledge the support of the EPSRC, grant no.: GR/S76823/01.
1
1 Introduction
In this paper we provide a new algebraic description of the structure on the se*
*t of
all unstable cohomology operations for a suitable generalised cohomology theory,
say E*(). The bigraded set of unstable operations for E*() is identified by
the usual Yoneda lemma argument with the bigraded set of cohomology groups
of the representing spaces, {Ek(E_l)}. This has considerable structure and it is
natural to ask how to best describe it.
To date, the most comprehensive work in this area is due to Boardman,
Johnson, and Wilson [BJW95 ]. They provide the following descriptions of this
structure.
1. The unstable operations of a suitable cohomology theory define a monad
on the category of complete, Hausdorff, filtered, graded, commutative,
unital E*algebras.
2. The unstable operations of a suitable cohomology theory are dual to the
enriched Hopf ring of the corresponding cooperations.
The Hopf ring part  i.e., ignoring the enrichment  of the second answer is,
of course, a wellestablished and important notion in algebraic topology. Since
the work of Ravenel and Wilson [RW77 ] the language of Hopf rings has been
widely used and the structures of the Hopf rings associated to many important
cohomology theories have been calculated; see, for example [CMS02 , RW77 ,
RW96 , Wil84]. A useful introduction to Hopf rings, with further references, can
be found in [Wil00].
In considering a Hopf ring, we have not yet taken into account one of the
most obvious pieces of structure: that operations may be composed. A Hopf ring
does not include any structure which dualises to composition of operations. The
enrichment in the second answer encodes the dual of composition. The enriched
Hopf ring structures of several important cohomology theories are described
in [BJW95 ].
The first answer, describing operations as a monad on a suitable category,
certainly includes the composition as a fundamental part of the structure. How
ever, this answer does not explicitly describe the internal structure on the set
of operations; instead it specifies the action of operations on some category.
In particular, this approach does not lend itself to specifying the structure of
operations via generators and relations.
Of these descriptions, that of an unenriched Hopf ring has proved to be more
amenable to further study than either that of an enriched Hopf ring or that of
a monad on a suitable category of algebras.
Our description of operations may be thought of as a monoidal reinterpre
tation of the first answer. It is algebraic in nature and we employ the language
of general or universal algebra to express it. One advantage of this approach
is that it describes the structure of unstable operations, including compositio*
*n,
directly. Another is that it allows for descriptions of unstable operations via
generators and relations. Such descriptions are expected to shed light on the
relationships between stable, additive, and unstable operations. For the Morava
Ktheories and related cohomology theories, these relationships were studied in
[SW06 ]; the results of this paper provide the foundations for a very explicit
description, via generators and relations, of the splitting of stable operations
2
from unstable operations given in [SW06 ]. We expect to develop this point of
view for familiar examples in future work.
Roughly speaking, our answer is that the unstable operations of a suitable
theory have the structure of a graded, completed TallWraith monoid; a term
that we shall now explain.
Let V be a variety of algebras, in the sense of general or universal algebra*
*. A
TallWraith Vmonoid is a set with precisely the algebraic structure required f*
*or
it to act on Valgebras. To make this precise, one considers the category of co*
*V
algebra objects in V. This is equivalent to the category of representable funct*
*ors
from V to V and so it has a monoidal structure corresponding to composition
of functors. A TallWraith Vmonoid is then defined to be a monoid in this
category. One example is very familiar: a ring is a TallWraith Vmonoid for
V the category of abelian groups. The case originally considered by Tall and
Wraith in [TW70 ], under the name biring triple, was for V the category of
commutative unital rings. Recently, Borger and Weiland [BW05 ] rediscovered
this and extended it to the case where V is the category of commutative unital
kalgebras, for a commutative unital ring k. They adopted the term plethory
in that situation; thus a plethory is thatwhichactsonalgebras. This is clea*
*rly
relevant to our purposes as unstable cohomology operations of multiplicative
cohomology theories act on the cohomology algebras.
It is also clear that there remains work to be done, because the cohomol
ogy theories that we are considering are graded and topologised. The grading
introduces no technical difficulties: varieties of graded algebras have been st*
*ud
ied for almost as long as varieties of ordinary algebras; they are also known as
manysorted algebras in the literature. We arrive at TallWraith V*monoids for
V* a variety of graded algebras. These are naturally bigraded, as are unstable
cohomology operations.
The main work of this paper comes in the extension of TallWraith monoids
to a suitably topologised context. For E*() a multiplicative cohomology theory
and X a CW complex, E*(X) is given the profinite topology; that is, the
filtration topology for the filtration by the ideals {keri*: E*(X) ! E*(Xf)}
for all inclusions of finite subcomplexes i: Xf ! X. We therefore develop
a theory of filtered TallWraith monoids so that our description of unstable
operations takes into account the profinite filtrations. While there are vario*
*us
notions of filtered objects in a (suitable) category in the literature, we have*
* not
been able to find a setup suited to our needs. Therefore we introduce a suitab*
*le
definition of filtered objects in a category to model the topology; in this set*
*ting
we introduce isofiltrations as the generalisation of complete,Hausdorff!space*
*s.!
This allows us to formulate the notion of a TallWraith V*monoid, where V*
denotes the category of isofiltered objects in the variety of graded algebras,*
* V*.
Once we have established the general theory of graded filtered TallWraith
monoids, the applications to suitable generalised cohomology theories are strai*
*ght
forward. We can now state precisely our monoidal reformulation of the first
description of unstable operations. We adopt standard notation related to a
cohomology theory E*(), so the representing spaces are denoted by E_k(for
k 2 Z), the corresponding homology theory is denoted by E*(), and the coef
ficient ring by E*.
Theorem A. Let E*() be a graded, commutative, multiplicative, cohomol
ogy theory. Let V* be the variety of graded, commutative, unital E*algebras.
3
Suppose that E*(E_k) is free as an E*module for each k. Then E*(E_*) is a
!*
TallWraith V monoid.
As noted in [BJW95 ], the freeness hypothesis of theorem A is satisfied for
HFp, BP , MU, K(n), and KU.
There is a natural notion of a module for a TallWraith monoid. The coho
mology of spaces provides examples of modules for the TallWraith monoid of
unstable operations.
Theorem B. Let E*() and V* be as in theorem A. Let X be a topological
space. The natural morphisms of sets
Ek(X) ! V*(E*(E_k), E*(X))
___
make the completed cohomology of X, E*(X), into a module for the TallWraith
!* *
V monoid of unstable operations E (E_*).
It turns out that the Hopf ring of cooperations is also a module for the
TallWraith monoid of operations, and it is this extra structure that is encoded
in the term enriched. Further examples come from considering operations and
cooperations from one theory to another.
Theorem C. Let E*() and V* be as in theorem A. Let F *() be an
other graded, commutative, multiplicative cohomology theory. Suppose that each
F*(E_k) is free as an F *module. Then the following statements are true.
1. The bigraded set F *(E_*) of unstable operations E*() ! F *() has the
!* *
structure of a module for the TallWraith V monoid E (E_*).
2. The Hopf ring F*(E_*) is a module for E*(E_*).
3. Let X be a space such that F*(X) is free as an F *module. Let CF* be the
category of F *coalgebras. Then the natural morphisms of sets
Ek(X) ! CF*(F*(X), F*(E_k))
extend to a morphism of modules for E*(E_*).
This paper is organised as follows. Section 2 covers background material
from general algebra, in both the ungraded and graded contexts. The main aim
of this section is to establish the necessary conditions to consider TallWraith
monoids and certain important related concepts. Section 3 is concerned with
setting out the necessary details of the theory of filtered objects in a catego*
*ry. In
this section we consider several types of filtrations and the relationships bet*
*ween
them. As we shall want to apply the constructions of general algebra in such
categories we are also concerned with establishing the categorical properties
of these categories of filtered objects. The main technical work of this paper
is in this section and concerns functors between categories of filtered objects.
Section 4 brings together the work of the preceding sections by considering
TallWraith monoids in the filtered context. Section 5 applies the results to t*
*he
examples arising in algebraic topology from suitable generalised cohomology
theories, thus proving the theorems stated above.
4
The reader more interested in the results than the method by which they
are demonstrated may prefer to read this paper in reverse.
Finally, we wish to acknowledge the work of Boardmann, Johnson, and Wil
son in understanding unstable operations. Although this paper does not depend
on [BJW95 ] mathematically, it was an invaluable resource as a guide to deter
mine the form of our final answer.
5
2 General Algebra
In this section we shall expand a little on the basic constructions of general
algebra. The results quoted in this section are all standard results from that
field. For ungraded algebra objects in Set, these results can be found in any
good introduction to the subject, for example [Ber98]. The more general cases
can be found in the wider literature, for example in [KPMS82 ]. We record these
results here without proofs to establish notation and as a quick reference for
the rest of the paper. For those initiates of the deeper secrets of general alg*
*ebra
we mention that we are only considering algebras of finite arity and so we can
assume that our identities are specified by finite sets.
We start by summarising the results that we need in the arena of ungraded
algebras, also known as singlesorted or homogeneous algebras. We shall then
explain how this generalises to graded algebras, also known as manysorted or
heterogeneous algebras.
2.1 Ungraded Algebra
Definition 2.1 Let D be a category with finite products.
1. A type is a pair ( , n) where   is a set and n:  ! N0 is a
morphism of sets called the arity morphism.
2. An algebra object, H, in D consists of an object in D, H, together
with, for each ! 2  , a Dmorphism !H :Hn(!) ! H; these mor
phisms are called the operations of the algebra object in D. A mor
phism of algebra objects in D is a morphism of the underlying objects
in D which intertwines the operations.
3. An algebra is an algebra object in Set.
4. We denote the category of algebra objects in D by D and the category
of algebra objects in Set by . We refer to the functor D ! D which
assigns to an algebra object in D the underlying object in D as the
forgetful functor. We write the underlying object in D of an algebra
object, H, in D as H.
We trust to context to distinguish between the type and the resulting
category of algebras.
One of the key initial results in general algebra is the following character*
*i
sation of algebra object structures.
Proposition 2.2 To give an object, H, in D the structure of an algebra ob
ject is equivalent to giving a lift of the contravariant homfunctor D(,_H):*
* D !
Setto a functor D ! . __
If H is an algebra object in D and D is an object in D we shall write
D(D, H)
for the corresponding algebra with underlying set D(D, H). The operations
on D(D, H) are straightforward; let ! 2  be an element of arity n and let
6
!H :Hn ! H be the operation on V . Then the corresponding operation on
D(D, H) is the Dmorphism
n f1x...xfn !H
D ! Dn ! Hn ! H
where n :D ! Dn is the nfold diagonal morphism.
algebras and algebra objects simply have operations, they are not con
strained to satisfy any particular identities. To consider identities, we need *
*to
know about free algebras.
Proposition 2.3 Let D be a category with finite products which is closed under
filtered colimits and such that finite products commute with filtered colimit*
*s.
Then the forgetful functor D ! D has a left adjoint, F :D ! D , which is_
called the free algebra functor. __
For identities, we only need to know about free algebras in Set.
Definition 2.4 Let be a type. An identity for algebras is a triple (X, u, v)
where X is a (finite) set and u, v 2 F (X).
Let H be an algebra object in D. An identity for algebras, (X, u, v),
induces natural Dmorphisms
uH , vH :HX ! H.
These are defined as follows.
The canonical projections HX ! Hin D define a set morphism
X ! D(HX , H).
As H is an algebra object in D, the right hand side of this is the underlying
set of an algebra. Using the adjunction we therefore have a morphism of
algebras
F (X) ! D(HX , H)
and thus the elements u, v 2 F (X) define elements in the underlying set of
the algebra D(HX , H) which is D(HX , H). Thus we have the required
Dmorphisms uH , vH :HX ! H.
Definition 2.5 An algebra object, H, in D is said to satisfy the identity
(X, u, v) if the two induced Dmorphisms uH , vH :HX ! Hare the same.
Definition 2.6 A variety of algebras, V, is specified by a type, , and a set
of identities for algebras, J. It is the full subcategory of consisting of *
*all
algebras which satisfy all of the identities in J.
The pair ( , J) is a presentation of V.
Let D be a category with finite products, V a variety of algebras with prese*
*n
tation ( , J). The category of Valgebra objects in D, DV , is the full subcate*
*gory
of D consisting of all algebra objects in D which satisfy all of the identi*
*ties
in J.
As is wellknown, presentations are not unique.
Proposition 2.2 holds in the presence of identities.
Proposition 2.7 To give an object, H, in D the structure of a Valgebra object
is equivalent to giving a lift of the contravariant homfunctor D(, H):_D !
Setto a functor D ! V. __
7
To get free Valgebra objects we need to know that we can "impose" iden
tities on an algebra object.
Theorem 2.8 Let D be a complete category with finite products. Suppose that D
is an (E, M) category for some classes E of epimorphisms and M of monomor
phisms, that E is closed under taking finite products, and that D is Ecowell
powered. Then the inclusion functor DV ! D has a left adjoint, D ! DV ,_
called imposition of identities. __
Corollary 2.9 Under the conditions of theorem 2.8 and proposition 2.3, the
forgetful functor DV ! D has a left adjoint, FV :D ! DV , which is called_the
free Valgebra functor. __
Dual to Valgebra objects are coValgebra objects.
Definition 2.10 Let V be a variety of algebras, D a category with finite co
products. A coValgebra object in D is a Valgebra object in Dop. A morphism
of coValgebra objects in D is a morphism in D which intertwines the co
Valgebra structures. We denote the category of coValgebra objects in D by
DVc .
The morphisms are chosen such that there is an isomorphism of categories
DVc ~=(Dop V)op
and there is a natural forgetful functor DVc ! D. The analogue of proposi
tion 2.2 is the following.
Proposition 2.11 To give an object, G, in D the structure of a coValgebra
object is equivalent to giving a lift of the covariant homfunctor D(G,_): D*
* !
Setto a functor D ! V. __
We shall use similar notation: D(G, D) will denote the Valgebra with un
derlying set D(G, D).
The main tool of our analysis is the link between representable functors
and functors with adjoints. This result is a standard one from general algebra,
although one of its corollaries is perhaps the best known result.
Theorem 2.12 Let D be a category that has finite products, is cocomplete, is
cowellpowered, is an (E, M) category where E is closed under finite products,
and is such that its finite products commute with filtered colimits. Let V be a
variety of algebras. Let F be a category with coequalisers. Let G: F ! DV be
a covariant functor. Then the following statements are equivalent.
1. G has a left adjoint, G!.
2. The composition G:F ! D of G with the forgetful functor DV ! D has __
a left adjoint, G!. __
The relationship between the two adjoints is that there is a coequaliser
sequence in F, natural in R,
rR // pR
G!(FV(R))___sR//_G!(R)//_G!(R).
Working in the opposite category we obtain the corresponding result on
coalgebra objects.
8
Corollary 2.13 Let D be a category that has finite products, is cocomplete, is
cowellpowered, is an (E, M) category where E is closed under finite products,
and its finite products commute with filtered colimits. Let V be a variety of
algebras. Let F be a category with equalisers. Let G: F ! DV be a contravariant
functor. Then the following statements are equivalent.
1. G is one of a mutually right adjoint pair.
2. The composition G:F ! D of G with the forgetful functor DV ! D is __
one of a mutually right adjoint pair. __
Since a functor from a cocomplete category into Setis representable if and
only if it has a left adjoint, the following standard result of general algebra
readily follows from proposition 2.11.
Corollary 2.14 1.Let F be a cocomplete category, V a variety of alge
bras. For a covariant functor G: F ! V, the following statements are
equivalent.
(a)G has a left adjoint.
(b)G is representable by a coValgebra object in F.
(c)G is representable by an object in F.
2. Let F be a complete category, V a variety of algebras. For a contravariant
functor G: F ! V, the following statements are equivalent.
(a)G is one of a mutually right adjoint pair.
(b)G is representable by a Valgebra object in F.
(c)G is representable by an object in F. ___
We shall need to know various categorical properties of V.
Theorem 2.15 As a category, V is complete, cocomplete, wellpowered, ex
tremally cowellpowered, and is an (extremal epi, mono) category.
A morphism is an extremal epimorphism if and only if the underlying mor
phism of sets is an epimorphism. Moreover, all extremal epimorphisms are_
regular epimorphisms. __
2.2 Graded Algebras
We turn now to graded algebras. A graded algebra has components indexed by
some (fixed) set and its operations go from components to components rather
than being globally defined. The theory of graded algebras is very similar to
that of ungraded algebras.
We fix some (nonempty) grading set Z. We shall regard this both as a set
and as a (small) discrete category. We write DZ for the category of functors
Z ! D. As Z is a discrete category, there is no distinction between covariant
and contravariant functors from Z.
An object, D, in DZ represents both a covariant and a contravariant functor
D ! SetZ via
0
D*(D0) = z 7! D(D(z), D ) ,
0
D*(D0) = z 7! D(D , D(z)) .
9
To define a graded algebra, we first need to define the graded version of a
type.
Definition 2.16 A Zgraded type, *, is a triple ( *, i, o) where  *is a
set, and a
i:  *! Zm and o:  *! Z
m2N0
are morphisms of sets. For an operation !, we call i(!) the input and o(!) the
output of !. We define the arity function, n:  *! N0 by the composition
a a
 *i! Zm ! {*} ~=N0.
m2N0 m2N0
`
We think of m2N0Zm as the set of finite ordered sets of elements of Z and
so we interpret the element in Z0 as representing the empty subset of Z. Under
the assumption that D has finite products, for an element (z1, . .,.zm ) and an
object, D, in DZ , we write
mY
D(z1, . .,.zm ) = D(zj)
j=1
with D(;) = TD , the terminal object in D.
Definition 2.17 Let D be a category with finite products.
1. An *algebra object, H, in D consists of an object, H, in DZ together
with, for each ! 2  *, a Dmorphism
!H :H(i(!)) ! H(o(!)).
A morphism of *algebra objects in D is a morphism of the underlying
objects in DZ which intertwines the operations.
2. An *algebra is an *algebra object in Set.
3. We denote the category of *algebra objects in D by D * and the category
of *algebra objects in Set by *. We refer to the functor D * ! DZ
which assigns to an *algebra object in D the underlying object in DZ
as the forgetful functor. We write the underlying object in DZ of an *
algebra object, H, in D as H.
All of the results for ungraded algebras have their counterparts in graded
algebras.
Proposition 2.18 To give an object, H, in DZ the structure of an *
algebra object is equivalent to giving a lift of the contravariant homfunctor_
D(, H): D ! SetZ to a functor D ! *. __
As before, if H is an *algebra object in D and D is an object in D we
shall write
D(D, H)
for the corresponding *algebra with underlying object in SetZ,
z 7! D(D, H(z)).
Free *algebras exist under suitable circumstances.
10
Proposition 2.19 Let D be a category with finite products which is closed under
filtered colimits and such that finite products commute with filtered colimit*
*s.
Then the forgetful functor D * ! DZ has a left adjoint, F * :DZ ! D * , __
which is called the free *algebra functor. __
Identities are defined by modifying the ungraded definition in the obvious
way.
Definition 2.20 Let * be a graded type. An identity for` *algebras is a
triple (X, u, v) where X is an object in SetZ such that z2ZX(z) is a finite s*
*et
and u, v 2 F *(X)(z) for some z 2 Z.
Let (X, u, v) be an identity for *algebras with u, v 2 F *(X)(z1). Let H
be an *algebra object in D. Consider the object in SetZ
Y X(z)
z0 7! D H(z) , H(z0) .
z2Z
`
As z2ZX(z) is finite, the product on the left is finite. For each z0 2 Z, the*
*re
is an obvious projection morphism in D
Y
H(z)X(z)! H(z0)X(z0)
z2Z
and thus for x 2 X(z0) we have a canonical projection morphism in D
Y
H(z)X(z)! H(z0).
z2Z
This yields a Setmorphism
Y X(z)
X(z0) ! D H(z) , H(z0)
z2Z
and thus a natural transformation of functors, equivalently a SetZmorphism,
Y X(z)
X ! D H(z) , H.
z2Z
The same adjunction argument as in the ungraded case now produces Dmor
phisms Y
uH , vH : H(z)X(z)! H(z1).
z2Z
Definition 2.21 An *algebra object, H, in D is said to satisfy the identity
(X, u, v) if the two induced morphisms uH , vH are the same.
Definition 2.22 A variety of graded algebras, V*, is specified by a graded type,
*, and a set of identities for *algebras, J. It is the full subcategory of *
* *
consisting of all *algebras which satisfy all of the identities in J.
The pair ( *, J) is a presentation of V*.
Let D be a category with finite products, V* a variety of graded algebras wi*
*th
presentation ( *, J). The category of V*algebra objects in D, DV* , is the full
subcategory of D * consisting of all *algebra objects in D which satisfy all*
* of
the identities in J.
11
Proposition 2.18 holds in the presence of identities.
Proposition 2.23 To give an object, H, in DZ the structure of a V*algebra
object is equivalent to giving a lift of the contravariant homfunctor_D(,_H*
*): D !
SetZ to a functor D ! V*. __
The same conditions as in the ungraded case allow us to impose identities
and so get free V*algebra objects.
Theorem 2.24 Let D be a complete category with finite products. Suppose
that D is an (E, M) category for some classes E of epimorphisms and M of
monomorphisms, that E is closed under taking finite products, and that D is
Ecowellpowered. Then the inclusion functor DV* ! D * has a left adjoint,_
D * ! DV* , called imposition of identities. __
Corollary 2.25 Under the conditions of theorem 2.24 and proposition 2.19,
the forgetful functor DV* ! DZ has a left adjoint, FV* :DZ ! DV* , which is_
called the free V*algebra functor. __
Dual to V*algebra objects are coV *algebra objects.
Definition 2.26 Let V* be a variety of graded algebras, D a category with
finite coproducts. A coV *algebra object in D is a V*algebra object in Dop.*
* A
morphism of coV *algebra objects in D is a morphism in DZ which intertwines
the coV *algebra object in D structures. We denote the category of coV *
algebra objects in D by DV* c.
The analogue of proposition 2.18 is the following.
Proposition 2.27 To give an object, G, in DZ the structure of a coV *alge
bra object is equivalent to giving a lift of the covariant homfunctor_D(G,_*
*): D !
SetZ to a functor D ! V*. __
We shall use similar notation: D(G, D) will denote the V *algebra with
underlying object D(G, D) in SetZ.
Theorem 2.12 easily generalises to the graded case.
Theorem 2.28 Let D be a category that has finite products, is cocomplete, is
cowellpowered, is an (E, M) category where E is closed under finite products,
and its finite products commute with filtered colimits. Let V* be a variety of
graded algebras. Let F be a category with coequalisers. Let G: F ! DV* be a
covariant functor. Then the following statements are equivalent.
1. G has a left adjoint, G!.
2. The composition G:F ! DZ of G with the forgetful functor DV* ! DZ __
has a left adjoint, G!. __
We have the same relationship between the two adjoints as in the ungraded
case: there is a coequaliser sequence in F, natural in H,
rH // pH
G!(FV*(H))___sH//_G!(H)//_G!(H).
The graded version of corollary 2.13 follows immediately. To get the graded
version of corollary 2.14 we need to understand the relationship between ad
junctions and representability in the graded case.
Lemma 2.29 Let D be a cocomplete category. A covariant functor G: D !
SetZ has a left adjoint if and only if it is representable by an object in DZ .
12
Proof.Suppose that G has a left adjoint, say H: SetZ ! D. We extend this
to a functor HZ :(SetZ)Z ! DZ in the obvious way. Let I be the object in
(SetZ)Z defined by
_ ( 0 !
z 7! z07! {*} ifz = z .
; otherwise
Then for an object, X, in SetZ we have isomorphisms in SetZ
Z
SetZ (I, X)= z 7! Set (I(z), X)
~= z 7! Y Set(I(z)(z0), X(z0))
z02Z Y
~= z 7! Set({*}, X(z)) x Set(;, X(z0))
z06=z
~= z 7! X(z) x Y {*}
z06=z
~= z 7! X(z)
~= X,
all natural in X. Hence for D an object in D there are natural isomorphisms
G(D) ~=SetZ(I, G(D)) ~=D(HZ (I), D)
and so G is represented by the object, HZ (I), in DZ .
Conversely, suppose that G is represented by the object, G, in DZ . Let X
be an object in SetZ. We have the following natural isomorphisms of sets
SetZ(X, G(D)) ~=SetZ(X, D(G, D))
~=Y Set(X(z), D(G(z), D))
z2ZY
~= D(G(z), D)X(z)
z2ZY a
~= D G(z), D
z2Z X(z)
~=D a a G(z), D .
z2ZX(z)
Therefore we define the functor H: SetZ! D on objects by
a a
H(X) = G(z)
z2ZX(z)
and in the obvious way on morphisms. This is the required left adjoint. __*
*_
As a corollary we deduce the graded version of corollary 2.14.
Corollary 2.30 1.Let D be a cocomplete category, V* a variety of graded
algebras. For a covariant functor G: D ! V*, the following statements
are equivalent.
13
(a)G has a left adjoint.
(b)G is representable by a coV *algebra object in D.
(c)G is representable by an object in DZ .
2. Let D be a complete category, V* a variety of graded algebras. For a
contravariant functor G: D ! V*, the following statements are equivalent.
(a)G is part of a mutually right adjoint pair.
(b)G is representable by an V*algebra object in D.
(c)G is representable by an object in DZ . ___
The categorical properties of V* are the same as those of V.
Theorem 2.31 As a category, V* is complete, cocomplete, wellpowered, ex
tremally cowellpowered, and is an (extremal epi, mono) category.
A morphism is an extremal epimorphism if and only if the underlying mor
phism of objects in SetZ is an epimorphism. Moreover, all extremal epimor_
phisms are regular epimorphisms. __
The work of the following sections can be viewed simply as applications of
corollaries 2.14 and 2.30.
2.3 The TallWraith Monoidal Structure
The categories VVc and V* V*c have a monoidal structure corresponding to
composition of (representable) functors. The first trace of this that we have
discovered in the literature is [Fre66] where it is referred to as the tensor p*
*roduct
of algebras. The first study of the corresponding monoids that we have found is
[TW70 ] which deals with the category of commutative, unital rings. As we are
similarly interested in the monoids, we have elected to call them TallWraith
monoids. For consistency, and because the terminology of tensor products is
already somewhat overused, we name the monoidal structure on VVc and V*V*c
the TallWraith monoidal structure.
Although, as we have just said, the ideas in this section go back at least to
[Fre66], we have not been able to find a reference which covers all that we wish
to do; in particular, theorem 2.34 and the extensions to graded algebras. On the
other hand, these results are not central to this paper but rather are a guide *
*to
what to expect in the filtered context and so we have not included their proofs
here. The missing proofs can be found in [SW07b ].
In addition to [Fre66] and [TW70 ] mentioned above, similar ideas occur in
[BW05 ] and [BH96 ].
Theorem 2.32 Let V be a variety of algebras. The category VVc has a monoidal
structure which we write as (VVc , , I). The functor VVc ! CovFun (V, V)
given by sending a coValgebra object in V to the covariant functor that it
represents, is strong monoidal. The underlying object in V of the unit,_I, is
isomorphic to FV({*}). __
We shall not give a full proof of this result here; the idea can be found in
[Fre66] and a full proof is in [SW07b ]. We shall give a description of the pr*
*oduct
pairing as this will be important later.
For a coValgebra object, B, in V let us write B*: V ! V for the covariant
functor that it represents. By corollary 2.14, this functor has a left adjoint
14
which we denote by B!. Now coValgebra objects in V are objects in V with
extra structure; this extra structure involves morphisms from the underlying
object in V to iterated coproducts of it. As B!is a left adjoint, it preserves*
* co
products and thus lifts to a functor Bc!:VVc ! VVc. The assignment B 7! Bc!
is functorial in B. The pairing on VVc is, up to natural isomorphism, given on
objects by
(B1, B2) 7! B2c!(B1).
It has the property that we have natural isomorphisms
V(B1, V(B2, B3)) ~=V(B1 B2, B3).
Theorem 2.32 readily adapts to the following situations.
Proposition 2.33 Let D be a cocomplete category, V and W varieties of al
gebras. There are products
VVc x DVc ! DVc ,
(VVc )opx DV ! DV ,
VWc x VVc ! VWc ,
all compatible with the monoidal structure of VVc and with composition of_rep
resentable functors. __
We write all of the pairings using the notation  .
There are two things to note from this generalisation. Firstly that there are
two pairings which involve VVc and V. The first views V as SetV and so comes
from the middle pairing above with D = Set; in terms of functors we have
(B V )*(X) = Set(X, B V ) ~=V(B, Set(X, V )).
The second views V as VSetcand so comes from the third pairing with W = Set;
in terms of functors we have
(V B) *(V 0) = V(V B, V 0) ~=V(V, V(B, V 0)).
This latter pairing was the one considered in [TW70 ] with V the category of
commutative, unital rings.
The second thing to note from this generalisation is the annoyance of having
a partially contravariant pairing. Providing D is sufficiently structured this *
*can
be countered; again, the details can be found in [SW07b ].
Theorem 2.34 Let D be a category satisfying the conditions of theorem 2.12.
Then there is a pairing
VVc x DV ! DV , (B, R) 7! B " R,
which is covariant in both arguments and satisfies
DV (B " R, R0) ~=DV (R, B R0)
naturally in all arguments. ___
In a monoidal category it is natural to consider monoids.
Definition 2.35 Let V be a variety of algebras. A TallWraith Vmonoid is a
monoid in VVc . We write the category of such monoids as VVc T .
15
These were discussed briefly in [BH96 , chs 63, 64], though without explicit
reference to the underlying monoidal structure on VVc .
With a monoid one can consider modules for that monoid. Since the monoidal
category VVc acts on other categories we can consider modules that are not co
Valgebra objects in V. That is, if P is a TallWraith Vmonoid and D is a
cocomplete category then we can consider coValgebra objects, G, in D for
which there is a DVc morphism
P G ! G
satisfying the obvious coherences.
In [BH96 ] the authors show that the category of objects in V with an action
of a TallWraith Vmonoid is again a variety of algebras. Extending this, we
easily see that a Valgebra object in D or coValgebra object in D is a module
for a TallWraith Vmonoid if and only if the corresponding functor D ! V
factors through the category of objects in V with an action of the TallWraith
Vmonoid.
Two remarks are worth making at this juncture. Firstly, if W is another
variety of algebras then the structure of a P module on a coWalgebra object
in V does not have such an interpretation since a coWalgebra object in V
represents a functor out of V. Secondly, due to the variance shift, a P module
in DV is better thought of as a P comodule as the required morphism is
R ! P R.
We can surmount this using the product " since the adjunction turns a coaction
as above into a more normallooking action. That is to say, if a Valgebra obje*
*ct,
R, in D is a P comodule for with coaction morphism
R ! P R
then it is a P module for " with action morphism
P " R ! R.
The adaptation of all this to the graded situation is straightforward.
Theorem 2.36 Let V* be a variety of graded algebras. The category V*V* chas
a monoidal structure which we write as (V* V*c, , I). The functor V*V* c!
CovFun (V* , V*), given by sending a coV *algebra object in V* to the covaria*
*nt_
functor that it represents, is strong monoidal. __
As before we shall give a description of the product. A coV *algebra objec*
*t,
B, in V* represents a functor B*: V* ! V*. By corollary 2.30 this functor has
a left adjoint B!:V* ! V*. We extend this to a functor BZ!:V*Z ! V*Z in the
obvious way. Coproducts in a graded category are formed by taking component
bycomponent coproducts, whence BZ!preserves coproducts because B!does.
Hence it lifts to a functor Bc!:V*V*c ! V*V*c. This has the property that the
adjunction isomorphism lifts to an isomorphism of V*algebras
V*(B1, V*(B2, V )) ~=V*(B2 c!(B1), V ).
Thus there is a natural isomorphism of coV *algebra objects in V*
B1 B2 ~=B2c!(B1).
16
The other part of the structure that needs a word of explanation is the
representing object for the unit of the monoidal structure. We saw in the proof*
* of
lemma 2.29 that the identity functor SetZ! SetZis representable by an object
in (SetZ)Z, labelled I in that proof. The free V*algebra on the components of
this object in (SetZ)Z represents the identity on V*.
Proposition 2.37 Let D be a cocomplete category, V* and W* varieties of
graded algebras. There are products
V*V*cx DV* c ! DV* c,
(V* V*c)opx DV* ! DV* ,
V*W* c x V*V*c ! V*W* c,
all compatible with the monoidal structure of V*V* c and with composition_of
representable functors. __
We write all of the pairings using the notation  .
We remark that the varieties of graded algebras, V* and W* , could be graded
by different indexing sets. This allows us to take, for example, W* = Setand
so get the obvious pairing
V* x V*V*c ! V*.
We can remove the variance switch in the middle product by means of the
graded analogue of theorem 2.34.
Theorem 2.38 Let D be a category satisfying the conditions of theorem 2.12.
Then there is a pairing
V*V* cx DV* ! DV*
which is covariant in both arguments and satisfies
DV* (B " H, H0) ~=DV* (H, B H0)
naturally in all arguments. ___
Definition 2.39 Let V* be a variety of graded algebras. A TallWraith V*
monoid is a monoid in V* V*c. We write the category of such monoids as
V*V* cT .
The remarks regarding modules (and comodules) for a TallWraith Vmonoid
carry over to the graded case.
17
3 Filtered Categories
The purpose of this section is to introduce a categorical version of a very spe*
*cific
type of topology. What we wish to generalise is the following situation: we have
a topological space whose topology is the projective topology for a family of
maps into discrete spaces. This particular case is easy to put into a general
categorical situation and we do not need any of the usual machinery used to
meld topology and category theory.
In the first part we introduce the basic idea: filtered objects. To give a s*
*et,
X, a topology in this fashion it is sufficient to give a family of maps with so*
*urce
X. Putting this into a categorical context leads to projectively filtered objec*
*ts
in an arbitrary category. We shall also define inductively filtered objects sin*
*ce
we shall need to consider how contravarient functors transform filtered objects
in one category into filtered objects in another category.
In the example of topological spaces, we only need to consider surjective
morphisms and we can reduce an arbitrary filtration to one in which all the
morphisms are surjective. We cannot mirror this reduction in all categories
and, moreover, the condition that a functor preserve epimorphisms is more
restrictive than we wish to impose. However, in certain categories there is a
reduction functor and we shall examine the extra features of the theory that
this introduces.
When we can consider these reduced filtrations it makes sense to consider
variations on the themes of being complete and being Hausdorff. Completion
is not a purely topological concept, rather it is a notion from the theory of
uniform spaces. The correct generalisation of these two notions to reduced
filtered objects involves examining the projective limit of the filtration. The*
*re
is a morphism from the underlying object to this limit and we can ask whether
this morphism is a monomorphism, epimorphism, or isomorphism. Being a
monomorphism corresponds to the topology being Hausdorff whilst being an
epimorphism generalises the notion of completeness.
We start by introducing the most general form of filtrations before moving on
to the reduced version. Once we have that then we can consider the projective
limit.
3.1 Projective Filtrations
We start with the general case of a filtration on an object in a category.
Definition 3.1 Let D be a category, D an object in D. We define D# to be the
quasiordered class whose elements are Dmorphisms with source D and whose
ordering is given by d1 d2 if there is a Dmorphism h such that hd1 = d2.
A projective filtration, Q, on D is a nonempty, saturated, directed subclass
of D#.
We say that Q1 is stronger than Q2 if Q2 Q1.
If we are given a projective filtration Q on an object in D without having
specified the underlying object in D we shall write it as Q. An element of Q *
*is
a Dmorphism which we shall write as q :Q ! Qq.
In this context, saturated means that if d1 d2 and d1 2 Q then d2 2 Q.
Let Q be a projective filtration on D. Let f :D0 ! D be a Dmorphism.
The family of all elements of D0#of the form df for d an element of Q is a
18
nonempty, saturated, directed subclass of D0#and hence a projective filtration
on D0.
Definition 3.2 We refer to this filtration as the pull back of Q by f and write
it as f*(Q).
This construction is strictly associative.
Lemma 3.3 Given Dmorphisms D1 f!D2 g!D3 and a projective filtration Q
on D3, the projective filtrations f*(g*(Q)) and (gf)*(Q) on D1 are the_same.
__
With these notions we can define a category of projective filtrations on ob
jects in D.
Definition 3.4 We define the category of projectively filtered objects of D, .K*
*D.
Its objects are projective filtrations on objects in D. A .KDmorphism f :Q1 ! *
*Q2
is a Dmorphism f: Q1 ! Q2 with the property that Q1 is stronger than
f*(Q2).
By construction, the obvious functor .KD! D is faithful.
Any projective filtration is completely determined by an initial subclass,
which per force is directed, and any nonempty directed subclass of D# de
termines a projective filtration by saturation; that is, if d1, d2 are in D# wi*
*th
d1 d2 and d1 is in the specified class then we include d2. It is clear that t*
*he
original directed class is initial for the resulting projective filtration. If *
*D has
finite products then any subclass of D#, directed or not, determines a projecti*
*ve
filtration: first we include all finite products and then we saturate it. Ther*
*e
fore we could choose to work with directed subclasses of D#, or even arbitrary
subclasses, but the above formulation of saturated subclasses is most directly
analogous to a topology on a set. The correspondences are: projective filtra
tion and topology, directed subclass and a basis of the topology, subclass and a
subbasis of the topology.
We shall find it useful to have a characterisation of when a Dmorphism
lifts to a .KDmorphism in terms of choices of initial subclasses of the projec
tive filtrations involved. Let Q1 and Q2 be projective filtrations in D and let
f :Q1 ! Q2 be a Dmorphism on the underlying objects in D. Suppose that
we have initial subclasses of Q1 and Q2 indexed by 1 and 2 respectively.
Then f lifts to a .KDmorphism if and only if for each ~2 in 2 there is a ~1 in
1 and a Dmorphism f~1,~2:Q1,~1! Q2,~2such that the following diagram
of Dmorphisms commutes
Q1___f___//Q2
q1,~1 q2,~2
fflfflf~1,fflffl~2
Q1,~1_____//Q2,~2.
Lemma 3.5 The assignment D ! D.Kunderlies a 2functor of 2categories
Cat ! Cat.
.K .K
Proof.For a covariant functor G: D ! E we define a covariant functor G :D !
.KEin the obvious way: for Q an object in D.K, G.K(Q) is the saturation of the
19
nonempty, directed subclass of G(Q)# consisting of G(q) for q 2 Q. For a
.KDmorphism f :Q .K
.K 1 ! Q2, G (f) has underlying Emorphism G(f). That this
is an Emorphism is obvious. This construction is compatible with composition
...K .K.K
in that GH = GH . It is obvious that identity functors map to identity functors.
Similarly, if :G ! H is a natural transformation of covariant functors
.K .K .K
D ! E, we define a natural transformation ..K:G! H. For an object, Q, in D ,
the .KEmorphism ..KQhas underlying Emorphism Q. Again, this construction
is obviously compatible with composition and identity natural transformations._
__
An important consequence of this is the following result.
Proposition 3.6 Let G: D ! E be a covariant functor and suppose that it has
.K .K __
a left adjoint, say H: E ! D, then H is left adjoint to G . _*
*_
If D has a terminal object then the class of all projective filtrations of a*
* fixed
object in D is a (large!) complete lattice. Its top and bottom elements provide
adjoints to the forgetful functor .KD! D.
Proposition 3.7 The forgetful functor .KD! D has a left adjoint D: D ! .KD.
If D has a terminal object then the forgetful functor .KD! D has a right adjoint
I: D ! .KD.
For an object, D, in D, D(D) is D# whilst I(D) is the subclass of D# con
sisting of all Dmorphisms from D to terminal objects in D.
Proof.Clearly the descriptions given of D(D) and I(D) do produce projective
filtrations on D and if Q is another projective filtration on D then we have
I(D) Q D(D); the second inclusion by definition and the first as Q is
nonempty.
From this, it is clear that if f :D1 ! D2 is a Dmorphism then it underlies
.KDmorphisms D(D
.K 1) ! D(D2) and I(D1) ! I(D2). As the forgetful functor
D ! D is faithful, this is sufficient to define D and I on morphisms.
Clearly, if we apply either D: D ! .KDor I: D ! .KDand the forgetful functor
.KD! D then the resulting composition is the identity functor on D.
Finally, from above we see that the identity on D lifts to morphisms D(D) !
Q ! I(D). These provide the required natural transformations for the adjunc_
tions. __
Definition 3.8 For an object, D, in D we refer to D(D) as the discrete (pro
jective) filtration on D and I(D) as the indiscrete (projective) filtration on D
(assuming that D has a terminal object).
These two functors are very simple examples of a more general type of func
tor.
Definition 3.9 To filter a category is to give a functor D ! .KDwhich is right
inverse to the forgetful functor. We call such a functor a projective filtration
functor.
Examples 3.10 1. The first example is of a profinite filtration. Let F be
a nonempty full subcategory of D which is closed under finite products.
We refer to objects in F as finite objects.
20
Let D be an object in D. We define a projective filtration on D as follows.
We start with the subclass of D#consisting of all Dmorphisms with target
a finite object. Our assumption on F ensures that this is directed. We
saturate it to produce a projective filtration.
It is straightforward to show that the assignment to an object in D of its
profinite filtration is functorial. We therefore have the profinite filt*
*ration
functor on D.
2. The second example of a category that can be filtered is that of a filtered
category. We shall define a filtration functor
.KD! .K.KD.
Let Q be an object in .KD. We start by observing that for q 2 Q, the D
morphism q :Q ! Qq is the underlying Dmorphism of a .KDmorphism
Q ! D(Qq). Let us write .Kq:Q ! D(Qq) for the resulting .KDmorphism.
The subclass of Q# consisting of the elements .Kqis directed, as the origi
nal projective filtration was directed, and hence saturates to a projective
filtration.
It is clear from its construction that this is functorial in Q.
3.2 Reduced Filtrations
Having looked at general filtrations, we now turn to a particular type of filtr*
*a
tion. Let us consider the example of a topology on a set defined by a projective
filtration. The structure of the category of sets means that we can ensure that
each of the maps in the filtration is a surjection. This has certain advantages
which we wish to mirror in our more general filtered categories. Although the
definition below does not depend on any additional properties of the underlying
category, in order to do anything useful with it we need to assume that this
category is an extremally cowellpowered (extremal epi, mono) category. We
also want to know that the forgetful functor .KD! D has a right adjoint; for th*
*is
we need to know that D has a terminal object.
Definition 3.11 Let D be a category. A projective filtration Q on an object
in D is said to be reduced if Q has an initial subclass consisting of extremal
epimorphisms.
We write DK for the full subcategory of D.Kconsisting of reduced projective
filtrations. Let S: KD! .KDbe the inclusion functor.
We could broaden our definition by first choosing a reasonable class of epi
morphisms and considering those filtrations with morphisms in that class, but
we shall only be interested in extremal epimorphisms and so we confine our
attention to those. K .K
Under the right conditions, the inclusion functor S: D ! D has a right
adjoint.
Proposition 3.12 Let D be an (extremal epi, mono) category. Then there is
a reduction functor R: .KD! KDwhich is faithful. The composition RS: KD! KD
is the identity functor. The functor R is right adjoint to S.
21
Proof.Let Q be a projective filtration in D. We define another projective
filtration with the same underlying object in D as follows. Each element q 2 Q
is a Dmorphism q :Q ! Qq. By assumption on D this has a factorisation as
mqqK where qKis an extremal epimorphism and mq a monomorphism. Let us
write KQqfor the target of Kq. We claim that the class of morphisms consisting *
*of
these Kqis directed. This follows from the diagonal property of an (extremal ep*
*i,
mono) category. Suppose that q1 q2 in Q. Then there is some Dmorphism
f :Qq1! Qq2such that q2 = fq1. Thus the following is a commutative diagram
in D.
9Kq1 K mq1
Q_____//_Qq1____//Qq1
""
9Kq2 """
fflffl """
K ""
Qq2 ""f"
""
mq2 """
ff"""""lffl
Qq2
The diagonal property of the (extremal epi, mono)factorisations now implies
the existence of a Dmorphism QKq1! QKq2which fits into the above diagram.
Hence in Q# we have 9Kq1 9Kq2. Thus as Q is directed, the class { Kq} is al*
*so
directed. Its saturation is thus a projective filtration which, by construction*
*, is
reduced. Let us write the result as QK. We define R on objects by R(Q) = KQ.
To define R on morphisms we need to examine its interaction with pull
backs. Let Q be a projective filtration on an object in D, let D be an object
in D, and let f :D ! Q be a Dmorphism. We wish to compare f*(QK) with
9 9 9K K
f*(Q). We obtain an initial subclass for f*(Q ) by taking the extremal epimor
phisms appearing in the (extremal epi, mono)factorisations of elements of Q
and composing with f. That is, it consists of the Dmorphisms qKf :D ! 9KQq
where q :Q ! Qq is an element of Q with (extremal epi, mono)factorisation
mqqK and intervening object, 9KQq, in D. On the other hand, an initial subclass
9 9 9K
of f*(Q) consists of the extremal epimorphisms appearing in the (extremal epi,
mono)factorisations of the Dmorphisms qf for q in Q. For q in Q we therefore
have the diagram
f 9Kq 9K
D _____//_Q___//Qq

e mq
fflffl m fflffl
Dq ______________//Qq
where me is the (extremal epi, mono)factorisation of qf. As D is an (extremal
epi, mono) category there is a Dmorphism Dq ! 9KQqwhich fits into this dia
9 9 9K
gram. Hence f*(QK) f*(Q).
The proof that the defining subclass of KQis directed easily extends to show
that if Q1 and Q2 are projective filtrations on the same underlying object in
D with Q1 Q2 then 9KQ1 9KQ2. Putting this together with the above, we see
22
9 9*9 9K 9K
that if f :Q1 ! Q2 is a .KDmorphism then f*(Q9K2) f(Q2) Q1 and so
f also underlies a DKmorphism 9KQ1! 9KQ2which we denote by R(f). As f and
R(f) have the same underlying Dmorphism, this assignment clearly respects
composition and identities whence we have a functor R: .KD! KD.
As (extremal epi, mono)factorisations in D are unique up to canonical iso
morphism, the composition DK ! D.K! DK is clearly the identity (saturation
ensures that it is actually the identity, rather than just isomorphic to the id*
*en
tity). The obvious inclusion Q QKprovides the other natural transformation
in the adjunction.
Since both forgetful functors .KD! D and DK! D are faithful, and since the
reduction functor .KD! KDcovers the identity on D, the reduction functor_must
be faithful. __
The discrete filtration functor, D: D ! .KD, factors through KDwith no modif*
*i
cation since D(D) contains the initial subclass {D 1!D} and every isomorphism
is an extremal epimorphism. This provides us with a left adjoint to the forgetf*
*ul
functor DK! D. The indiscrete filtration functor, I: D ! .KD, (assuming that D
has a terminal object) is not so fortunate. It is not even true that the termin*
*al
morphism tD :D ! TD is always an epimorphism. Thus we need to define
K K .K K
I:D ! D as the composition of I with the reduction functor D ! D. This is
right adjoint to the forgetful functor DK! D.
3.3 IsoFiltrations
In this section we consider those filtered objects that are analogous to Hausdo*
*rf
spaces and to complete uniform spaces. In order to work with these objects
we need to make additional assumptions on our underlying category, namely
that it be complete and extremally cowellpowered. As these filtered objects
are a subclass of the reduced filtered objects we also retain the assumptions of
the previous section. Thus we assume that D is a complete, extremally cowell
powered, (extremal epi, mono) category. Note that completeness implies the
existence of a terminal object.
We start by defining another functor KD! D by taking the limit of a reduced
filtration. We need to be in KDrather than .KDto ensure that the definition mak*
*es
sense.
We start with the most general definition. Let D be a category and Q an
object in DK. We define a category (Q, D) by
Objects elements of Q,
Morphisms a morphism q1 ! q2 is a Dmorphism, d, from the target of q1 to
the target of q2 such that dq1 = q2.
An alternative description of this, which explains the notation, is as the full
subcategory of the comma category (Q, D) with class of objects the same as
Q.
Note that the quasiordered class Q when viewed as a category is a quotient
of (Q, D) under the relation f ~ g if f and g have the same source and target.
23
There is an obvious functor (Q, D) ! D which sends an object of (Q, D) to
its target, and which regards a morphism in (Q, D) as a Dmorphism. Although
the category (Q, D) is usually large, the functor (Q, D) ! D still might have a
limit. By abuse of notation, we shall refer to the limit of the functor (Q, D) *
*! D
as the limit of Q and write it as limQ.
When this limit exists, it is obvious that there is a natural Dmorphism
Q ! limQ.
The standard properties of limits show that if f :Q1 ! Q2 is a DKmorphism
and Q1 and Q2 are such that both of the appropriate limits exist, then there is
a corresponding Dmorphism f0: limQ1 ! limQ2 compatible with the above
natural morphisms.
Definition 3.13 If Q is an object in DKsuch that limQ exists and the natural
morphism Q ! limQ is an isomorphism then we say that Q is an isofiltration
on Q. We write !Dfor the full subcategory of DKconsisting of all such objects
in DK. We write B: !D! KDfor the inclusion functor.
Under reasonable conditions on D we can show that, in fact, every object in
DKhas a limit.
Proposition 3.14 If D is an extremally cowellpowered complete category,
then every object in DKhas a limit.
Proof.To prove the required result, we observe that if Q is an object in DK, it
has an initial subclass consisting of extremal epimorphisms with source Q. As
D is extremally cowellpowered, we can take this subclass to be small. It is, *
*per
force, directed. We claim that this is an initial full subcategory of (Q, D). L*
*et q1
and q2 in Q be extremal epimorphisms. Then as q1 is an epimorphism, there can
be at most one Dmorphism with the property that dq1 = q2. Comparison of the
ordering on elements of Q with the definition of morphisms in (Q, D) now shows
that (Q, D)(q1, q2) has at most one element and it has precisely one element if
and only if q1 q2. Hence our initial subclass of Q is a full subcategory of
(Q, D). It is clearly initial in (Q, D).
As D is complete we can therefore find a limit of the functor (Q, D) ! D by
taking a limit of its restriction to our small initial subclass of Q. This is u*
*nique
up to canonical isomorphism. It also depends functorially on Q since if we have
a DKmorphism Q1 ! Q2 then the pull back of Q2 is contained in Q1, whence __
we get a canonical Dmorphism on the limits, as they exist. __
By the remarks preceeding definition 3.13, these limits fit together to form
a functor KD! D and there is a natural transformation of functors KD! D from
the forgetful functor to this limit functor.
Definition 3.15 We shall refer to the functor constructed above as the projec
tive limit functor and write it as L: KD! D.
Using the fact that a category of filtered objects is itself naturally filte*
*red,
we obtain the following important construction.
Proposition 3.16 Let D be a complete, extremally cowellpowered (extremal
epi, mono) category. Then there is a natural lift of the projective limit funct*
*or
24
L: KD! D to a functor C: KD! !Dwhich is left adjoint to the inclusion functor
B: !D! DK. Moreover, the compositionCB:!!D! !Dis naturally isomorphic
to the identity functor on D .
Since the forgetful functoris!naturally isomorphic to the projectivelimit!
functor when restricted to D we do not need to specify which functor D ! D
we are lifting along.
Proof.Let Q be an object in DK. Let ': Q ! L(Q) be the canonical Dmor
phism. Let q be an element of Q. By the definition of a limit, there is a canon*
*ical
Dmorphism
!q:L(Q) ! Qq
such that q = !q'.
We claim that the subclass of L(Q)# consisting of the morphisms !qwhich
arise in this fashion is a projective filtration. It is obviously nonempty and
directed. To show that it is saturated, let d: L(Q) ! D be a Dmorphism and
suppose that it factors through !qfor some q in Q, say d = f!q. Then d' = fq
whence d' is in Q. The Dmorphism f defines a morphism in (Q, D) from q to
!
d'. Hence, by the definition of a limit, the canonical morphisms (d'):L(Q) ! D
! ! !
and !q:L(Q) ! Qq satisfy (d')= f q, whence (d')= d and so d is in our chosen
subclass. Hence this subclass is saturated and we have a projective filtration.
This construction clearly depends functorially on Q.
Let us show that this is, in fact, a reduced projective filtration. We need *
*to
show that it has an initial subclass of extremal epimorphisms. It certainly has
an initial subclass consisting of elements !qwhere q :Q ! Qq is an extremal
epimorphism. Since q is an epimorphism, so is !q. Suppose that we have a
factorisation !q= mf where m is a monomorphism; let D be the intervening
object in D. As D is an (extremal epi, mono) category, the morphism f': Q !
D has an (extremal epi, mono)factorisation, say f' = m0e0. Then mm0e0 =
mf' = !q' = q. As both m and m0 are monomorphisms, their composition
is still a monomorphism whence as q is an extremal epimorphism, mm0 is an
isomorphism. The monomorphism m is therefore a retraction, with right inverse
m0(mm0)1, and so is an isomorphism. Hence !qis an extremal epimorphism
and so the projective filtration that we have defined on L(Q) is reduced.
Let Q0denote this reduced projective filtration on L(Q). We now claim that
Q0 is an isofiltration. Consider the categories (Q, D) and (Q0, D). As Q0 =
! !
L(Q) is the limit of (Q, D) ! D, the assignment q 7! !qsatisfies (fq) = f q
and thus we have a covariant functor (Q, D) ! (Q0, D) which on objects is
q 7! !qand which leaves morphisms alone (when viewed as Dmorphisms). The
Dmorphism ': Q ! L(Q) defines a covariant functor (Q0, D) ! (Q, D). Since
!q' = q, the composition is the identity functor on (Q, D). The argument above
which showed that the family {!q} is saturated shows that the composition
in the other direction is the identity functor on (Q0, D). Since these functors
do not change the targets of the objects when viewed as Dmorphisms and do
not change the morphism sets it is clear that this isomorphism intertwines the
two functors (Q, D) ! D and (Q0, D) ! D. Hence as L(Q) is the limit of
(Q, D) ! D it is also the limit of (Q0, D) ! D and thus Q0is an isofiltration.
We therefore have a functor C: KD! !Das required.
25
It is clear from this construction that if we start with an object in !Dthen
all we do is replace the underlying object in D by an isomorphic one (with a
specified isomorphism), whence the composition CB is naturally isomorphic to
the identity functor.
It is also clear from the construction that the natural transformation from
the forgetful functor to L underlies a natural transformation from the identity
on DKto the composition BC.
It is obvious that these natural transformations produce the adjunction_as
stated. __
Definition 3.17 We shall refer to the functor C: KD! !Das the filtered pro
jective limit functor.
We have already noted that the discrete filtration functor, D: D ! .KD, fact*
*ors
through DK; it is equally easy to see that it factors through !D. For comparis*
*on,
K K !
I: D ! D does not factor through D , and there is little point in considering
K
the composition CI.
3.4 Categorical Properties
We wish to determine the categorical properties of the various categories of fi*
*l
tered objects. Certain results on .KD, KD, and !Dhave depended on the categori*
*cal
.KK
properties of D. We shall want to work with categories such as D and so we
need to know whether the various properties of D lift to, say, DK. We shall not
K.K
consider categories such as D where the second type of filtration is more restr*
*ic
tive than the first. We also want to be able to apply the results of section 2
and therefore we need to know other categorical properties to ensure that these
apply.
In summary, we want to know the following.
1. Conditions on D to ensure that DKis an extremally cowellpowered (ex
tremal epi, mono) category with a terminal object.
2. Conditions on D to ensure that !Dis a complete extremally cowellpow
ered (extremal epi, mono) category.
3. Conditions on D to ensure that !Dis cocomplete.
4. How to form products and (finite) coproducts in each of DKand !D.
Let us state all the conditions on D that we need so that they are collected
in one place. We assume that D is
1. complete,
2. cocomplete,
3. an (extremal epi, mono) category, and
4. extremally cowellpowered.
26
Let us illustrate the various functors that we have. We denote the forgetful
functor .KD! D by U, though we shall still use the notation Q for U(Q). We
shall also find it usefulto!have a notation for the forgetful functor !D! D so
we denote this by Y: D ! D. Let us write the discrete filtration functor D
as a functor into !Drather than .KD. We have the following (noncommuting!)
diagram.
!D___B____//_OOKS____//_.K
oo______D_goo______D_gOOOO
OOOOOCOOO R
OOOOOLOOOO 
OOOOYOOOI U
 OOOOOO 
fflfflOOO''fflfflOOOD
D D
We have the following identities and adjunctions
RS = 1, CB ~=1, YD = 1, UI = 1,
Y = USB USR = U, LBC ~=L, D pY,
C pB, S pR, SBD pU, UpI.
The functors B, S, D, and I are fully faithful; U is faithful, whence also R and
Y are faithful. The category !Dis a reflective subcategory of DK.
We shall use some results from [HS73 , xX] to transfer results from DKto !D.
These results refer to reflective subcategories that are also full subcategories
and closed under isomorphisms. These conditions are satisfied by !Dinside DK.
We note in passing that in what follows we are essentially proving that KDis
topological over D. This is mildly reassuring since our intention was to model a
specific type of topological space.
Let us start by defining push forward filtrations.
Lemma 3.18 Let {QI} be a class of objects in DK. Let (D, fI) be a sink for the
underlying class of objects in D. Then there is a reduced projective filtratio*
*n,
say Q0, on D such that
K 0
1. each Dmorphism fi:Qi ! D lifts to a DKmorphism fi:Qi! Q , and
2. if Q00is an object in DKand h: D ! Q00 is a Dmorphism then h lifts to
9K 0 00 00
a DKmorphism h :Q ! Q if and only if each hfi:Qi ! Q  lifts to a
DKmorphism.
In the case that the class of objects in DKhas only one element, say Q, and
so the sink is just f :Q ! D then we shall refer to the resulting object in DK
as the push forward of Q along f and write it as f*(Q).
When the class of objects in DKis empty, clearly the discrete filtration has
the required properties.
Proof.We define Q0 as follows: it consists of all Dmorphisms, g, with source
D such that for each i 2 I, gfi is in Qi.
Firstly, Q0 is not empty as D has a terminal object and so the terminal
morphism from D to this is in Q0.
27
Secondly, Q0is directed. To see this, suppose that g1: D ! D1 and g2: D !
D2 are in Q0. Consider the Dmorphism (g1 x g2) : D ! D1 x D2 (which
exists as D is complete). If we can show that this lies in Q0 then we are done
as it preceeds both g1 and g2. Let i 2 I. As Qi is directed and saturated,
(g1fix g2fi) is in Qi. By the functorality of products, this is (g1 x g2) fi.
As this holds for all i, (g1 x g2) is in Q0.
Thirdly, Q0is saturated. To see this, suppose that g1 is in Q0and g2 = kg1.
Then for i 2 I, g1fi is in Qi and so as this is saturated, kg1fi is in Qi. Hence
g2 is in Q0.
We therefore have a projective filtration on D. We shall now show that it is
reduced. Let g :D ! D1 be in Q0. As D is an (extremal epi, mono) category it
has an (extremal epi, mono)factorisation g = me with intervening object, D2,
in D , say. Let i 2 I. The Dmorphism gfi:Qi ! D1 is in Qiand so since Qiis
reduced it factors through an extremal epimorphism in Qi, say gfi= hiei with
intervening object, D3, in D and ei in Qi. We therefore have a commutative
diagram in D.
Qi__fi__//D__e__//D2
__
ei g__m__
fflfflhifflffl""__
D3 _____//_D1
As D is an (extremal epi, mono) category, we can find a Dmorphism D3 ! D2
which fits into this diagram. Hence efi is in Qi. Thus e is in Q0 which is now
shown to be reduced. K
By construction the Dmorphisms fi:Qi ! D lift to D morphisms Qi !
Q0.
Let h: D ! Q00 be a Dmorphism. Let q00be an element of Q00. Then q00h
is in Q0if and only if q00hfi is in Qi for all i 2 I. Hence h*(Q00) Q0if and_*
*only
if (hfi)*(Q00) Qi for all i 2 I. __
We already have the notion of pull back filtrations in the notnecessarily
reduced case and it is easy to see that this generalises.
Lemma 3.19 Let {QI} be a class of objects in DK. Let (D, fI) be a source for
the underlying class of objects in D. Then there is a reduced projective filtra*
*tion,
say Q0, on D such that
K 0
1. each Dmorphism fi:D ! Qi lifts to a DKmorphism fi:Q ! Qi, and
2. if Q00is an object in DKand h: Q00 ! D is a Dmorphism then h lifts to
9K 00 0 00
a DKmorphism h :Q ! Q if and only if each fih: Q  ! Qi lifts to a
DKmorphism.
In the case that the class of objects in DK has one element we obtain the
reduced pull back filtration, R(f*(Q)). If the class of objects in KDis empty, *
*we
obtain the reduced indiscrete filtration on D.
Proof.Each Dmorphism fi:D ! Qi defines a pull back filtration fi*(Qi) on
D. The union of these is a subclass of D#. As D is complete, we can find a
smallest projective filtration containing this subclass: first we include all f*
*inite
28
products to ensure that it is directed and nonempty (via the empty product)
and then we saturate it. Considered as an object in .KD, this clearly has the
required properties.
We then apply R to this object in .KD. The properties then follow from the
fact that R is right adjoint to the inclusion S: KD! .KDand that both of these_
functors cover the identity on D. __
As the forgetful functor DK! D is faithful and has both a left and a right
adjoint, it reflects and preserves monomorphisms and epimorphisms. Extremal
epimorphisms are easy to characterise.
Corollary 3.20 A DKmorphism f :Q1 ! Q2 is an extremal epimorphism if
and only if f is an extremal epimorphism in D and Q2 = f*(Q1).
Proof.Let us show the "only if" part first, so that we suppose that f :Q1 ! Q2
is an extremal epimorphism. We need to show two things: that f is an extremal
epimorphism and that Q2 = f*(Q1).
As the forgetful functor KD! D has a right adjoint, namely the reduced indis
crete functor, f: Q1 ! Q2 is an epimorphism. Suppose that f = mg with
m a monomorphism. Let D be the intervening object in D. We put the reduced
pull back filtration on D via m. Then by lemma 3.19, m lifts to a DKmorphism
mK:R(m*(Q2)) ! Q2 and g lifts to a DKmorphism gK:Q1 ! R(m*(Q2)) with
mKKg= f. As the forgetful functor is faithful, mK is a monomorphism. Since f
is an extremal epimorphism, mK is thus an isomorphism. Hence m = mK is an
isomorphism. Thus f is an extremal epimorphism.
From lemma 3.18, the identity on Q2 underlies a DKmorphism f*(Q1) !
Q2 and so f factorises as Q1 ! f*(Q1) ! Q2. As the forgetful functor is
faithful, the DKmorphism f*(Q1) ! Q2 is a monomorphism. Hence as f is an
extremal epimorphism, it is an isomorphism. As it covers the identity on Q2,
f*(Q1) and Q2 must in fact be the same projective filtrations.
Now let us show the "if" part. Let f :Q1 ! Q2 be such that f is an
extremal epimorphism and Q2 = f*(Q1). As the forgetful functor is faithful,
f is per force an epimorphism. Suppose that we have a factorisation of f as
mg with m a monomorphism and intervening object, Q0, in DK. We therefore
have a factorisation of f as mg. As the forgetful functor has a left adjo*
*int,
m is a monomorphism. Hence, as f is an extremal epimorphism, m is an
isomorphism. Consider the Dmorphism m1f: Q1 ! Q0. This simplifies
to g which lifts to a DKmorphism. Hence, by lemma 3.18 since Q2 = f*(Q1),
m1 lifts to a DKmorphism and thus, as the forgetful functor is faithful,_m *
*is
an isomorphism. Hence f is an extremal epimorphism. __
This characterisation helps us prove the required extremallity properties of
DK.
Corollary 3.21 DK is an (extremal epi, mono) category.
Proof.Let f :Q1 ! Q2 be a DKmorphism. The Dmorphism f has an (ex
tremal epi, mono)factorisation, say f = me. We can lift this to a factorisat*
*ion
29
of f as Q1 ! e*(Q1) ! Q2. This is an (extremal epi, mono) factorisation by
corollary 3.20 and as monomorphisms lift to monomorphisms.
To show uniqueness it is sufficient to show that we have the diagonal prop
erty. That is, suppose that we have a commutative square in DK,
Q1 _____//Q2
e m
fflffl fflffl
Q3 _____//Q4
with e an extremal epimorphism and m a monomorphism. The underlying
square in D has the same properties and thus there is a (unique) Dmorphism
h: Q3 ! Q2 which fits into the corresponding diagram in D. Then he lifts
to a DKmorphism so as Q3 = e*(Q1), h lifts to a DKmorphism and thus DKhas
the (extremal epi, mono)diagonalisation property. Thus KDis an (extremal_epi,
mono)category. __
Corollary 3.22 DK is extremally cowellpowered.
Proof.Let Q be an object in KD. From the characterisation of extremal epimor
phisms in DKwe see that the forgetful functor DK! D defines a bijection from
the class of isomorphism classes of extremal epimorphisms in DKwith source Q
to the class of isomorphism classes of extremal epimorphisms in D with source
Q. Hence the property of being extremally cowellpowered lifts from D to
DK. ___
From lemmas 3.18 and 3.19 we can deduce that DK is both complete and
cocomplete.
Proposition 3.23 DK is complete and cocomplete.
Proof.This is a standard proof. We form limits and colimits in DKby forming
the limit or colimit first in D and then putting the appropriate reduced filtr*
*ation
on the resulting object: the pull back filtration for the limit and the push_fo*
*rward
for the colimit. __
We therefore have all our required properties of DK. We now turn to !D.
Completeness and cocompleteness follow directly from proposition 3.23.
Corollary 3.24 !Dis complete and cocomplete.
Proof.It is a reflective, full subcategory of KDwhich is closed under isomorphi*
*sm.
Hence by [HS73 , corollaries 36.14,18], both completeness and cocompleteness
descend from DKto !D. ___
Note that colimits in !Dare not simply the colimits of the corresponding
family in KDand therefore do not necessarily project down to the corresponding
colimit in D. Rather we form the colimit in DKand then apply the functor C
to the resulting object.
30
Extremal epimorphisms in !Daremore!complicated than in DK and so we
need to work harder to prove that D is an (extremal epi, mono) category and
is extremally cowellpowered.
Proposition 3.25 !Dis an (extremal epi, mono) category.
Proof.The proof of the factorisation property is an adaptation of the standard
proof that every morphism in a complete wellpowered category is (extremal
epi, mono)factorisable.
Let f :K1 ! K2 be a !Dmorphism. We consider the class of all factorisa
tions f = mh with m a monomorphism. This is not empty as it contains the
factorisation (1, f). This is a quasiordered class with (m1, h1) (m2, h2) if
there is a !Dmorphism from the source of m1 to the source of m2 making the
obvious diagram commute. If this morphism exists, it is obviously unique and
amonomorphism.!There is an obvious functor from this quasiordered class to
D .
We wish to show that this functor has a limit. We shall do this by showing
that the class has a small initial subclass. Let f = mD eD be the (extremal e*
*pi,
mono)factorisation of f in D with intervening object, D, in D. Let f = mh
bea!factorisation of f with m a monomorphism and intervening object, K, in!
D . Then f = mh is a factorisation of f. As the forgetful functor D ! D
has a left adjoint (the discrete filtration functor) it takes monomorphisms to
monomorphisms and so m is a monomorphism. Hence as D is an (extremal
epi, mono) category there is a Dmorphism g :D ! K making the following
diagram commute.
h
K1 _____//_K<<
yy
eDgyyyy m
fflfflyyyfflfflmD
D _____//_K2
We pull back and reduce the projective filtration K2 on K2 via mD to one
on D; let us write this as Q. The above diagram then lifts to DKwith Q in the
lower left corner. Via the adjunction C pB, the DKmorphisms with source
Q factor through the natural morphism Q ! BC(Q). We therefore have the
diagram in DK
B(h)
B(K1) __________________//B(K)
!g)t99t
9KeD B( tttt B(m)
fflffl' tttB(!mDfflffl)
Q _______//_BC(Q)____//B(K2),
where ' is the canonical morphism.
We claim that !gis a monomorphism. It is necessary and sufficient to show
that !g is a monomorphism as the forgetful functor is faithful and has a left
adjoint. Thus let d1, d2: D0! C(Q) be Dmorphisms such that !gd1 = !gd2.
As C(Q) is the underlying object in D of an object in !D, it is a limit and *
*so
d1 and d2 are completely determined by their compositions with the morphisms
into the appropriate family. This family is the projective filtration Q and for*
* q
in Q we have a Dmorphism "q:C(Q) ! Qq such that "q' = q.
31
The projective filtration Q was defined as the reduction of the pull back
of K2 via mD . It therefore has an initial family as follows: for each k in K2
the fact that D is an (extremal epi, mono) category implies the existence of a
commutative diagram, unique up to canonical isomorphism,
mD
D _____//_K2
kD k
fflfflmkfflffl
Dk _____//_Kk,
with kD an extremal epimorphism and mk a monomorphism. The family kD is
initial for Q. We therefore have the following commutative diagram.
' m!g
D _____//EC(Q)___//K2
EE
EEE "kD k
kD EE""Efflffl fflffl
Dk __mk__//_Kk
Thus d1 and d2 are completely determined by the compositions "kDdi for k in
K2. Now as they satisfy !gd1 = !gd2, for each k in K2 we have km!gd1 =
km!gd2 whence mk"kDd1 = mk"kDd2. As mk is a monomorphism, we therefore
have "kDd1 = "kDd2 and thus d1 = d2. Hence !g is a monomorphism and thus
so is !g.!
The D morphism !mDis equal to m!gand so is a monomorphism. Let
!eD:K1 ! C(Q) be the !Dmorphism which, under the inclusion !D KD, maps
to 'e9KD. The factorisation of f in !Das !mD!eDis thus in our class of fact*
*orisations
and it preceeds the factorisation f = mh.
The key property of this factorisation is that the morphism !eDis obtained *
*by
applying the functor C to a KDmorphism 9KeD:B(K1) ! Q such that e9KD = eD ;
note that eD depends only on f and not on the factorisation that we were
trying to dominate. This factorisation is therefore completely determined by
the reduced projective filtration Q on D.
As D is extremally cowellpowered, the class of all reduced projective fil
trations on a specified object in D is actually a set. To see this, observe that
a reduced projective filtration is completely determined by its subclass of ex
tremal epimorphisms. Moreover, this subclass is closed under isomorphism and
so is a union of equivalence classes of extremal epimorphisms, whence a reduced
projective filtration is completely determined by an element of the power class
of the class of equivalence classes of extremal epimorphisms emanating from the
original object in D. As D is extremally cowellpowered, the class of equiva
lence classes of extremal epimorphisms is actually a set and so its power set is
also a set. Hence the class of reduced projective filtrations on a given object*
* in
D is a set. !
Thusour!class of factorisations of the D morphism, f, has an initial set a*
*nd
so, as D is complete, has a limit. The proof that this limit is an (extremal
epi, mono)factorisation of f proceeds exactly as in the analogous proof for
a morphism in a complete wellpowered category. See, for example, [HS73 ,
17.8,17.16].
32
The proof that !Dis in fact an (extremal epi, mono) category now follows __
since it is complete. See, for example, [HS73 , 34.1]. _*
*_
It is worth pointing out that even if D were, in fact, a (regular epi, mono)
category then it would not necessarily be true that !Dwas a (regular epi, mono)
category.
Buried within the above proof are all the necessary pieces to prove the final
property that we want.
Proposition 3.26 !Dis extremally cowellpowered.
Proof.It is easy to see from the proof that !Dis an (extremal epi, mono) categ*
*ory
that every extremal epimorphism is obtained by applying C to a DKmorphism
of the form B(K) ! Q with underlying Dmorphism an extremal epimorphism
(this is a necessary, but not sufficient, condition). To specify the isomorphism
class of an extremal epimorphism in the category of isofiltered objects of D
it is therefore sufficient to specify the isomorphism class of the corresponding
extremal epimorphism in D and areduced!projective filtration on the target.
Hence for an object, K, in D , the class of isomorphism classes of extremal
epimorphisms emanating from K injects into
a K
Iso(D D)
Iso(ex epid:K!D)
where Iso(ex epid: K ! D) is the class of isomorphism classes of extremal
epimorphisms with source K and Iso(DKD) is the class of isomorphism classes
of the fibre category of DK! D at D; that is, the class of reduced projective
filtrations on D.
As D is extremally cowellpowered, all of the classesin!this coproduct
are small. Hence the coproduct is small and thus D is extremally cowell_
powered. __
3.5 The Canonical Filtration Functor
*
*!!
Inthis!section we shall construct a right adjoint to the forgetful functor Y: *
*D !
D . The indiscrete filtration functor, and its reduction, provide right adjoint*
*s to
the forgetful functors .KD! D and DK! D but in general the forgetful functor
!D! D does not have a right adjoint. In the specific case !!D! !D, however,
we are able to construct one. It is a straightforward adaptation of the filtrat*
*ion
functor for a filtered category as described in example 3.10(2). We assume that
D has the properties of section 3.4.
It is simple to adapt the definition of example 3.10(2) to define a functor
!D! ..K!D. For an object, Q, in .KDthe canonical filtration was defined by ta*
*king
the projective filtration on Q with initial subclass the family of .KDmorphisms
..Kq:Q ! SBD(Qq) (recall that we now regard D as a functor into !D). Similarly,
..K!
we define a functor !D!D!by takingthe!projective filtration on K with initial
subclass the family of D morphisms k :K ! D(Kk).
33
!!
Proposition 3.27 This functor factors through D .
Proof.We need to show first that the filtration defined above is reduced and
then that it is an isofiltration. ..K
Let K be an object in !Dand let P be the resulting object in !D. By !
construction, an initial subclass for the filtration P is given by taking the D*
* 
morphisms !
k :K ! D(Kk)
for k in K. It is obvious that we may refine this further and take k in an init*
*ial
subclass of K.
In particular, we can take thosek!which are extremal epimorphisms. Let k be
one of these. We wish to show that k is an extremalepimorphism.!Since k is an
epimorphismand!the forgetful functor is faithful, k is an epimorphism. Now let
k = mf be afactorisation!with m a monomorphism. Let K0 be the intervening
object in D . By applying the forgetfulfunctor!we obtain a factorisation of k
as mf. The forgetful functor D ! D has a left adjoint, namely the discrete
filtration functor, so preserves monomorphisms. Hence m is a monomorphism
and thus, as k is an extremal epimorphism, m is an isomorphism. Its inverse
is a Dmorphism Kk ! K0 and hence lifts to a !Dmorphism D(Kk) ! K0.
Thislift!is inverse to m because the forgetful functor !D! D is faithful. Hen*
*ce
k is an extremal epimorphism and so P is a reduced projective filtration.
To show that it is an isofiltration we need to show that the limit of P is
isomorphic toK!via the canonical morphism. This follows from the description
of limits in D : they are formed by taking the underlying limit in D and putting
the reduced pull back filtration on the resulting object in D. In our case, the
resulting object in D is (naturally isomorphic to) K and it is obvious that_t*
*he_
reduced pull back filtration of the family K ! D(Kk) is again K. __
Definition 3.28 We shall refer to the functordefined!above as the canonical
filtration functor and denote it by X: !D! !D.
Let us now show that this functor is the required adjoint.
!!
Proposition 3.29 The canonical filtration functorX:!!D! D is right adjoint
and right inverse to the forgetful functor Y: !D! !D.
!! !
Proof.In this proof, Y will refer exclusively to the forgetfulfunctor!D ! D
and we will use the notation  for the forgetful functor D ! D.
By construction, YX is the identity functor on !D. This provides the natural
transformation which willbe!the counit of the adjunction.
Let L be an object in !D. Both L and XY(L) are isofiltrations on the
same underlying object in !D, namely Y(L). We shall show that XY(L) L
as projective filtrations on Y(L). By construction, XY(L) is the projective
filtration on Y(L) with initial subclass
!
k :Y(L) ! D(Y(L)k)
34
for k in Y(L). Now L is an isofiltration on Y(L) and so the canonical morphism
Y(L) ! limLl
l
isan!isomorphism of objects in !D, where the limit is over l in L. Limits in
D are formed by taking the corresponding limit in D and then putting the
reduced pull back filtration on the resulting object. Thus an initial subclass
of Y(L) consists of the extremal epimorphisms coming from the (extremal epi,
mono)factorisation of Dmorphisms of the form
0
Y(L) l!Ll l!Ll,l0
for l in L and l0 in Ll. We can assume that l and l0 are themselves in initial
subclasses of their respective filtrations and so we can assume that they are
extremal epimorphisms in their respective categories.
Let us show that the composition l0l is itself an extremal epimorphism. It*
* is
an epimorphism because l0and l are both epimorphisms. Let l0l = me be the
(extremal epi, mono)factorisation in D of l0l with intervening object, D, in*
* D.
By the above, e is in Y(L) and thus lifts to a !Dmorphism "e:Y(L) ! D(D).
The Dmorphism m: D ! Ll,l0lifts to a !Dmorphism D(m): D(D) ! D(Ll,l0).
This is again a monomorphismas!the forgetful functor is faithful. Since l0 is *
*in
Ll it also lifts to a D morphism "l0:Ll ! D(ll,l0). As lifts of Dmorphisms to
!Dmorphisms are unique, we therefore have the following commutative diagram
in !D.
Y(L) ___l___//_Ll
"e "l0
fflfflD(m) fflffl
D(D) _____//D(Ll,l0)
Since !Dis an (extremal epi, mono) category there is a !Dmorphism g :Ll !
D(D) which fits into the above diagram. Applying the forgetful functor we
see that l0 = mg. As l0 is an extremal epimorphism we see that m is an
isomorphism. Hence l0l is isomorphic to e and thus is an extremal epimorphism.
Thus an initial subclass for Y(L) is the family of Dmorphisms
0
Y(L) l!Ll l!Ll,l0
with l in L and l0in Ll. Thus an initial subclass of XY(L) consists of the fami*
*ly
of !Dmorphisms
"l0
L l!Ll! D(Ll,l0).
As each of these factors through l it is in L. Hence XY(L) L and so the
!!
identity !Dmorphism on Y(L) lifts to a D morphism L ! XY(L).
These lifts fittogether!to define a natural transformation of functors from
theidentity!on !Dto XY: all the necessary diagrams commute because they do
in D . This will be the unit of our adjunction.
35
Our functors, Y andX,!are both lifts of the identity functor on !Dalong
the forgetful functor !D! !D, one lifting the source and the other the target.
The natural transformations, YX ! I and I ! XY, are both lifts of the !
identity natural transformation I ! I on !D. Thereforefor!an object, L, in !D
and object, K, in !Dthe forgetful functor Y: !D! !Dinduces a commutative
diagram of morphisms of homsets.
!D(Y(L), K)_____//!! ____//_!
D (L, X(K)) D (Y(L), K)
 
= Y =
! fflffl= ! fflffl= ! fflffl
D (Y(L), K) _____//D(Y(L), K)____//_D(Y(L), K)
As the forgetful functor is faithful, the morphisms in the upper line are_isomo*
*r_
phisms and hence X is right adjoint to Y. __
3.6 Lifts of Functors
In this section we shall examine certain lifts of functors involving filtered c*
*ate
gories. The two lifts that we shall consider are described in the next definiti*
*on.
Recall that for a functor G: D ! E we defined, in section 3.1, a corresponding
.K .K .K
functor G :D ! E.
The conditions that we impose on our categoriesin!the following theorems
are not minimal. Recall that we write Y: !D! !Dfor the forgetful functor.
Definition 3.30 Let D and E be complete, cocomplete,extremally!cowell
powered, (extremal epi, mono) categories. Let G: D ! E be a covariant functor.
! ! !
We define G : D ! E by
! .K ! X !!B 9K!S ..K!.KG.KRK C !
G : = CRG SBX: D ! D ! D ! D ! E! E! E.
! ! !
Let H: E ! !Dbe a covariant functor. We define H : E ! D by
! .K ! B K S .K.KH..K!R9K!C!!Y!
H : = YCRH SB: E ! E! E! D ! D ! D ! D .
We trust that there will be no confusion with using the same notation for
two different constructions. Note that in these definitionswe!have two differe*
*nt
instancesof!various9functors.KIn the definition of G the inclusion functor B *
*is
! ! K
from !Dto !Dwhereas in the definition of H it is from E to E. We trust that
this also will not cause confusion.
We wish to prove two results about these constructions. The first gives a
condition whereby the first construction is associative. The second relates to
adjunctions.
Theorem 3.31 Let D, E, and F be complete, cocomplete,extremally!cowell!
powered, (extremal epi, mono) categories. Let G: D ! E and H: E ! F be
covariant functors. Thenif!Hpreserves!monomorphisms, there is a natural
isomorphism of functors D ! F
!! !!
HG ~=H G
36
satisfying the obvious coherence for triples.
Proof.Let us expand out the two sides to make clear what we have to prove.
!! ....K! .K..K!
HG = CRHG SBX = CRH G SBX,
!! .K .K .K !
H G = CRH SBXCRG SBX = CRH SBXG .
..K! !
From this it is clear that the first step is to compare G SBX with SBXG .
..K! ..K! !
These are functors !D! E . Using the fact that G () = G () we see that
..K! ! ! ..K!!
both G SBX and SBXG are lifts of G along the forgetful functor E ! E.!
To compare these lifts we need to examine the resulting filtrations on G (K) for
an object, K, in !D.
Firstly, let us establish some notation. Applying R to an object, Q, in .KDd*
*oes
not change the underlying object in D, it merely alters the filtration. An init*
*ial
class for R(Q) is given by taking the extremal epimorphisms which come from
the (extremal epi, mono)factorisations of the Dmorphisms in Q. As before,
for q :Q ! Qq in Q let us write
9Kq:Q ! 9KQq
for the corresponding extremalepimorphism.!!
Let K be an object in D . Let us examine G (K). An initial subclass of the
filtration X(K) is given by the family
K ! D(Kk)
for k in K. Here and henceforth we will suppress the label for the morphism
as the notation rapidly becomes unwieldy; in each case it will be the obvious
morphism derived from k.
.K
An initial subclass of the filtration GSBX(K) is thus given by the family
G(K) ! GD(Kk) (3.1)
.K
and of RG SBX(K) by
9 9 9 9 K
G(K) ! GD(Kk) .
!
To get G (K) we apply C which replaces the source of these morphisms by the
!
appropriate limit. This produces an initial family for G (K) consisting of the
..K!
E morphisms
! 9 9 9 9 K
G (K) ! GD(Kk) .
!
We can read off from this an initial family for SBXG (K). It consists of the
!Emorphisms
! 9 9 9 9 K
G (K) ! D(GD(Kk) ).
!
By applying the above as far as (3.1)with G in place of G we can also read off
..K! !
an initial family for G SBX(K). It consists of the E morphisms
! !
G (K) ! G D(Kk).
37
!
It is obviousfrom!the construction that G D = DGD. Thus we can rewrite the
above E morphisms as !
G(K) ! DGD(Kk).
9 9 9 9 K
The objects, GD(Kk) , in E are defined (up to canonical isomorphism) by
the (extremal epi, mono)factorisations
9 9 9 9 K
G(K) ! GD(Kk) ! GD(Kk).
!
From the construction of G as a limit (via the functor C) we see that we can
! !
replace G(K) by G (K) in this. Thus, as projective filtrations on G (K),
..K! !
G SBX(K) is contained in SBXG (K). From this we deduce that the identity
! ..K!
on G (K) lifts to a E morphism
! ..K!
SBXG (K) ! G SBX(K).
! !
As this is a lift of the identity on G it defines a natural transformation SBXG*
* !
..K!
G SBX.
We claim that this becomes a natural isomorphismafter!reduction. To prove
this claim, consider the following diagrams in E .
! _______//
G (K) DGD(Kk)88r
 rrrrr
 rrr
fflfflrr
9 9 9 9 K
DGD(Kk)
! ______//_99 9 9 9_K___//
G (K) DGD(Kk) DGD(Kk)99s
 ssss
 sss
fflffl sss
9 99999999KK ssss
DGD(Kk) sssss
 ssss
 sss
fflfflss
9 9 9 9 K
DGD(Kk)
The second is derived from the first by taking the (extremal epi, mono)
factorisations of the vertical and horizontal morphisms. The diagonal arrow
in each diagram is a monomorphism since it is D applied to a monomorphism
and D preserves monomorphisms. Thus by the diagonalisation property of an
(extremal epi, mono) category, there is an isomorphism
999999999KK9 9 9 9 9 K
DGD(Kk) ~=DGD(Kk)
fitting in to the second diagram.
38
Thus the natural transformation
! ..K!
SBXG ! G SBX
induces a natural isomorphism
! ..K!
RSBXG ! RG SBX.
This is still not quite what is needed as there is an occurence of the funct*
*or
R which is not present in the expansions of the two functors that we wish
to compare. What we shall now see is that we could easily insert R at the
appropriate juncture without changing the result; actually we insert SR.
There is a natural transformation SR ! 1 coming from the adjunction
.K .K .K
S pR. Applying RH yields a natural transformation RH SR ! RH . We
..K!
wish to show that this is a natural isomorphism. Let Y be an object in E.
Using notation as above, we have the following initial classes of the projective
filtrations
9 9 9K .K
H(Y ) ! H(Yy) for RH (Y ),
9 999KK .K
H(Y ) ! H(Yy) for RH SR(Y )
and the natural transformation is the identity on H(Y ). This natural transfo*
*r
mation comes from the fact that the diagonalisation property of an (extremal
epi, mono) category allows us to add in the required morphism on the inner
diagonal of this diagram:
H(Y )____//999KH(Yy)//_H(Yy)
<