RIGIDIFICATION OF ALGEBRAS OVER MULTI-SORTED
THEORIES
JULIA E. BERGNER
Abstract.We define the notion of a multi-sorted algebraic theory, which *
*is
a generalization of an algebraic theory in which the objects are of diff*
*erent
"sorts." We prove a rigidification result for simplicial algebras over t*
*hese the-
ories, showing that there is a Quillen equivalence between a model categ*
*ory
structure on the category of strict algebras over a multi-sorted theory *
*and an
appropriate model category structure on the category of functors from a *
*multi-
sorted theory to the category of simplicial sets. In the latter model st*
*ructure,
the fibrant objects are homotopy algebras over that theory. Our two main
examples of strict algebras are operads in the category of simplicial se*
*ts and
simplicial categories with a given set of objects.
1.Introduction
Algebraic theories are useful in studying many standard algebraic objects, su*
*ch
as monoids, abelian groups, and commutative rings. An algebraic theory provides*
* a
functorial means of describing particular algebraic objects without specifying *
*gen-
erating sets for the operations to which the objects are subject, or for the re*
*lations
between these operations [12]. Given a category C of algebraic objects, the ass*
*oci-
ated algebraic theory TC (if it exists) is a small category with products satis*
*fying the
property that specifying an object of C is equivalent to giving a product-prese*
*rving
functor TC ! Sets.
Consider a category C with an associated algebraic theory TC. If a functor fr*
*om
TC to the category of simplicial sets preserves products, then it is essentiall*
*y a
simplicial object in C and is thus a combinatorial model for a topological obje*
*ct
in C, such as a topological group when C is the category of groups. We call such
a functor a strict T-algebra (Definition 2.3). If the functor preserves product*
*s up
to homotopy, we call it a homotopy T-algebra (Definition 2.4). A homotopy T-
algebra can be viewed as a simplicial set with the appropriate algebraic struct*
*ure
"up to homotopy," in a higher-order sense. Using an appropriate notion of weak
equivalence on homotopy T-algebras [2, 5.6], the following result due to Badzio*
*ch
relates strict and homotopy T-algebras:
Theorem 1.1. [2, 1.4] Let T be an algebraic theory. Any homotopy T-algebra is
weakly equivalent as a homotopy T-algebra to a strict T-algebra.
As a motivation for the work in this paper, consider the category of monoids.
There is an associated algebraic theory TM , and thus a simplicial monoid can
be specified by a TM -algebra. However, the notion of simplicial monoid can be
____________
Date: August 8, 2005.
2000 Mathematics Subject Classification. Primary: 18C10; Secondary 18G30, 18*
*E35, 55P48.
Key words and phrases. algebraic theories, model categories, operads, simpli*
*cial categories.
1
2 J.E. BERGNER
generalized to that of a simplicial category, by which we mean a category enric*
*hed
over simplicial sets, since a simplicial monoid is a simplicial category with o*
*ne
object. We would like to have a generalization of Badzioch's theorem which appl*
*ies
to simplicial categories. From the point of view of algebraic structure, the m*
*ain
difference between a simplicial monoid and a simplicial category with more than
one object is that in the latter case the description of the algebraic structur*
*e is
more complicated, in that two morphisms can be combined by the composition
operation only if they satisfy certain compatibility conditions on the domain a*
*nd
range. Therefore, we would like to describe a more general notion of theory whi*
*ch is
capable of describing algebraic structures in which the elements have various s*
*orts
or types, and in which the operations which can be used to combine a collection*
* of
elements depend on these sorts.
There is in fact such a "multi-sorted" theory, TOCat, such that a product-
preserving functor TOCat! Sets is essentially a category with object set O (Exa*
*m-
ple 3.5). A simplicial category, analogously, can be viewed as a product-preser*
*ving
functor TOCat! SSets.
A simpler example of an algebraic structure which requires the use of a multi-
sorted theory, which we will describe in more detail in Example 3.2, is the cas*
*e of
a group acting on a set. There are two sorts of elements, namely, the elements *
*of
the group and the elements of the set. Two elements of the group can be combined
via multiplication, or an element can be inverted. An element of the group and *
*an
element of the set can be combined via the group action. However, the elements
of the set cannot be combined with one another in any nontrivial way, so the
operations which we allow depend on the sort of element involved. The example of
a module over a ring is constructed similarly in Example 3.3.
Another application of the notion of a multi-sorted theory gives a convenient
description of an operad. In Example 3.4, we characterize the theory Toperadof
operads. An operad in the category of sets is then a product-preserving functor
from Toperadto the category of sets.
A multi-sorted theory T is a category with products, so we can define strict *
*and
homotopy T-algebras as before (see Definitions 3.6 and 3.7). Using a definition
of weak equivalence for homotopy T-algebras (Proposition 4.12), the main result
which we prove for multi-sorted theories is the following generalization of The*
*orem
1.1:
Theorem 1.2. Let T be a multi-sorted algebraic theory. Any homotopy T-algebra
is weakly equivalent as a homotopy T-algebra to a strict T-algebra.
As Badzioch does, we will actually prove a stronger statement in terms of a
Quillen equivalence of model category structures (Theorem 5.1).
Using our example of the theory Toperadof operads, an operad in the category
of simplicial sets is a strict Toperad-algebra. A homotopy operad, or sequence*
* of
simplicial sets with the structure of an operad only up to homotopy, is then a
homotopy Toperad-algebra and can be rigidified to a strict operad using this th*
*eorem.
Returning to the example of simplicial categories, let O be a set and SCO the
category of simplicial categories with object set O in which the morphisms are *
*the
identity on the objects. In [3], we use Theorem 1.2 to prove a relationship bet*
*ween
SCO and the category of Segal categories with the same set O in dimension zero.*
* In
[4], we use the ideas of this proof to prove an analogous relationship between *
*the
category of all small simplicial categories and the category of all Segal categ*
*ories.
MULTI-SORTED THEORIES 3
Throughout this paper, we frequently work in the category of simplicial sets,
SSets. Recall that a simplicial set is a functor op ! Sets, where denotes
the cosimplicial category whose objects are the finite ordered sets [n] = (0, .*
* .,.n)
and whose morphisms are the order-preserving maps. The simplicial category op
is then the opposite of this category. Some examples of simplicial sets are, f*
*or
each n 0, the n-simplex [n], its boundary `[n], and, for any 0 k n, the
simplicial set V [n, k], which is [n] with the kth face removed. More informat*
*ion
about simplicial sets can be found in [8, I.1].
In this paper, we begin by recalling the definition of an algebraic theory and
stating some of its basic properties. Using this definition as a model, we then*
* define
a multi-sorted theory. We should note here that this notion is not a new one; s*
*imilar
definitions are given by Ad'amek and Rosick'y [1, 3.14] and by Boardman and Vogt
[5, 2.3]. (The still more general definition of a finite limit theory is used b*
*y Johnson
and Walters [11].) Because our perspective is slightly different, however, we *
*will
give a precise definition followed by some examples. Given a multi-sorted theory
T, we define strict and homotopy T-algebras over a multi-sorted theory T and sh*
*ow
that the existence of a model category structure on the category of all T-algeb*
*ras.
We also show the existence of a model category structure on the category of all
functors T ! SSets in which the fibrant objects are the homotopy T-algebras. We
then show that there is a Quillen equivalence between these two model categorie*
*s.
Acknowledgments. I am grateful to Bill Dwyer for suggesting this approach to
studying simplicial categories and operads. I would also like to thank Bernard
Badzioch and Michael Johnson for helpful conversations about this work.
2. A Summary of Algebraic Theories
We first recall the definition of an ordinary algebraic theory. More details *
*about
algebraic theories can be found in chapter 3 of [6].
Definition 2.1. An algebraic theory T is a small category with finite products
and which has as objects Tn for n 0 together with, for each n, an isomorphism
Tn ~=(T1)n. In particular, T0 is the terminal object in T.
We can use theories to describe certain algebraic categories, namely those wh*
*ich
are determined by sets with n-ary operations for each n 2. Consider a category
C such that there exists a forgetful functor
: C ! Sets
taking an object of C to its underlying set, and its left adjoint (a free funct*
*or)
L : Sets ! C.
In other words, C is required to have free objects. If the category C and the a*
*djoint
pair ( , L) satisfy some additional technical conditions (see [6, 3.9.1] for de*
*tails),
we will call C an algebraic category.
Given an object X of an algebraic category C, we have a natural map
jX : L (X) ! X
and given a set A, we have another map
"A : A ! L(A).
4 J.E. BERGNER
In order to discuss a theory over the algebraic category C, consider a set A
together with a map mA : L(A) ! A satisfying two conditions: the composite
map
A _"X__// L(A)mA_//_A
is the identity map on A, and the diagram
L(mA) // mA
( L)2A __________//_ L(A)__//_A
jLAL
is a coequalizer. These maps define an algebraic structure on the set A, specif*
*ically
the structure possessed by the objects of C [12].
For example, if C = G, the category of groups, is the forgetful functor tak*
*ing
a group to its underlying set, and L is the free group functor taking a set to *
*the
free group on that set, then these two conditions are precisely the ones defini*
*ng a
group structure on the set A.
We would like to discuss the algebraic theory T corresponding to C to simplify
this way of talking about algebraic structure. Let X be an object of C. We cons*
*ider
natural transformations of functors C ! Sets
_(-)_x_._.x.-(-)z_______"! (-).
n
Using the adjointness of and L, we have that
(X) ~=Hom Sets({1}, (X)) ~=Hom C(L{1}, X)
where {1} denotes the set with one object, and we can think of L{1} as the free
object in C on one generator, since L is the free functor. Hence, we have
(X)n = Hom Sets({1}, (X))n
a
= Hom Sets( {1}, (X))
n
= Hom Sets({1, . .,.n}, (X))
= Hom C(L{1, . .,.n}, X).
Now, by Yoneda's Lemma we have a bijection between the set of natural maps
(X)n ! (X) and the set Hom C(L{1}, L{1, . .,.n}). The objects
L{OE} = T0, L{1} = T1, . .,.L{1, . .,.n} = Tn, . . .
are the objects of the algebraic theory T corresponding to C. The morphisms are
the opposites of the ones in C between these objects. More precisely stated, T *
*is the
opposite of the full subcategory of representatives of isomorphism classes of f*
*initely
generated free objects of C.
Given an object X of C, define a functor HX : T ! Sets such that
HX (L{1, . .,.n}) = Hom C(L{1, . .,.n}, X) = (X)n.
Now, the algebraic category C is equivalent to the category of the functors HX ,
namely, the full subcategory of the category of functors A : T ! Sets whose obj*
*ects
preserve products, or those for which the canonical map A(Tn) ! A(T1)n induced
by the n projection maps is an isomorphism of sets for all n 0 [12].
MULTI-SORTED THEORIES 5
Example 2.2. Let G denote the category of groups. Consider the full subcategory
of G whose objects Tn are the free groups on n generators for n 0 (where T0 is
the trivial group). The opposite of this category is TG, the theory of groups. *
*It can
be shown that the category of product-preserving functors TG ! Sets is equivale*
*nt
to the category G.
Product-preserving functors from the theory T to Sets are called algebras over
T. We would also like to consider functors from an algebraic theory to the cate*
*gory
SSets of simplicial sets. To do so, we must first define a simplicial algebra *
*over
a theory T. For simplicity, we will also use the term "algebra" to refer to th*
*ese
simplicial algebras.
Definition 2.3. [2, 1.1] Given an algebraic theory T, a (strict simplicial) T-a*
*lgebra
A is a product-preserving functor A : T ! SSets. Namely, the canonical map
A(Tn) ! A(T1)n,
induced by the n projection maps Tn ! T1, is an isomorphism of simplicial sets.
In particular, A(T0) is the one-point space [0].
The category of all T-algebras will be denoted AlgT. Similarly, we have the
notion of a homotopy algebra, for which we only require products to be preserved
up to homotopy:
Definition 2.4. [2, 1.2] Given an algebraic theory T, a homotopy T-algebra is a
functor X : T ! SSets which preserves products up to homotopy. The functor X
preserves products up to homotopy if for each n the canonical map
X(Tn) ! X(T1)n
is a weak equivalence of simplicial sets. In particular, we assume that X(T0) *
*is
weakly equivalent to [0].
There exists a forgetful functor, or evaluation map,
UT : AlgT ! SSets
such that UT(A) = A(T1). This functor has a left adjoint, the free T-algebra fu*
*nctor
FT : SSets ! AlgT
where, if Y is any simplicial set,
a
FT(Y )(T1) = Hom T(Tn, T1) x Y n= ~
n 0
where the identifications come from the structure of the algebraic theory [2, 2*
*.1].
3.Multi-Sorted Algebraic Theories
We now generalize the definition of an algebraic theory to that of a multi-so*
*rted
theory.
Definition 3.1. Given a set S, an S-sorted algebraic theory (or multi-sorted th*
*eory)
T is a small category with objects Tff_nwhere ff_n=< ff1, . .,.ffn > for ffi 2 *
*S and
n 0 varying, and such that each Tff_nis equipped with an isomorphism
Yn
Tff_n~= Tffi.
i=1
6 J.E. BERGNER
For a particular ff_n, the entries ffi can repeat, but they are not ordered. In*
* other
words, ff_nis a an n-element subset with multiplicities. There exists a termin*
*al
object T0 (corresponding to the empty subset of S).
Notation. Lower-case Greek letters (with or without subscripts), say ff or ffi,*
* will
be used to denote objects of S, whereas underlined ones, say ff_nor simply ff_,*
* will
denote an n-element subset of objects of S (with multiplicities) for n 1.
Notice that a theory with a single sort is a theory in the sense of the previ*
*ous
section.
We would like to speak of multi-sorted theories corresponding to categories w*
*hich
are analogous to the algebraic categories which we had in the ordinary case. Ho*
*w-
ever, because we have several objects (or "sorts") Tffwhere we only had the obj*
*ect
T1 in an ordinary theory, we have many pairs of adjoint functors, one for each *
*sort.
Let C be a category with coproducts such that given any element fi 2 S, we have
a forgetful functor
fi: C ! Sets
and its left adjoint, the free functor
Lfi: Sets ! C.
For each object X in C and element fi 2 S, we have a map
jX,fi: Lfi fi(X) ! X
and, for each set A a map
"A,fi: A ! fiLfi(A).
As before, in order to make sense of the notion of theory, we consider a set A
together with, for each fi 2 S, a map
mA,fi: fiLfi(A) ! A
satisfying two conditions: the composite map
"A,fi mA,fi
A ____//_ fiLfi(A)_//_A
is the identity map on A, and the diagram
_fiLfi(mA,fi)//_ mA,fi
( fiLfi)2A_________// fiLfi(A)_//_A
fijLfiA,fiLfi
is a coequalizer. These maps define a "multi-sorted algebraic structure" on C. *
*In
particular, we have a notion of composition for certain elements of C depending
on their sorts. Given this structure, we can now construct the S-sorted theory
corresponding to the category C.
Given ffi, fi 2 S, we consider the natural transformations of functors C ! Se*
*ts
ff1(-) x . .x. ffn(-) ! fi(-).
As before, we can apply these functors to an object X of C and rewrite to obtain
a map
Hom Sets({1}, ff1(X)) x . .x.Hom Sets({1}, ffn(X)) ! Hom Sets({1}, fi(X))
which, by adjointness, is equivalent to
Hom C(Lff1{1}, X) x . .x.Hom C(Lffn{1}, X) ! Hom C(Lfi{1}, X).
MULTI-SORTED THEORIES 7
Since C has coproducts, we can rewrite this map as
Hom C(Lff1{1} q . .q.Lffn{1}, X) ! Hom C(Lfi{1}, X).
Then, by Yoneda's Lemma, there is a bijection between the set of natural transf*
*or-
mations
ff1(-) x . .x. ffn(-) ! fi(-)
and the set a
Hom C (Lfi{1}, Lffk{1}).
k
The objects of the theory T corresponding to C are given by finite coproducts of
"free" objects Lffk{1} of C for all choices of ffk, and the morphisms are the o*
*pposites
of those of C. Let X be an object of C and (ff1, . .,.ffn, fi) 2 Sn+1 an (n + 1*
*)-tuple
of elements in S. We define the map HX,ff1,...,ffn,fi: Top ! Sets such that
an an
HX,ff1,...,ffn,fi( Lffk{1}) = Hom C( Lffk{1}, X) = ff1(X) x . .x. ffn(X).
k=1 k=1
If the category C satisfies analogous conditions to those of [6, 3.9.1], then C*
* is
equivalent to the category of all such functors.
We now consider with some examples.
Example 3.2. Consider pairs (G, X), where G is a group and X is a set. We
can obtain two different 2-sorted theories from these pairs, one corresponding *
*to
the category of unstructured pairs, and the other corresponding to the category*
* of
pairs (G, X) with a given action of the group G on the set X.
In each case, we have two forgetful functors and their respective left adjoin*
*ts.
We begin with the category of unstructured pairs, which we denote P. The objects
are the pairs (G, X) and the morphisms (G, X) ! (H, Y ) consist of pairs (', f)
where ' : G ! H is a group homomorphism and f : X ! Y is a map of sets. For
each sort i = 1, 2 we have a forgetful map
i: P ! S
and its left adjoint
Li: S! P.
When i = 1, we have, for any group G and set X,
1(G, X) = G
(where on the right hand side G denotes the underlying set of the group G) and
for any set S
L1(S) = (FS, OE)
where FS denotes the free group on the set S.
Similarly, when i = 2, we define
2(G, X) = X
and
L2(S) = (e, S)
where e denotes the trivial group.
In order to determine the objects of our theory, consider functors
Fi,j: P ! Sets
8 J.E. BERGNER
such that Fi,j(G, X) = Gix Xj. In other words,
Fi,j(G, X) = Hom P(L1(i) q L2(j), (G, X))
where i denotes the set with i elements and similarly for j. The objects of the
theory will be representatives of the isomorphism classes of the L1(i) q L2(j) *
*for
all choices of i and j. This coproduct in P is defined to be the coproduct of e*
*ach
element in the pairs. Thus we have
(G, X) q (G0, X0) = (G * G0, X x X0)
where G * G0denotes the free product of groups. So, our corresponding theory is
the opposite of the full subcategory of P whose objects are of the form L1(i)qL*
*2(j).
When we equip each pair (G, X) with an action of G on X to obtain another
category which we denote PA , the process is identical until we have to specify*
* the
coproduct, since in this case we need to take the group actions into account. We
then have the coproduct in PA
(G, X) q (G0, X0) = (H, (H xG X) q (H xG0X0))
where H = G * G0and we have defined
H xG X = {(h, x)|h 2 H, x 2 X}= ~
when (hg, x) ~ (h, gx) for any g 2 G. We can now take the opposite of a full
subcategory of PA as above to obtain the corresponding theory.
Example 3.3. A very similar example is the case of a commutative ring R and an
R-module A. Again, we have two different 2-sorted theories: one where we simply
have a ring R and regard A merely as an abelian group, and the other where we
consider the R-module structure on A.
As before, we begin with PR , the category of pairs with no additional struct*
*ure.
We have the forgetful map
1 : PR ! Sets
where 1(R, A) = R for any ring R and abelian group A, where on the right side
R is the underlying set of the ring R. Its left adjoint is the functor
L1 : Sets ! PR
where for any set S, L1(S) = (Z[S], e), where Z[S] is the free commutative ring*
* on
the set S and e denotes the trivial (abelian) group. Then we have the map
2 : PR ! Sets
such that 2(R, A) = A, where again on the right hand side A is the underlying
set of the abelian group A. Its left adjoint is the map
L2 : Sets ! PR
where L2(S) = (Z, F AS) where F AS denotes the free abelian group on the set S.
To know what the objects of this 2-sorted theory are, we need to know what the
coproduct is. We have that
(R, A) q (R0, A0) = (R Z R0, A A0),
and from there we can obtain a theory as in the previous example.
MULTI-SORTED THEORIES 9
Now consider the category PM whose objects are pairs (R, A) where R is a ring
and A is a module over A. If A and A0 are modules over R and R0, respectively,
we have a coproduct similar to that in the group action example. So, we say that
(R, A) q (R0, A0) = (R Z R0, (R0 Z A) (R Z A0)).
Example 3.4. Another example of a multi-sorted theory is the N-sorted theory
of operads. Recall that an operad in the category of sets is a sequence of sets
{P (k)}k 0, a unit map 1 2 P (1), and operations
P (k) x P (j1) x . .x.P (jk) ! P (j1 + . .+.jk)
satisfying associativity, unit, and equivariance conditions [14, II.1.4].
There is a notion of a free operad on n generators at levels m1, . .,.mn [14,
xII.1.9]. Specifically, such a free operad has, for each 1 i n, a generato*
*r in
P (mi). Note that the values of mi can repeat. For example, one can think of the
free operad on n generators, each at level 1, as the free monoid on n generator*
*s.
In the category of operads, consider the full subcategory of free operads. Ea*
*ch
object in this category, then, can be described as the free operad on n generat*
*ors at
levels m1, . .,.mn for some n 0 and m1, . .,.mn. The opposite of this categor*
*y is
the theory of operads. Using the notation we have set up for multi-sorted theor*
*ies,
we have that Tfffor ff 2 N is just the free operad on one generator at level ff*
* and
for ff_n=< ff1, . .,.ffn >, we have that Tff_nis the free operad on n generator*
*s at
levels ff1, . .f.fn.
There is also a notion of non- operads, where we no longer have an action of
the symmetric group or an equivariance condition [14, II.1.14]. We can define t*
*he
theory of non- operads analogously, taking the opposite of the full subcategor*
*y of
free non- operads in the category of all non- operads.
Example 3.5. Consider the category OCat whose objects are the categories with
a fixed object set O and whose morphisms are the functors which are the identity
map on the objects. There is a theory TOCat associated to this category. The
objects of the theory are categories which are freely generated by directed gra*
*phs
with vertices corresponding to the elements of the set O. This theory will be s*
*orted
by pairs of elements in O, corresponding to the morphisms with source the first
element and target the second. In other words, this theory is (O x O)-sorted.
In particular, consider ff = (x, y) 2 O x O. Then, if x 6= y, Tffis the categ*
*ory
with object set O and one nonidentity morphism with source x and target y. If
x = y, then Tffis the category freely generated by one morphism from x to itself
and no other nonidentity morphisms.
In general, if ff_=< ff1, . .,.ffn >, then Tff_is the category with object se*
*t O and
morphisms freely generated by the morphisms given for each ffk as in the previo*
*us
case.
Consider the forgetful functor ff: OCat ! Sets where, for any object X in C,
ff(X) = Hom X(x, y).
Its left adjoint then is the free functor Lffdefined by, for a set A,
(
Lff(A) = C withHom C (x, y) = A ifx 6= y
C withHom C (x, y) = FA ifx = y
where FA is the free monoid generated by the set A and where in each case there
are no other nonidentity morphisms in the category C.
10 J.E. BERGNER
As with ordinary algebraic theories, we can define strict and homotopy T-alge*
*bras
for a multi-sorted theory T.
Definition 3.6. Given an S-sorted theory T, a (strict simplicial) T-algebra is a
product-preserving functor A : T ! SSets. Here, product-preserving means that
the canonical map
Yn
A(Tff_n) ! A(Tffi)
i=1
induced by the projections Tff_n! Tffifor all 1 i n is an isomorphism of
simplicial sets.
As before, we will denote the category of strict T-algebras by AlgT.
Definition 3.7. Given an S-sorted theory T, a homotopy T-algebra is a functor
X : T ! SSets which preserves products up to homotopy. The functor X preserves
products up to homotopy if the canonical map
Yn
X(Tff_n) ! X(Tffi)
i=1
induced by the projection maps Tff_n! Tffifor all 1 i n is a weak equivalen*
*ce
of simplicial sets.
We would like to prove a rigidification result similar to Theorem 1.1 above. *
*We
begin by finding model category structures for T-algebras and homotopy T-algebr*
*as.
We then find a Quillen equivalence between these model category structures T-
algebras for any multi-sorted theory T.
4. Model Category Structures
In this section, we define, given a multi-sorted theory T, model category str*
*uc-
tures on the category of diagrams T ! SSets and on the category of T-algebras.
We begin with a review of model category structures.
Recall that a model category structure on a category C is a choice of three d*
*is-
tinguished classes of morphisms: fibrations, cofibrations, and weak equivalence*
*s. A
(co)fibration which is also a weak equivalence will be called an acyclic (co)fi*
*bration.
With this choice of three classes of morphisms, C is required to satisfy axioms
MC1-MC5 [7, 3.3].
An object X in C is fibrant if the unique map X ! * from X to the terminal
object is a fibration. Dually, X is cofibrant if the unique map OE ! X from the*
* initial
object to X is a cofibration. The factorization axiom MC5 guarantees that each
object X has a weakly equivalent fibrant replacement bXand a weakly equivalent
cofibrant replacement eX. These replacements are not necessarily unique, but th*
*ey
can be chosen to be functorial in the cases we will use [10, 1.1.3].
The model category structures which we will discuss are all cofibrantly gener*
*ated.
A cofibrantly generated model category C is a model category for which there are
two sets of morphisms, one of generating cofibrations and one of generating acy*
*clic
cofibrations, such that a map is a fibration if and only if it has the right li*
*fting
property with respect to the generating acyclic cofibrations, and a map is an a*
*cyclic
fibration if and only if it has the right lifting property with respect to the *
*generating
cofibrations [9, 11.1.2]. To describe such model categories, we make the follow*
*ing
definition.
MULTI-SORTED THEORIES 11
Definition 4.1. [9, 10.5.2] Let M be a category and I a set of maps in C. Then
an I-injective is a map which was the right lifting property with respect to ev*
*ery
map in I. An I-cofibration is a map with the left lifting property with respect*
* to
every I-injective.
We are now able to state the theorem that we will use to prove our model
category structures in this paper.
Theorem 4.2. [9, 11.3.1] Let M be a category which has all finite limits and co*
*l-
imits. Suppose that M has a class of weak equivalences which satisfies the "two*
* out
of three property" (model category axiom MC2) and which is closed under retract*
*s.
Let I and J be sets of maps in M which satisfy the following conditions:
(1) Both I and J permit the small object argument [9, 10.5.15].
(2) Every J-cofibration is an I-cofibration and a weak equivalence.
(3) Every I-injective is a J-injective and a weak equivalence.
(4) One of the following conditions holds:
(i)A map that is an I-cofibration and a weak equivalence is a J-cofibra*
*tion,
or
(ii)A map that is both a J-injective and a weak equivalence is an I-
injective.
Then there is a cofibrantly generated model category structure on M in which I *
*is
a set of generating cofibrations and J is a set of generating acyclic cofibrati*
*ons.
We will refer to the standard model category structure on the category SSets *
*of
simplicial sets. In this case, a weak equivalence is a map of simplicial sets f*
* : X ! Y
such that the induced map |f| : |X| ! |Y | is a weak homotopy equivalence of
topological spaces. The cofibrations are inclusions, and the fibrations are the*
* maps
with the right lifting property with respect to the acyclic cofibrations [8, I.*
*11.3].
This model category structure is cofibrantly generated; the generating cofibrat*
*ions
are the maps `[n] ! [n] for n 0, and the generating acyclic cofibrations are
the maps V [n, k] ! [n] for n 1 and 0 k n.
We will also need the notion of a simplicial model category M, or one for whi*
*ch
an object X K is defined for any object X of M and simplicial set K. In particu*
*lar,
M is a simplicial category with a model structure which is required to satisfy *
*several
axioms [9, 9.1.6].
Definition 4.3. For any objects X and Y in a simplicial model category M, the
function complex is the simplicial set Map (X, Y ).
It is important to note that a function complex is only homotopy invariant in
the case that X is cofibrant and Y is fibrant. For the general case, we have t*
*he
following definition:
Definition 4.4. [9, 17.3.1] A homotopy function complex Map h(X, Y ) in a sim-
plicial model category M is the simplicial set Map (Xe, bY) where Xe is a cofib*
*rant
replacement of X in M and bYis a fibrant replacement for Y .
Several of the model category structures that we will use will be obtained by
localizing a given model category structure with respect to a map or a set of m*
*aps.
Suppose that P = {f : A ! B} is a set of maps with respect to which we would
like to localize a model category M.
12 J.E. BERGNER
Definition 4.5. A P -local object X is a fibrant object of M such that for any
f : A ! B in P , the induced map on homotopy function complexes
f* : Map h(B, W ) ! Map h(A, W )
is a weak equivalence of simplicial sets. A map g : X ! Y in M is then a P -loc*
*al
equivalence if for every local object W , the induced map on homotopy function
complexes
g* : Map h(Y, W ) ! Map h(X, W )
is a weak equivalence of simplicial sets.
Given a multi-sorted theory T, let SSetsT denote the category of functors T !
SSets. Note that the category AlgT of strict T-algebras is a full subcategory *
*of
SSetsT.
The category SSetsT is an example of a category of diagrams. In general, given
any small category D, there is a category SSetsD of D-diagrams in SSets, or fun*
*ctors
D ! SSets. We can obtain two model category structures on SSetsD by the
following results.
Theorem 4.6. [8, IX 1.4] Given the category SSetsD of C-diagrams of simplicial
sets, there is a simplicial model category structure SSetsDfin which the weak e*
*quiva-
lences and fibrations are objectwise and in which the cofibrations are the maps*
* which
have the left lifting property with respect to the maps which are both fibratio*
*ns and
weak equivalences.
Theorem 4.7. [8, VIII 2.4] There is a simplicial model category SSetsDcin which
the weak equivalences and the cofibrations are objectwise and in which the fibr*
*ations
are the maps which have the right lifting property with respect to the maps whi*
*ch
are cofibrations and weak equivalences.
We now return to the situation where our small category is a multi-sorted the*
*ory
T. We would like to have an evaluation map and its left adjoint as in the ordin*
*ary
case (see the end of section 2 above), but here we will have one for each fl 2 *
*S.
These evaluation maps look like
Ufl: AlgT ! SSets
such that
Ufl(A) = A(Tfl)
for any T-algebra A.
Each functor Uflhas a left adjoint, the free functor
Ffl: SSets ! AlgT
such that, given a simplicial set Y and object Tfi_in T,
a
Ffl(Y )(Tfi_) = (Hom (Tfl,...,fl, Tfi_) x Y n)= ~
n 0
where the equivalence is as in the ordinary case (see the end of section 2 abov*
*e).
Given a theory T (regular or multi-sorted), define a weak equivalence in the
category AlgT of T-algebras to be a map which induces a weak equivalence of
simplicial sets after applying the evaluation functor Ufffor each sort ff. Simi*
*larly,
define a fibration of T-algebras to be a map f such that Uff(f) is a fibration *
*of
MULTI-SORTED THEORIES 13
simplicial sets. Then define a cofibration to be a map with the left lifting pr*
*operty
with respect to the maps which are fibrations and weak equivalences.
The following theorem is a generalization of a result by Quillen [15, II.4].
Theorem 4.8. Let T be an S-sorted theory. There is a cofibrantly generated model
category structure on AlgT with the weak equivalences, fibrations, and cofibrat*
*ions
as defined above.
We first need to define sets I and J which will be our candidates for generat*
*ing
sets of cofibrations and acyclic cofibrations, respectively. We first define I *
*to be the
set of maps Uff`[n] ! Uff [n] for each n 0 and ff 2 S. Similarly, define J to*
* be
the set of all maps UffV [n, k] ! Uff [n] for each n 1, 1 k n, and ff 2 S*
*. We
now use these sets to prove our model category structure.
Proof.We need to show that the conditions of 4.2 are satisfied for these sets I*
* and
J. The existence of limits and colimits and the conditions on the weak equivale*
*nces
follow just as they do in the case where T is an ordinary theory [15, II.4].
We now show that I and J satisfy the small object argument. Consider some T-
algebra X, which can be written as a directed colimit colimm(Xm ) and can there*
*fore
be computed objectwise. Thus, we can show that `[n] is small:
Hom AlgT(Fff`[n], colimm(Xm=))HomSSets( `[n], Uffcolimm(Xm ))
= Hom SSets( `[n], colimm(UffXm ))
= colimmHom SSets( `[n], UffXm )
= colimmHom AlgT(Fff`[n], Xm ).
The object V [n, k] can be shown to be small analogously, so we have proved sta*
*te-
ment (1).
We first prove statements (3) and (4)(ii), namely that an I-injective is prec*
*isely
a J-injective and a weak equivalence. An I-injective f : X ! Y by definition has
the right lifting property with respect to the maps in I, but using the adjoint*
*ness
of Uffand Fff, this fact is equivalent to f's being an acyclic fibration. But t*
*hen f
is a weak equivalence and has the right lifting property with respect to the ma*
*ps
in J, again by adjointness.
It remains to prove statement (2). Suppose i : A ! B is a J-cofibration. Then
it has the right lifting property with respect to the fibrations. Another adjo*
*int-
ness argument shows that i therefore is an I-cofibration and a weak equivalence,
completing the proof.
We now need a model category structure on the category of homotopy T-algebras.
However, the category of homotopy T-algebras does not have all finite limits and
colimits (axiom MC1). Thus, we instead define a model category structure on all
diagrams T ! SSets in such a way that the fibrant objects are homotopy T-algebr*
*as.
The following theorem holds for model categories M which are left proper and
cellular. We will not define these conditions here, but refer the reader to [9,*
* 13.1.1,
12.1.1] for more details. It can be shown that SSetsT satisfies both these cond*
*itions
[9, 13.1.14, 12.5.1].
Theorem 4.9. [9, 4.1.1] Let M be a left proper cellular model category and P a *
*set
of morphisms of M. There is a model category structure LP M on the underlying
category of M such that:
14 J.E. BERGNER
(1) The weak equivalences are the P -local equivalences.
(2) The cofibrations are precisely the cofibrations of M.
(3) The fibrations are the maps which have the right lifting property with r*
*espect
to the maps which are both cofibrations and P -local equivalences.
(4) The fibrant objects are the P -local objects.
To localize the model structure SSetsTf, we first need an appropriate map. To
do so for ordinary algebraic theories, Badzioch [2, 2.9] uses free diagrams whi*
*ch
are corepresented by the objects Tn of the theory T. In particular the n projec*
*tion
maps Tn ! T1 induce maps
a
Hom (T1, -) ! Hom (Tn, -).
n
He defines his localization with respect to these maps. We would like to define
similar free diagrams in a multi-sorted theory.
For each ff_n=< ff1, . .,.ffn > and 1 i n, there exists a projection map
Tff_n! Tffiinducing a map
Hom T(Tffi, -) ! Hom T(Tff_n, -).
Taking the coproduct of all such maps results in a map
an
~ff_n: Hom T(Tffi, -) ! Hom T(Tff_n, -).
i=1
These maps are the ones which we will use to localize SSetsT. We define P to be
the set of all such maps ~ff_nfor each ff_nand n 0.
Proposition 4.10. There is a model category structure LSSetsT on the category
SSetsT with weak equivalences the P -local equivalences, cofibrations as in SSe*
*tsTf,
and fibrations the maps which have the right lifting property with respect to t*
*he
maps which are cofibrations and weak equivalences.
Proof.This proposition is a special case of Theorem 4.9.
The following propositions are proved by Badzioch for ordinary theories, and *
*his
proofs follow for multi-sorted theories.
Proposition 4.11. [2, 5.5] An object Z of LSSetsT is fibrant if and only if it *
*is a
homotopy T-algebra which is fibrant as an object of SSetsTf.
Proposition 4.12. [2, 5.6] If X and X0 are homotopy T-algebras in SSetsT and
there is a P -local weak equivalence f : Z ! X0, then f is also a weak equivale*
*nce
in SSetsTf, i.e. an objectwise weak equivalence.
Proposition 4.13. [2, 5.8] A map f : X ! X0 is a P -local equivalence if and
only if for any T-algebra Y which is fibrant in SSetsTc, the induced map of fun*
*ction
complexes
f* : Map(X0, Y ) ! Map(X, Y )
is a weak equivalence of simplicial sets.
These results can actually be stated in more generality; they are really just
statements about the fibrant objects in a localized model category structure (s*
*ee
chapter 3 of [9] for more details).
Hence, we can consider the category LSS etsT to be our homotopy T-algebra
model category structure.
MULTI-SORTED THEORIES 15
5.Rigidification of Algebras over Multi-Sorted Theories
We are now able to prove the following statement, which is a stronger version*
* of
Theorem 1.2:
Theorem 5.1. There is a Quillen equivalence of model categories between AlgT
and LSSetsT.
We begin with the necessary definitions. Recall that an adjoint pair
F : C____//Do:oR_
(where F is the left adjoint and R is the right adjoint) is defined by a map
' : Hom D(F X, Y ) ! Hom C(X, RY )
and is sometimes written as the triple (F, R, ') [13, IV.1].
Definition 5.2. [10, 1.3.1] If C and D are model categories, then the adjoint p*
*air
(F, R, ') is a Quillen pair if one of the following statements is true:
(1) F preserves cofibrations and acyclic cofibrations.
(2) R preserves fibrations and acyclic fibrations.
Definition 5.3. [10, 1.3.12] A Quillen pair is a Quillen equivalence if for all*
* cofi-
brant X in C and fibrant Y in D, a map f : F X ! Y is a weak equivalence in D
if and only if the map 'f : X ! RY is a weak equivalence in C.
We need to find an adjoint pair of functors between AlgT and LSSetsT and prove
that it is a Quillen equivalence. Let
JT : AlgT ! SSetsT
be the inclusion functor. We need to show we have an adjoint functor taking an
arbitrary diagram in SSetsT to a T-algebra. We first make the following definit*
*ion.
Definition 5.4. Let C be a small category and SSetsD the category of functors
C ! SSets. Let P be a set of morphisms in SSetsD . An object Y of SSetsD is
strictly P -local if for every morphism f : A ! B in P , the induced map on fun*
*ction
complexes (Definition 4.3)
f* : Map (B, Y ) ! Map (A, Y )
is an isomorphism of simplicial sets. A map g : C ! D in SSetsD is a strict P -*
*local
equivalence if for every strictly P -local object Y in SSetsD , the induced map
g* : Map (D, Y ) ! Map (C, Y )
is an isomorphism of simplicial sets.
Now, given a category of C-diagrams in SSets and the full subcategory of stri*
*ctly
P -local diagrams for some set P of maps, we have the following result.
Lemma 5.5. Consider two categories, the category of all diagrams X : C ! SSets
and the category of strictly local diagrams with respect to the set of maps P =*
* {f :
A ! B}. Then the forgetful functor from the category of strictly local diagrams*
* to
the category of all diagrams has a left adjoint.
16 J.E. BERGNER
Proof.Without loss`of generality, assume that we have just one map f in P ; oth*
*er-
wise replace f by fffff. Given an arbitrary diagram X, we would like to const*
*ruct
a strictly local diagram from X. So, suppose that X is not strictly local, i.e.*
* the
map
f* : Map (B, X) ! Map (A, X)
is not an isomorphism. First suppose that f* fails to be surjective. Then we ob*
*tain
an object X0 as the pushout in the following diagram:
` //
A!X A _____X
| |
| |
` |fflffl fflffl|
A!X B _____//X0
where each coproduct is taken over all maps A ! X. If X fails to be injective, *
*we
obtain X0 by taking the pushout
` `
(B A B) _____//X
| |
| |
`|fflffl |fflffl
B ________//X0
`
again where the coproduct is over all maps B A B ! X, and where the map
a
B B ! B
A
is the fold map. (If f* is neither injective nor surjective, apply one of the a*
*bove
pushouts, then apply the other to the new object X0 rather than to the original
X.)
In the first case, i.e. where f* is not surjective, for any strictly local ob*
*ject Y
we obtain a commutative diagram
Map (X0, Y )___//_Map(B, Y )
|~=| |~=|
fflffl| fflffl|
Map (X, Y )____//_Map(A, Y )
showing that the map X ! X0 is a strict local equivalence since f : A ! B is.
In the second case, where f* is not injective, we obtain a similar diagram, b*
*ut
it takes more work to show that the map X ! X0 is a strict local equivalence. We
first obtain the diagram
Map (X0, Y )______//_Map(B, Y )
| |
| |
fflffl| fflffl|`
Map (X, Y )____//Map(B A B, Y )
It then suffices to show that the right hand vertical arrow is an isomorphism.
MULTI-SORTED THEORIES 17
`
Recall that the object B A B is defined as the pushout in the diagram
A _______//_B
| |
| |
fflffl| fflffl|`
B ____//_B A B
which enables us to look at the diagram
`
Map(B A B, Y )____//_Map(B, Y )
| ~|
| =|
fflffl| fflffl|
Map (B, Y )______//_Map(A, Y ).
Hence the map a
B ! B B
A
is a strict local equivalence. But, this map fits into a composite
*
* *
* *
* `
B_____//__________33__________________________________*
*_____________________________________________________________________________*
*_____________________________________________________________________________*
*_______________________________B//A_BB
id
Since the identity map is a strict local equivalence, it follows that the map
a
B B ! B
A
is a strict local equivalence, since it can be shown that the strictly local eq*
*uivalences
satisfy the "two out of three property" (model category axiom MC2).
Therefore, in either case, the map X ! X0is a strict local equivalence. Howev*
*er,
we still do not know that the map
Map (B, X0) ! Map (A, X0)
is an isomorphism. So, we repeat this process, taking a (possibly transfinite) *
*colimit
to obtain a local object eXsuch that there is a local equivalence X ! eX.
It suffices to show that the functor which takes a diagram X to the local dia*
*gram
eXis left adjoint to the forgetful functor. So if J is the forgetful functor fr*
*om the
category of strictly local diagrams to the category of all diagrams and K is the
functor we have just defined, we need to show that
Map (X, JY ) ~=Map (KX, Y )
for any diagram X and strictly local diagram Y . But, proving this statement is
equivalent to showing that
Map (X, Y ) ~=Map (Xe, Y )
which was shown above for each step, and it still holds for the colimit.
We consider the category SSetsT of T-diagrams and the strict localization with
respect to the set of maps
an
P = { Hom T(Tffi, -) ! Hom T(Tff_, -)}
i=1
defined in the last section to obtain the model category structure of homotopy
T-algebras.
18 J.E. BERGNER
Recall that we defined the inclusion map (or forgetful functor)
JT : AlgT ! SSetsT.
Applying the above lemma to this functor of diagrams, we obtain its left adjoint
functor
KT : SSetsT ! AlgT.
The following proposition holds in the more general situation of an arbitrary
diagram category.
Proposition 5.6. The adjoint pair of functors
KT : SSetsT____//AlgTo:oJT._
is a Quillen pair.
Proof.As categories, AlgT is a subcategory of SSetsT, and the map JT is an incl*
*u-
sion. Since in both cases, the fibrations and weak equivalences are defined obj*
*ect-
wise, JT preserves fibrations and acyclic fibrations.
Lemma 5.7. Each map KT(~ff_) is an isomorphism in AlgT.
Proof.Let A be a T-algebra. Notice that by Yoneda's Lemma we have that
Map SSetsT(Hom T(Tff_, -), A) ' A(Tff_).
Then we have the following weak equivalences of simplicial sets:
a a
Map AlgT(KT( Hom T(Tffi, -)),'A)MapSSetsT( Hom T(Tffi, -), A)
i Y i
' MapSSetsT(Hom T(Tffi, -), A)
Yi
' A(Tffi)
i
' A(Tff_n)
' Map SSetsT(Hom T(Tff_n, -), A)
' Map AlgT(KT(Hom T(Tff_n, -)), A)
It then suffices to show that the map KT(~) actually induces this isomorphism.
This fact follows from the commutativity of the diagram
KT // T
SSetsT_____AlgOO::u
uu
JTOFfl|| uuu
|uuu Ffl
SSets
which follows since
a a a
KT( JT(Hom T(Tff_, -))) ' KTJT(Hom T(Tff_, -)) ' Hom T(Tff_, -).
Now, we need to show that the same adjoint pair is still a Quillen pair when *
*we
replace the model structure SSetsT with the model structure LSSetsT.
MULTI-SORTED THEORIES 19
Proposition 5.8. The adjoint pair
KT : LSSetsT_____//AlgTo:oJT_
is a Quillen pair.
Proof.Consider again the set of maps
a
P = {~ff_: Hom T(Tffi, -) ! Hom T(Tff_, -)}.
i
The model category structure LSS etsT is obtained by localizing with respect to
these maps. Then using Lemma 5.7, we have that each map KT(~ff_n) is an isomor-
phism in AlgT. Hence, it follows from [9, 3.3.20] that the pair of adjoints for*
*ms a
Quillen pair even after the localization on SSetsTf.
Before stating the main theorem, that the above Quillen pair is actually a Qu*
*illen
equivalence, we first need a lemma. Badzioch's proof [2, 6.5] for ordinary theo*
*ries
is slightly different than the one here, but it would follow for multi-sorted t*
*heories
as well.
Lemma 5.9. If X is cofibrant in LSS etsT, then the unit map j : X ! KTX =
JTKTX is a weak equivalence in LSSetsT.
Proof.Case 1: The cofibrant object X is a free diagram. Then X looks like
a
Hom T(Tff_n, -).
ff_
The proof for such an object follows from the proof of Lemma 5.7.
Case 2: Let X be any cofibrant diagram. Then X ' hocolim opXi where each
Xi is a free diagram. It then suffices to show that Map (KTX, Y ) ' Map (X, Y )*
* for
any T-algebra Y which is fibrant in SSetsTcof. Using case 1, we have the follow*
*ing:
Map(X, Y )' Map (hocolim opXi, Y )
' holim Map (Xi, Y )
' holim Map (KTXi, Y )
' Map (hocolim opKTXi, Y )
' Map (KTX, Y ).
The lemma follows.
Now, the proof of the main theorem follows from this lemma exactly as it does
for ordinary theories in [2, 6.4].
Theorem 5.10. The Quillen pair of functors
KT : LSSetsT____//_AlgTo:oJT._
is a Quillen equivalence.
Proof.Let X be a cofibrant object in LSS etsT, A a fibrant object in AlgT, and
f : X ! A = JTA a map in LSS etsT. We need to show that f is a P -local
20 J.E. BERGNER
equivalence if and only if its adjoint map g : KTX ! A is a weak equivalence in
AlgT. There is a commutative diagram
j
X ____//_EEKTX
EEE |g
fEEE""Efflffl||
A
First assume that f is a P -local equivalence. Then g must also be a P -local
equivalence since j is, by the previous lemma. However, g is a map in AlgT, and
so it is an objectwise weak equivalence, or a weak equivalence in AlgT.
Conversely, suppose that g is a weak equivalence in AlgT. Then it is a P -loc*
*al
equivalence. Hence, f = g O j is also a P -local equivalence.
Hence, we have a Quillen equivalence of model categories between strict T-
algebras and homotopy T-algebras.
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Kansas State University, 138 Cardwell Hall Manhattan, KS 66506
E-mail address: bergnerj@member.ams.org