A MODEL CATEGORY FOR LOCAL PO-SPACES
PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
Abstract.Locally partial-ordered spaces (local po-spaces) have been used
to model concurrent systems. We provide equivalences for these spaces by
constructing a model category containing the category of local po-spaces*
*. We
show the category of simplicial presheaves on local po-spaces can be giv*
*en
Jardine's model structure, in which we identify the weak equivalences be*
*tween
local po-spaces. In the process we give an equivalence between the categ*
*ory of
sheaves on a local po-space and the category of 'etale bundles over a lo*
*cal po-
space. Finally we describe a localization that should provide a good fra*
*mework
for studying concurrent systems.
1. Introduction
The motivation for this paper stems from the study of concurrent processes a*
*c-
cessing shared resources. Such systems were originally described by discrete mo*
*dels
based on graphs, possibly equipped with some additional information [Mil80]. The
precision of these models suffers, however, from an inaccuracy in distinguishing
between concurrent and non-deterministic executions. It turned out that a satis-
factory way to organize this information can be based on cubical sets, giving r*
*ise
to the notion of Higher-Dimensional Automata or HDA's [Gou96 , Gou02]. HDA's
live in slice categories of cSet, the category of cubical sets and their morphi*
*sms.
A different view, which has its origins in Dijkstra's notion of progress gra*
*phs
[Dij68], takes the flow of time into account. The difficulty here is to adequa*
*tely
model the fact that time is irreversible as far as computation is concerned. On*
* the
other hand, one would like to identify execution paths corresponding to (at lea*
*st)
the same sequence of acquisitions of shared resources. However, in order not to
lose precision, this notion of homotopy is also subject to the constraint above*
* of the
irreversibility of time. There are two distinct approaches, both based on topol*
*ogical
spaces.
One approach, advocated by P. Gaucher, is to topologize the sets of paths be-
tween the states of an automaton, which technically amounts to an enrichment
with no units [Gau03 ]. The intuition behind the setup is to distinguish betwe*
*en
spatial and temporal deformations of computational paths. The related framework
of Flows has clear technical advantages from a (model-)categorical point of vie*
*w.
____________
2000 Mathematics Subject Classification. Primary 55U35, 18G55, 68Q85; Second*
*ary 18F20,
55U10.
Key words and phrases. local po-spaces (local pospaces), abstract homotopy t*
*heory, model
categories, concurrency, simplicial presheaves, sheaves, 'etale bundles, direct*
*ed homotopy (diho-
motopy), context.
This research was partially funded by the Swiss National Science Foundation *
*grant 200020-
105383.
1
2 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
The other approach, advocated by Fajstrup, Goubault, Raussen and others, is to
topologize partially ordered states of automata. Such objects are called partia*
*lly-
ordered spaces or po-spaces (also pospaces)1. The advantage of using po-spaces
is that there is a very simple and intuitive way to express directed homotopy or
dihomotopy [Gou03 , FGR99 ].
However, the price paid is that po-spaces cannot model executions of (concur-
rent) programs with loops. The solution is to order the underlying topological
space only locally. Such objects are called local po-spaces and the notion of d*
*iho-
motopy becomes more intricate in this context. Nevertheless, practical reasons *
*like
tractability call for a good notion of equivalence in the category of local po-*
*spaces.
Put differently, it would be useful to be able to replace a given local po-spac*
*e model
with a simpler local po-space which nevertheless preserves the relevant compute*
*r-
scientific properties.
In this paper, we study these questions in the framework of Quillen's (closed)
model categories [Qui67, Hov99, Hir03]. Briefly, a model category is a category
with all small limits and colimits and three distinguished classes of morphisms
called weak equivalences, cofibrations and fibrations. Weak equivalences that *
*are
also cofibrations or fibrations are called trivial cofibrations and trivial fib*
*rations,
respectively. These morphisms satisfy four axioms that allow one to apply the
machinery of homotopy theory to the category. This machinery allows a rigorous
study of equivalences. We remark that there are other frameworks for studying
equivalence. However model categories have the most developed theory, and have
succeeded in illuminating many diverse subjects.
Our aim is to construct a model category of locally partial-ordered spaces as*
* a
foundation for the study of concurrent systems. This is technically difficult b*
*ecause
locally partial-ordered spaces are not closed under taking colimits. We will de*
*fine
a category LPS of local po-spaces, which embeds into the category sPre(LPS )of
simplicial presheaves on local po-spaces. The objects of sPre(LPS )are contrava*
*ri-
ant functors from LPS to the category of simplicial sets and the morphisms are
the natural transformations. This embedding is given by a Yoneda embedding (see
Definition 2.17),
y~: LPS ! sPre(LPS ).
We now briefly describe some technical conditions on model categories which
strengthen our theorems. For more details see Definitions 8.2 and 8.4 and [Hov9*
*9 ,
Hir03]. A model category is proper if the weak equivalences are closed under bo*
*th
pushouts with cofibrations and pullbacks with fibrations. It is left proper if *
*the first
condition holds. A model category is cofibrantly generated if the model category
structure is induced by a set of generating cofibrations and a set of generatin*
*g trivial
cofibrations, both of which permit the small object argument. A cellular model
category is a cofibrantly generated model category in which the cell complexes *
*are
well behaved. A simplicial model category M is a model category enriched over
simplicial sets, which for any X 2 M and any simplicial set K has objects X K
and XK which satisfy various compatibility conditions.
Theorem 1.1. The category sPre(LPS ) has a proper, cellular, simplicial model
structure in which
____________
1M. Grandis uses a related approach [Gra03] in which the underlying topologi*
*cal space comes
with a class of directed paths. However these spaces are not partially-ordered,*
* even locally.
A MODEL CATEGORY FOR LOCAL PO-SPACES 3
o the cofibrations are the monomorphisms,
o the weak equivalences are the stalkwise equivalences, and
o the fibrations are the morphisms which have the right lifting property w*
*ith
respect to all trivial cofibrations.
Furthermore among morphisms coming from LPS (using the Yoneda embedding
LPS ,! sPre(LPS )), the weak equivalences are precisely the isomorphisms.
The model structure on sPre(LPS )is Jardine's model structure [Jar87, Jar96]
on the category of simplicial presheaves on a small Grothendieck site. We show *
*that
Shv(LPS ) is a Grothendieck topos which has enough points. Under this condition,
Jardine showed that the weak equivalences are the stalkwise equivalences.
This model category can be thought of as a localization of the universal inje*
*c-
tive model category of local po-spaces [Joy84, Dug01 , DHI04 ]. While in gener*
*al
the weak equivalences are interesting and nontrivial [Jar87], this is not true *
*for
those coming from LPS . To obtain a more interesting category from the point of
view of concurrency we would like to localize with respect to directed homotopy
equivalences. In [Bub04 ] it is argued that the relevant equivalences are the d*
*irected
homotopy equivalences relative to some context. The context is a local po-space*
* A
and the directed homotopy equivalences rel A are a set of morphisms in A # LPS .
We combine this approach with Theorem 1.1 as follows. First we remark that A
embeds in sPre(LPS ) as ~y(A). Next the model structure on sPre(LPS ) induces
a model structure on the coslice category ~y(A) # sPre(LPS.)Finally one can take
the left Bousfield localization of this model category with respect to the dire*
*cted
homotopy equivalences rel A.
Theorem 1.2. Let I = {~y(f) | f is a directed homotopy equivalenceArel}. Then
the category ~y(A) # sPre(LPS )has a left proper, cellular model structure in w*
*hich
o the cofibrations are the monomorphisms,
o the weak equivalences are the I-local equivalences, and
o the fibrations are those morphisms which have the right lifting property*
* with
respect to monomorphisms which are I-local equivalences.
Recall that, given a topological space Z, 'etale bundles over Z are maps W !*
* Z
which are local homeomorphisms. Let O(Z) be Z's locale of open subsets and reca*
*ll
that sheaves over Z are functors O(Z)op! Set that enjoy a good gluing property.
There is a well-known correspondence between 'etale bundles and sheaves. We
establish a directed version of this correspondence, which may be of independent
interest.
Theorem 1.3. Let Z 2 LPS . Let Etale(Z) be the category of di-'etale bundles
over Z, i.e. the category of bundles which are local dihomeomorphisms. Let O(Z)
be the category of open subobjects of Z. There is an equivalence of categories:
: Etale(Z) o Shv(O(Z)) : .
Acknowledgments. The authors would like to thank Eric Goubault, Emmanuel
Haucourt, Kathryn Hess and Phil Hirschhorn for helpful discussions and sugges-
tions.
Contents
1. Introduction 1
4 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
2. Background 4
3. Local po-spaces 10
4. The open-dicover topology 12
5. Equivalence of sheaves and di-'etale bundles 14
6. Points 17
7. Stalkwise equivalences 18
8. Model categories for local po-spaces 20
Appendix A. Hypercovers 24
References 25
2.Background
This section contains some known definitions and facts we build on. We start
by stating the definition of a model category in subsection 2.1. Next we review*
* the
basics on presheaves in subsection 2.2 and on sheaves in subsection 2.3. We then
recall the notions of topoi and geometric morphisms in subsection 2.4 and of st*
*alks
in subsection 2.5. Our main reference for this materialoisp[MLM92 ]. Subsectio*
*n 2.6
is devoted to some important model structures on sSetC , the category of simpli*
*cial
presheaves over a category C. The material is drawn from [Jar87, Jar96, DHI04].
2.1. Model categories. Recall that a morphism i : A ! B has the left lifting
property with respect to a morphism p : X ! Y if in every commutative diagram
A _____//X
i|| p||
fflffl| fflffl|
B _____//Y
there is a morphism h : B ! X making the diagram commute. Also f is a retract
of g if there is a commutative diagram:
A ________AA>>
| AA__"""|
f|| X |f|
|fflfflg|fflffl||__
B ___|___AAB>>"
A_fflffl|_""
Y
Definition 2.1. A model category is a category with all small limits and colimi*
*ts
that has three distinguished classes of morphisms: W, called the weak equivalen*
*ces;
C, called the cofibrations; and F, called the fibrations, which together satisf*
*y the
axioms below. We remark that morphisms in W \ C, and W \ F, are called trivial
cofibrations and trivial fibrations, respectively.
(1)Given composable morphisms f and g if any of the two morphisms f, g,
and g O f are in W, then so is the third.
(2)If f is a retract of g and g is in W, C or F, then so is f.
(3)Cofibrations have the left-lifting property with respect to trivial fibr*
*ations,
and trivial cofibrations have the left-lifting property with respect to *
*fibra-
tions.
A MODEL CATEGORY FOR LOCAL PO-SPACES 5
(4)Every morphism can be factored as a cofibration followed by a trivial fi*
*bra-
tion, and as a trivial cofibration followed by a fibration. These factor*
*izations
are functorial.
op
2.2. Presheaves. Recall that a presheaf P on C is just a functor P 2 SetC . In
particular, "hom-ing"
C(_, C) :Cop ! Set
X 7! C(X, C)
gives rise to a presheaf and further to the Yoneda embedding
op
y : C ae Set C
C 7! C(_, C).
This embedding is dense, i.e.
P ~=colim(y O ss)
canonically for any presheaf P , where ss : (y # P ) ! C is the projection from
the comma-category y # P . Recall that a presheaf in the image of the Yoneda-
embedding (up to equivalence) is called representable.
2.3. Sheaves.
Definition 2.2. A sieve on M 2 C is a subfunctor S C(_, M). A Grothendieck
topology J on C assigns to each M 2 C a collection J(M) of sieves on M such that
(i)(maximal sieve) C(_, M) 2 J(M) for all M 2 C;
(ii)(stability under pullback) if g : M ! N and S 2 J(N), then (g O _)*(S) 2
J(M) as given by
(g O _)*(S)_____//_S
fflffl|_ fflffl
| |
fflffl| fflffl|
C(_, M)_(gO_)//_C(_, N)
(iii)(transitivity) if S 2 J(M) and R is a sieve on M such that (fO_)*(R) 2 *
*J(U)
for all f : U ! M in the image of S, then R 2 J(M);
We say that a sieve S on M is a covering sieve or a cover of M whenever
S 2 J(M).
Remark 2.3. Unwinding definition 2.2 pinpoints a sieve as a right ideal, i.e. a*
* set of
arrows S with codomain M such that f 2 S =) f Oh 2 S whenever the codomain
of h, cod(h) = dom (f), the domain of f. From this point of view, pulling back a
sieve S on M by an arrow N f-!M amounts to building the set
f*(S) def={h| cod(h) = N, f O h 2 S}.
It is then immediate how to rephrase a Grothendieck topology in terms of right
ideals.
op
Definition 2.4. Let J be a Grothendieck topology on C. A presheaf P 2 SetC is
a sheaf with respect to J provided any natural transformation ` : S ) P uniquely
extends through y(M) as in
6 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
S ___`__//_P<<__
fflffl________
| _______
fflffl|____
y(M)
for all S 2 J(M) and all M 2 C. J is subcanonical if the representable presheav*
*es
are sheaves.
Remark 2.5. Let ` : S ! P be a natural transformation from a sieve S to a presh*
*eaf
P . If one sees S as a right ideal S = {uj : Mj ! M}, then ` amounts to a funct*
*ion
that assigns to every uj : Mj ! M 2 S an element aj 2 P (Mj) such that
P (v)(aj) = ak
for all v : Mk ! Mj and for all uk = uj O v 2 S. Such a function is called a
matching family for S of elements of P . A matching family aj 2 P (Mj) admits an
amalgamation a 2 P (M) if
P (uj)(a) = aj
for all uj : Mj ! M 2 S. From this point of view, the Yoneda lemma characterizes
a sheaf as a presheaf such that every matching family has a unique amalgamation
for all S 2 J(M) and all M 2 C.
A Grothendieck topology is a huge object. In practice, a generating device *
*is
used.
Definition 2.6. A basis K for a Grothendieck topology assigns to each object M
a collection K(M) of families of morphisms with codomain M such that
(i)all isomorphisms f : U ! M are contained in K(M),
(ii)given a morphism g : N ! M 2 C and {fi : Ui ! M} 2 K(M), then the
family of pullbacks {ss2 : UixM N ! N} 2 K(N), and
(iii)given {fi : Ui ! M} 2 K(M) and for each i, {hij: Aij! Ui} 2 K(Uj),
then the family of composites {fiO hij: Aij! M} 2 K(M).
Remark 2.7. Given a basis K for a Grothendieck topology one generates the cor-
responding Grothendieck topology J by defining
V 2 J(M) () there isU 2 K(M) such thatU V.
As expected, the sheaf condition can be rephrased in terms of a basis.
As an example, consider the case C = O(X) with X a topological space and
O(X) its locale of opens. The basis of the open-cover (Grothendieck) topology i*
*s,
as expected, given by open coverings of the opens.
op
Theorem 2.8. Let Shv (C, J) be the full subcategory of SetCopwhose objects are
sheaves for J. The inclusion functor i : Shv (C, J) ! SetC has a left adjoint
a called the associated sheaf functor or sheafification. This left adjoint pres*
*erves
finite limits.
Theorem 2.8 is listed as Theorem III.5.1 in [MLM92 ]. There are several equ*
*iv-
alent ways to construct the associated sheaf functor, the most classical one be*
*ing
the "plus-construction" applied twice.
Remark 2.9. A cover on M amounts to a cocone in C with vertex M. The associated
sheaf functor maps these cocones onto colimiting ones. Moreover, it is univers*
*al
with respect to this property.
A MODEL CATEGORY FOR LOCAL PO-SPACES 7
2.4. Topoi.
Definition 2.10. A category E has exponentials provided that for all X 2 E, the
functor _x X : E ! E has a right adjoint denoted (_)X , so that
E(Y x X, Z) ~=E(Y, ZX ).
Suppose now E has a terminal object 1, and has finite limits. A subobject class*
*ifier
is a monomorphism true: 1 ae such that for every monomorphism s : S ae X,
there is a unique morphism OES such that pullback of truealong OES yields s:
S______//1
fflffl|_fflffl
s| true|
fflffl| fflffl|
X _OES//_
The category E is a topos if it has exponentials and a subobject classifier.
A subobject classifier is obviously unique (up to isomorphism). Furthermore,*
* a
topos has all finite colimits, though this is not easy to prove. It would take *
*pages to
enumerate all the remarkable features of a topos, see [Joh77] for an introducti*
*on to
the lore of the material. Let us just say that topoi as introduced by Grothendi*
*eck
and his collaborators had a very strong algebro-geometrical flavor [AGV72 ], y*
*et the
rich structure is relevant not only for for algebraic geometers but for logicia*
*ns as
well [Law63 , Law64, Law73].
Definition 2.11. A site (C, J) is a small category C equipped with a Grothen-
dieck topology J. A Grothendieck topos is a category equivalent to the category
Shv (C, J) of sheaves on (C, J).
The following are well known.
Proposition 2.12. (1)A Grothendieck topos is a topos;
(2)Set isoaptopos;
(3)SetC is a topos for any C.
Definition 2.13. Let E and F be topoi. A geometric morphism g : F ! E is a
pair of adjoint functors
__g*_//
Eoo?__ F
g*
such that the left adjoint g* is left-exact (that is, it preserves finite limit*
*s). The
right adjoint is called direct image and the left one inverse image.
op
As an example, i : Shv(C, J) ,! SetC is the direct image part of a geometric
morphism. Notice that the convention for a geometric morphism is to have the
direction of its direct image part.
Definition 2.14. A (geometric) point in a topos E is a geometric morphism
p : Set! E
(we write p 2 E by abuse of notation). A topos E has enough points if given
f 6= g : P ! Q 2 E there is a point p 2 E such that p*f 6= p*g 2 Set.
8 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
2.5. Stalks and germs.
op
Definition 2.15. Let (C, J) be a site, a : Set C ! Shv (C, J) the associated
sheaf functor and x 2 Shv(C, J) a point. The stalk functor at x is given by
op
stalkx def=x* O a : SetC ! Set.
Given a presheaf F , we say that stalkx(F ) is the stalk of F at x. As an ex*
*ample,
consider again the case C = O(X) with X a (this time) Hausdorff topological
space and O(X) its locale of opens equipped with the open-cover topology. Let
Shv (X) be the corresponding topos of sheaves. It can be shown that any geometr*
*ic
point x : Seto!pShv (X) corresponds to a "physical" point x02 X. The stalk of
F 2 SetO(X) at x is then given by
stalkx(F ) := colim F (U).
U2O(X),x02U
Write germx,U: F (U) ! stalkx(F ) for the canonical map at U (germx when U is
clear from the context). We call the equivalence class germx,U(s) of s in stalk*
*x(F )
the germ of s at x. Obviously,
stalkx(F ) = {germx,U(s) | U 2 O(X), x02 U, s 2 F (U)}.
2.6. Simplicial Presheaves. For the rest of this section, let C be a small cate*
*gory
with a Grothendieck topology J such that Shv(C, J) has enough points.
Let be the simplicial category which has objects [n] = {0, 1, . .,.n} for *
*n 0,
and whose morphisms are the mapsosuchpthat x y implies that f(x) f(y).
Then sSet is the category Set . This category has a well-known model struc-
ture (see [Hov99 ] for example) where WsSetare the morphisms whose geometric
realization is a weakohomotopypequivalence and CsSetare the monomorphisms.
Objects of sSetC are called simplicial presheaves on C since
op i opjCop opxCop i Copj op
sSetC = Set ~=Set ~= Set .
There is an embedding
op Cop
~ : SetC ! sSet
F 7! ~F
where ~F is constant levelwise i.e. (~F )(C)ndef=F (C) for all n 2 N, and all t*
*he
face and degeneracy maps are the identity. There is a further embedding
op
fl :sSet ! sSetC
K 7! flK
where flK is constant objectwise i.e.oflKp(C) def=K for all C 2 C.
Recallothatpfor C 2 C and F 2 SetC , the Yoneda lemma gives the isomorphism
SetC (y(C), F ) ~=F (C), where y is the Yoneda embedding (see Section 2.2). In
the simplicial case we have the following variation, which can be proved using *
*the
same idea used in the proof of the Yoneda lemma.
op
Proposition 2.16. (Bi-Yoneda) Let C 2 C and F 2 sSetC . There is an isomor-
phism op
sSetC ~y(C)x fl [n], F~=F (C)n
natural in all variables.
A MODEL CATEGORY FOR LOCAL PO-SPACES 9
op
Definition 2.17. Using the Yoneda embedding y : C ! SetC for presheaves one
can define an embedding
op~ Cop
~y: C y-!SetC -!sSet
for simplicial presheaves. The functor ~yis also called a Yoneda embedding.
op
There are two Quillen equivalent model structures on sSetC which are in a
certain sense objectwise:
op
o the projective model structure sSetCprjwhere Wprjand Fprjare objectwise
(that is, f : P ! Q 2 Wprj(Fprj) if and only if for all C 2 C, f(C) :
P (C) ! Q(C) 2 WsSet(FsSet) ), and op
o the injective model structure sSetCinjwhere Winjand Cinjare objectwise.
These were studied by Bousfield and Kan [BK72 ] and Joyal [Joy84], respective*
*ly.
op Cop
Proposition 2.18. Both sSetCprjand sSetinjare proper, simplicial, cellular model
categories. All objects are cofibrant in the latter. The identity functor is *
*a left
Quillen equivalence from the projective model structure to the injective model *
*struc-
ture.
The injective one is more handy when it comes down to calculating homotopical
localizations, yet the fibrant objects are easier to grasp in the projective on*
*e2.
Using the stalk functor for presheaves, one can define a simplicial stalk fu*
*nctor
for simplicial presheaves.
Definition 2.19. The simplicial stalk functor at a point p in Shv(C) is given by
op
(_)p :sSetC ! sSet
P 7! {stalkp(Pn)}n 0.
op
A morphism f : P ! Q 2 sSetC is a stalkwise equivalence if fp : Pp ! Qp 2 sSet
is a weak equivalence for all points p in Shv(C).
Jardine [Jar87] proved the existence of a local version of Joyal's injective*
* model
structure. Since we will only be interested in the special case where Shv (C) h*
*as
enough points, we will not recall the definition of local weak equivalences.
Theorem 2.20 ([Jar87,oJar96]).pLet C be a small category with a Grothendieck
topology. Then sSetC the category of simplicial presheaves on C has a proper,
simplicial, cellular model structure in which
o the cofibrations are the monomorphisms, i.e. the levelwise monomorphisms
of presheaves,
o the weak equivalences are the local weak equivalences, and
o the fibrations are the morphisms which have the right lifting property w*
*ith
respect to all trivial cofibrations.
Furthermore, if the Grothendieck topos Shv (C) has enough points, then the local
weak equivalences are the stalkwise equivalences.
Jardine's model structure can be seen to be cellular since it can also be co*
*n-
structed as a left Bousfield localization of the injective model structure [DHI*
*04 ].
____________
2They are objectwise Kan.
10 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
3.Local po-spaces
The focus of this section is to provide the reader with the main definitions*
* and
constructions. We define a small category of local po-spaces LPS and state some
of the properties, most of which are proved in the later sections. We show that
Theorem 1.1 follows from these properties and a theorem of Jardine.
To simplify the analysis, we will only work with topological spaces which are
subspaces of Rn for some n, since this provides more than enough generality for
studying concurrent systems. The main technical advantage of this setting is th*
*at
we obtain small categories.
Definition 3.1. (i)Let Spaces be the category whose objects are subspaces
of Rn for some n, and whose morphisms are continuous maps.
(ii)Let PoSpaces be the category whose objects are po-spaces: that is U 2
Spaces together with a partial order (a reflexive, transitive, anti-symm*
*etric
relation) such that is a closed subset of U xU in the product topolo*
*gy.
(iii)For any M 2 Spaces define an order-atlas on M to be an open cover3
U = {Ui} of M indexed by a set I, where Ui 2 PoSpaces . These partial
orders are compatible: i agrees with j on Ui\ Uj for all i, j 2 I. We
will usually omit the index set from the notation.
(iv)Let U and U0 be two order atlases on M. Say that U0 is a refinement of
U if for all Ui 2 U, and for all x 2 Ui, there exists a U0j2 U0 such that
x 2 U0j Ui and for all a, b 2 U0j, a j0b if and only if a ib.
(v) Say that two order atlases are equivalent if they have a common refineme*
*nt.
This is an equivalence relation: reflexivity and symmetry follow from the
definition. For transitivity, if U and U0 have a refinement V = {Vi} and*
* U0
and U00have a refinement W = {Wj}, let T = {Vi\ Wj}. One can check
that T is an order atlas of M and that is a refinement of U0 and U00.
Any po-space (U, ) is a local po-space with the equivalence class of order *
*atlases
generated by the order atlas {U}. As a further example, we remark that any disc*
*rete
space has a unique equivalence class of order-atlases.
Definition 3.2. Let LPS be the category of local po-spaces described as follow*
*s.
The objects, called local po-spaces, are all pairs (M, U) where M is an object *
*in
Spaces and U is an equivalence class of order-atlases of M. The morphisms,
called dimaps are described as follows. f 2 LPS ((M, U), (N, V)) if and only *
*if
f 2 Spaces(M, N) and for all V = {Vj}j2J 2 V there is a U = {Ui}i2I2 U such
that for all i 2 I, j 2 J, for all x, y 2 Ui\ f-1 (Vj),
x Ui y =) f(x) Vj f(y).
Remark 3.3. This condition is not necessarily true for arbitrary U 2 U. For ex-
ample, take M = {-1, 1} with U the unique equivalence class of order atlases
generated by the order atlas U = {{-1}, {1}}. Let f = IdM : (M, U) ! (M, U).
Now let M+ be the po-space on M with the ordering -1 1 and let M- be the
po-space on M with the ordering 1 -1. Then {M+ } 2 U and {M- } 2 U (both
have U as a common refinement). However, even though -1, 1 2 M+ \ f-1 (M- ),
-1 M+ 1 butf(-1) M- f(1).
____________
3That is, for all i, Uiis open as a subspace of M and M = [iUi.
A MODEL CATEGORY FOR LOCAL PO-SPACES 11
Remark 3.4. It is easy to check that a dimap of po-spaces is also a dimap of lo*
*cal
po-spaces. Thus PoSpaces the category of po-spaces is a subcategory of LPS .
Remark 3.5. Subobjects in LPS .
If (M, U) 2 LPS , then a subspace L M 2 Spaces inherits local po-space
structure as follows. Let U = {Ui} 2 U and let W = {Wi} where Wi = L \ Ui
and Wi has the partial order inherited from Ui. Then W is an open cover of L
and the partial orders are compatible. That is W is an order atlas. Let W be the
equivalence class of W .
We claim that W does not depend on the choice of U. Let "U= {U"i} 2 U, let
W"i= L \ "Ui, and let "W = {W"i}. U and "Uhave a common refinement ^U= {U^i}.
Let W^i= L \ ^Uiand let W^ = {W^i}. Then one can check that W^ is a common
refinement of W and "W. So the equivalence class of "W is also W.
Next we claim that there is a dimap ' : (L, W) ! (M, U) given by the inclusi*
*on
' : L ,! M. Let U = {Uk} 2 U, let Wk = L \ Uk, and let W = {Wk}. Then
W 2 W. Let x, y 2 Wj \ '-1(Uk) = Wj \ L \ Uk = Wj \ Wk. Note that '(x) = x
and '(y) = y. Then
x Wj y () x Wk y () x Uk y.
Therefore when L M 2 Spaces, then there is an induced inclusion (L, W)
(M, U) 2 LPS .
The remark above will be used implicitly and without reference in Section 6.
Definition 3.6. A collection of dimaps {OEj : (Mj, Uj) ! (M, U)} LPS is an open
dicover if
(i){OEj : Mj ! M} is an open cover, and
(ii)for each j, Uj is the local po-space structure inherited from (M, U).
Remark 3.7. The local po-space structures inherited by the subspaces of (M, U) *
*are
compatible. So if {OEj : (Mj, Uj) ! (M, U)} is a open cover, then for each j, t*
*here
is a Uj = {Ujk} 2 Uj such that U0 = {Ujk}j,kis an order atlas for M and U0 2 U.
The following is easy to check.
Lemma 3.8. Spaces and LPS are small categories.
Define U : LPS ! Spaces to be the forgetful functor defined on objects and
morphisms as follows (M, U) 7! M and ' 7! '.
Define F : Spaces ! LPS as follows. If M is an object in Spaces , then let
F (M) = (M, ~MOE), where M~OEis the equivalence class of MOE= {M} with x M
y () x = y. If f : M ! N 2 Spaces, then F (f) = f : (M, ~MOE) ! (N, ~NOE).
This is a dimap since for any V = {Vj} 2 ~NOEwith x, y 2 f-1 Vj, x M y =) x =
y =) f(x) = f(y) =) f(x) Vj f(y).
Remark 3.9. Note that U is faithful and F includes Spaces as a full subcategory
of LPS .
Proposition 3.10. F : Spaces AE LPS : U is an adjunction.
Proof.Let M be an object in Spaces and (N, ~V) 2 LPS . We claim that there is
a natural bijection
LPS (F (M), (N, ~V)) ~=Spaces(M, U(N, ~V)).
12 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
We need to show that there is a natural bijection
~=
` : Spaces(M, N) -! LPS ((M, ~MOE), (N, ~V)).
If f 2 LPS ((M, ~MOE), (N, ~V)), then f 2 Spaces(M, N) such that for any V =
{Vj} 2 ~V, for all j, f|f-1(Vj)satisfies x M y =) f(x) Vj f(y). Since x M *
* y
if and only if x = y this last condition is vacuous. Thus the bijection is sim*
*ply
` : f 7! f.
To show naturality let ff : (N, ~V) ! (N0, ~V)02 LPS and , : M0 ! M 2 Spaces.
Then
`(U(ff) O f O ,) = ff O f O , = ff O `(f) O ,.
Remark 3.11. LPS does not have colimits.
Consider the product of the directed circle and an interval. Now collapse t*
*he
top circle of this cylinder. The vertex of the resulting cone does not have a l*
*ocal
partial order.
4. The open-dicover topology
We define the open cover Grothendieck topology for Spaces and the open dicov*
*er
Grothendieck topology for LPS in the following lemma. The proof of the lemma
follows directly from the definition of a basis for a Grothendieck topology.
Lemma 4.1. (1) Spaces has a Grothendieck topology whose basis is given
by the open covers. For M 2 Spaces let K(M) = {open covers ofM}.
Let J be the Grothendieck topology generated by K. Call J the open cover
topology.
(2)Analogously, LPS has a Grothendieck topology whose basis is given by the
open dicovers in LPS . Let K((M, U)) = {open dicovers of(M, U)}. Call
the Grothendieck topology generated by K the open-dicover topology.
In Section 3, we defined a Grothendieck topology to be subcanonical if every
representable presheaf a sheaf. In this section, we will prove that the open-di*
*cover
topology is subcanonical.
The following proposition shows that if a Grothendieck topology is generated*
* by
a basis K, then to see if a presheaf is a sheaf it suffices to check the basis.*
* For the
definition of matching families and amalgamations see Remark 2.5.
Proposition 4.2 ([MLM92 , Proposition III.4.1]). Let C be a small categoryowit*
*hpa
Grothendieck topology J generated by a basis K. Then a presheaf P 2 SetC is a
sheaf for J if and only if for every M 2 C and every cover {OEj : Mj ! M} 2 K(M*
*),
every matching family for {OEj} of elements of P has a unique amalgamation.
op
Example 4.3. Let N 2 Spaces and y(N) = Spaces(-, N) 2 SetSpaces . Let
OEj : Mj ! M be an open cover, and let ffj : Mj ! N be a matching family. Then
OEj has a unique amalgamation OE : M ! N. Therefore y(N) is a sheaf for the open
cover topology, and hence the open cover topology is subcanonical.
Proposition 4.4. In the open-dicover topology J for local po-spaces every repre-
sentable presheaf is a sheaf. That is J is subcanonical.
A MODEL CATEGORY FOR LOCAL PO-SPACES 13
Proof.Consider the representable presheaf
op
y((N, ~V)) = LPS (-, (N, ~V)) 2 SetLPS .
By Proposition 4.2, y((N, ~V)) is a sheaf if and only if for all open dicovers *
*{OEj} 2
K((M, ~U)), any matching family
{ffj : (Mj, ~Uj) ! (N, ~V)}
has a unique amalgamation ff : (M, ~U) ! (N, ~V). That is, there is a map ff su*
*ch
that the diagrams
OEj
(Mj, ~Uj)____//(M, ~U)
tt
ffj|| ttffttt
fflffl|yyttt
(N, ~V)
commute in LPS for all j.
Let {ffj} be such a matching family for an open dicover {OEj}. Since {OEj} is*
* an
open dicover, then by Remark 3.7 for each j there is a Uj = {Ujk} 2 ~Ujsuch that
U0 = {Ujk}j,kis an order atlas and U0 2 ~U.
By definition {OEj : Mj ! M} is a cover in Spaces and {ffj : Mj ! N} is a
matching family. Therefore there is a unique amalgamation ff : M ! N 2 Spaces.
That is, there is a map ff such that
OEj
Mj _____//M
_
ffj||__ff___
fflffl|""__
N
commutes in Spaces for all j. It remains to show that ff is a dimap. Let V =
{Vl} 2 ~V. Since ffj : (Mj, ~Uj) ! (N, ~V) 2 LPS , there is a "Uj= {U"jk}k 2 ~U*
*jsuch
that for all k, l,
for allx, y 2 "Ujk\ ff-1j(Vl), x U"jky =) ffj(x) Vlffj(y).
Now for each j, let ^Uj= {U^jk}k 2 ~Ujbe a common refinement of "Ujand Uj. Then
since ^Ujis a refinement of "Uj,
(1) for allx, y 2 ^Ujk\ ff-1j(Vl), x U^jky =) ffj(x) Vlffj(y),
and since ^Ujis a refinement of Uj, if we define U = {U^jk}j,k, then U 2 ~U.
Since ff is an amalgamation of {ffj} in Spaces if x 2 ^Ujk M, then ff(x) = f*
*fj(x)
and for all l, ^Ujk\ ff-1j(Vl) = ^Ujk\ ff-1(Vl). Therefore using (1)for all k, *
*l,
for allx, y 2 ^Ujk\ ff-1(Vl), x U^jky =) ff(x) Vlff(y).
That is ff is a dimap. Therefore ff : (M, ~U) ! (N, ~V) is a unique amalgamatio*
*n of
{ffj}.
14 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
5.Equivalence of sheaves and di-'etale bundles
In this section C is either Spaces or LPS with the Grothendieck topology ge*
*n-
erated by open (di)covers.
Notation 5.1. We will use A openB to denote that A is an open subset of B.
op
Notation 5.2. Let Z 2 C and let F 2 SetC . Choose x 2 U openZ and s 2 F (U).
Then for open subobjects of U, L ,i!U, we have F (i) : F (U) ! F (L) and we will
use the notation
s|L := F (i)(s).
Recall that stalkx(F ) = colimx2LopenUF (L) and germx(s) is the equivalence cla*
*ss
represented by s in stalkx(F ).
Definition 5.3. Given Z 2 C, a bundle over Z is just a morphism p : W ! Z 2 C.
An (di)'etale bundle is a bundle which is a local (di)homeomorphism. That is, g*
*iven
y 2 W there is some open set V W such that p(V ) is open in Z and p|V is an
isomorphism in C.
A morphism of ('etale) bundles p : W ! Z and p : W 0! Z is a morphism
` : W ! W 02 C such that the following diagram commutes:
W A_____`______//W 0
AAA ____
pAA__AA""p0____
Z
Let Etale(Z) denote the category of (di)'etale bundles over Z. In addition l*
*et
O(Z) denote the category of open subobjects of Z, where the objects are open
subobjects of Z and the morphisms are the inclusions.
Theorem 5.4 (Theorem 1.3). Let Z 2 C. Then there is an equivalence of cate-
gories
: Etale(Z) o Shv(O(Z)) : .
Proof.It is well known that the statement of Theorem 1.3 is true when C = Spaces
(see for example [MLM92 , Corollary II.6.3]). We will show that this equivalen*
*ce be-
tween 'etale bundles on topological spaces and sheaves on topological spaces ex*
*tends
to local po-spaces.
First we describe the functors and in the case where C = Spaces . The
functor assigns to each bundle W -p!Z the presheaf of cross-sections:
P : O(Z)op ! Set
U 7! {s : U ! W 2 C | p O s = IdU}
U ,`!V 7! `* (`*(t) = t O `).
One can check that if p is 'etale, then P is in fact a sheaf [MLM92 , p.79]. T*
*hus
restricts to a functor : Etale(Z) ! Shv(O(Z)).
Given a presheaf P : O(Z)op ! Set, (P ) is the bundle W -p!Z where
W = {germx s | x 2 U openZ, s 2 P (U)} and p : germxs 7! x.
A MODEL CATEGORY FOR LOCAL PO-SPACES 15
A basis for the topology on W is given by the sets `s(U), where U is an open se*
*t in
Z, s 2 P (U) and
`s: U ! (P )
x 7! germxs.
Using this topology, p : W ! Z is a continuous map. Again, one can check that if
P is a sheaf, then W -p!Z is in fact an 'etale bundle [MLM92 , p.85]. So res*
*tricts
to a functor : Shv(O(Z)) ! Etale(Z).
Now we will show that and can be similarly defined in the case where
C = LPS . Let p : (W, ~T) ! (Z, ~U) be an 'etale bundle of local po-spaces. T*
*he
definition of is exactly the same: ((W, ~T) p-!(Z, ~U)) is the sheaf of cros*
*s-sections.
Given a sheaf P on a local po-space (Z, ~U), (P ) = (W -p!Z) is an 'etale bu*
*ndle
of topological spaces. To extend to local po-spaces it remains to define a lo*
*cal
order on W and show that this makes p a dimap.
Lemma 5.5. W has a canonical local po-space structure such that p is a dimap.
Proof.Recall that the sets `s(U) defined above form a basis for the topology of*
* W .
Choose an order atlas {(Ui, i)} 2 ~Ufor Z. For each open sub-po-space V Ui
and each s 2 P (V ), `s(V ) W is a po-space under the relation
germxs `s(Vg)ermys if and onlyx iy.
This is well-defined since {Ui} is an order-atlas, and it makes `s(V ) a po-spa*
*ce since
s`: Ui! `s(Ui) is a homeomorphism.
We claim that
T := {s`(V ) | V openUi, s 2 P (V )}
is an order atlas on W . First we need to show that it is an open cover. Each of
the sets is open by construction. If U 2 O(Z) and s 2 P (U), consider germx s.
Since {Ui} is an open cover of Z, for some i, x 2 Ui. Let V = U \ Ui. Then
germx s = germxs|V 2 (s`|V()V ). Therefore T is an open cover of W .
Finally we need to show that the orders are compatible. For k = 1, 2 let Vk *
*open
UikopenZ, and sk 2 P (Vk). Assume g1, g2 2 `s1(V1)\s`(V2). That is, g1 = germx1*
*s1
= germx1s2 and g2 = germx2s1 = germx2s2. For k = 1, 2,
g1 s`k(Vk)g2 () x1 ikx2.
Since {Ui} is an order-atlas, the order i1and i2are compatible. Therefore the
orders s`1(V1)and s`1(V1)are compatible, and T is an order-atlas on W .
Let ~Tbe the equivalence class of order atlases of T . We claim that ~Tdoes *
*not
depend on the choice of U 2 ~U.
Let U, U0 2 ~U, then U and U0 have a common refinement U00. Let T, T 0, T 00*
*be
the corresponding order-atlases for W constructed as above. We will show that T*
* 00
is a refinement of T .
Let A openUj 2 U, s 2 P (A) and germxs 2 `s(A). Then there is some U00k2 U00
such that x 2 U00kand U00kis a sub-po-space of Uj. Let A00= A \ U00k. It follow*
*s that
(s|`A00)(A00) `s(A), and germxs = germx(s|A00) 2 (s`|A00)(A00) 2 T 00. Since *
*U00kis a
sub-po-space of Uj it follows that (s`|A00)(A00) is a sub-po-space of `s(A). Th*
*us T 00
is a refinement of T .
Similarly T 00is a refinement of T 0and is hence a common refinement of T and
T 0. Therefore ~Tdoes not depend on the choice of U 2 ~U.
16 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
Finally we will show that the projection p : W ! Z given by germx s 7! x
is a dimap. Let U 2 ~Ube an order-atlas on Z. Let T be the order-atlas on W
constructed above from U. Observe that T 2 ~T, since ~Tdoes not depend on the
choice of U 2 ~U. Let Uj 2 U, let A openUi2 U, and let s 2 P (A). Assume that
germx1s, germx2s 2 `s(A) \ p-1(Uj).
Then x1, x2 2 Ui\ Uj. By the construction of T and since U is an order atlas,
germx1s `s(A)germx2s () x1 Ui x2 () x1 Uj x2.
Therefore can be extended to local po-spaces.
Thus we have maps
: Etale(Z) o Shv(O(Z)) : .
To show that they give an equivalence of categories we will show that for a she*
*af
P and an 'etale space W -p!Z there are natural isomorphisms
fflW : W ! W and jP : P ! P.
Recall that elements of W are of the form `s(x) = germxs, where s : U ! W
satisfies p O s = IdU and x 2 U. Define fflW to be the map `sx 7! sx. We will
show this is an isomorphism by constructing an inverse `W . Let y 2 W~and let
x = py. Since W is 'etale there exists y 2 V openW such that p|V : V -=!p(V ). *
*Let
q = (p|V )-1. Then define `W (y) = germxq = `qx. Then we claim `W is an inverse
for fflW . Indeed
fflW `W y = fflW `qx = qx = y.
Also for all `sx 2 W , `W fflW `sx = `W sx = germxt, where t is a restriction*
* of s.
So germxt = germxs = `sx.
Finally we claim that fflW and `W are dimaps. First choose T = {Tk} 2 ~Tand
U = {Ui} 2 ~Usuch that p satisfies the dimap condition. >From T construct the
canonical order atlas of the form {s`V } for W as in the proof of Lemma 5.5. *
*Now
let `sx1, `sx2 2 `sV \ ffl-1W(Tk). Then by construction,
`sx1 `sV`sx2 () x1 Ui x2.
Since s satisfies the dimap condition this implies that sx1 Tk sx2 which is the
same as fflW `sx1 Tk fflW `sx2. Thus fflW is a dimap. Next let y1, y2 2 Tk \ *
*`-1W(s`V ) =
Tk\fflW (s`V ) = Tk\sV . Then there are x1, x2 2 V such that y1 = sx1 and y2 = *
*sx2.
Since p satisfies the dimap condition
y1 Tk y2 =) py1 Ui py2.
But this is the same as x1 Ui x2 which implies that `sx1 `sV`sx2. Therefore `W
is a dimap.
The proof that the morphism jP is a bijection is the same as the proof in the
case of topological spaces [MLM92 , Theorem II.5.1].
A MODEL CATEGORY FOR LOCAL PO-SPACES 17
6.Points
In this section C is either Spaces or LPS with the Grothendieck topology ge*
*n-
erated by openo(di)covers.p
Let SetC and Shv (C) be the topoi of presheavesoandpsheaves on C. Recall
that the inclusion functor i : Shv (C) ! SetC has a right adjoint a called the
associated sheafofunctor.pRecall from Definition 2.15 that if p is a point in S*
*hv(C)
and ff 2 SetC , then stalkp(F ) = p* O a(ff).
Let Z 2 C. Then Z is a topological space or a local po-space and we can choo*
*se
any point (in the usual sense) x 2 Z. Define
op
p*x: SetC ! Set
F 7! colim F (L)
x2LopenZ
where the colimit is taken over all open subsets of Z containing x. See Remark *
*3.5
for a discussion of subobjectsoinpLPS .
Given a functor p* : SetC ! Set there is an induced functor
opp*
A : C y-!SetC -! Set,
where y is the Yoneda embedding defined on objects and morphisms by Z 7!
C(-, Z) and ' 7! C(-, ').
Givenoapfunctor A : C ! Setoonepcan define induced adjoint functors p* :
SetC ! Setand p* : Set! SetC (p* = - C A and p* = C(A, -), see [MLM92 ,
Section VII.2] ).
Definition 6.1. (i)The functor A : C ! Set is flat if the corresponding p*
is left exact.
(ii)A is continuous if A sends each covering sieve to an epimorphic family of
functions. That is, if S is a covering sieve, then the family of functi*
*ons
{A(')|' 2 S} is jointly surjective.
Proposition 6.2 ([MLM92op, Corollary VII.5.4]). Using the correspondence above,
p is a point in SetC if and only if A is flat. Furthermore p descends to a poi*
*nt
in Shv (C) if and only if A is flat and continuous.
Proposition 6.3. px defined above descends to a point in Shv (C) .
p*x
(2) SetCop_____//_SetOO;;
v v
i||a||vv
|fflffl|v
Shv (C)
Proof.Let Ax = p*xO y, where y is the Yoneda embedding.
First we show thatop*xispleft exact, that is it preserves finite limits. Let*
* F xG H
be a pullback in SetC .
p*x(F xG H) = colimx2L(ZF xG H)(L)
= colimx2LFZ(L) xG(L)H(L)
= colimF (L) xcolimG(L)colimH(L)
= p*xF xp*xGp*xH
18 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
The third equality holds because colimcommutes with pullbacks in Set,oandpthe
others are by definition. Thus A is flat and p*xis a point in SetC .
Next we show that Ax is continuous. Let {Yi-'i!N} be a covering sieve for N
in C . Recall that Ax = p*xO y. Let ('i)* denote composition with 'i. For each
arrow in the covering sieve,
p*xO y(Yi-'i!N)= p*x(C(-, Yi) ('i)*---!C(-, N))
= colimx2L(ZC(L, Yi) ('i)*---!C(L, N))
= y(Yi)x ('i)*---!y(N)x.
We claim that this is an epimorphic family of functions in Set . Let f 2 y(N)*
*x.
Then there is an open subspace L such that x 2 L Z and f is represented by
a morphism f0 2 C(L, N). Since {Yi} covers N, f0(x) 2 Yk for some k. Let K =
(f0)-1(Yk). Then K is open and x 2 K L Z. Furthermore f0|K 2 C(K, Yk)
which represents an element f002 y(Yk)x, and ('k)*f00= f. Hence we have an
epimorphic family as claimed. Thus A is continuous and px descends to a point in
Shv(C) .
Abusing notation we will also denote the inducedofunctorpin diagram (2)by p*x.
With this abuse of notation, the stalk of F 2 SetC at x is given by stalkx(F )*
* =
p*xa(F ) = p*x(F ). Note that stalkx(F ) = {germx(s) | x 2 U openZ, s 2 F (U)}.
Theorem 6.4. The points px defined above provide enough points for Shv (C).
That is, given f 6= g : P ! Q 2 Shv(C), there is an Z 2 C and a x 2 Z such that
p*xf 6= p*xg : p*xP ! p*xQ 2 Set.
Proof.Given Z 2 C and either P 2 Shv (C) or f 2 Mor Shv (C), let PZ or fZ
denote the restriction to Shv(O(Z)).
Assume that f 6= g : P ! Q 2 Shv (C). Thus there is some Z 2 C such that
fZ 6= gZ : PZ ! QZ 2 Shv(O(Z)).
By Theorem 1.3, this is equivalent to saying that the corresponding maps be-
tween 'etale spaces are not equal. That is,
fZ 6= gZ : PZ ! QZ 2 Etale(Z).
Thus there is some point y 2 PZ such that fZ(y) 6= gZ(y).
By the definition of , y = germxs for some x 2 U openZ and s 2 PZ(U). That
is y 2 stalkx(P ) = p*xP . Therefore p*xf 6= g*xg : p*xP ! p*xQ.
7. Stalkwise equivalences
Let (C, o) be a site with a subcanonicaloGrothendieckptopology such that Shv*
*(C)
has enough points and let ~y: C ! sSetC be the Yoneda embedding. Recall
the definition of stalkwise equivalence in Definition 2.19 which usesothepsimpl*
*icial
stalk functor (.)p. Also recall the Yoneda embedding ~y: C ! sSetC given in
Definition 2.17. Let ' : X ! Y 2 C.
Lemma 7.1. y~(') is a stalkwise equivalence if and only if for all points p in
Shv (C), p*ay(') 2 Set is an isomorphism.
A MODEL CATEGORY FOR LOCAL PO-SPACES 19
Proof.Let p be a point in Shv(C) . Recall that the simplicial stalk of ~y(') at*
* p is
given by
(~y('))p = {stalkp(~y(')n)}n 0 = {p*ay(')}n 0,
which is simplicially constant. Thus ~y(')p 2 sSetis an isomorphism if and only*
* if
p*ay(') 2 Set is an isomorphism.
Lemma 7.2. If theoGrothendieckptopology o is subcanonical, then the composite
functor C y-!SetC a-!Shv(C) is faithful.
Proof.By the Yoneda lemma, y is full and faithful. Since o is subcanonical im(y)
Shv (C). Furthermore a O i : Shv (C) ! Shv (C) is naturally isomorphic to the
identity functor [MLM92 , Corollary III.5.6]. Thus ay is naturally isomorphic *
*to y
which is faithful.
Theorem 7.3. Let ' : X ! Y 2 C and assume that ~y(') is a stalkwise equivalence.
Then ' is bijective.
The proof of this theorem is split into the following two propositions.
Proposition 7.4. Let ' : X ! Y 2 C and assume that ~y(') is a stalkwise
equivalence. Then ' is epi.
Proof.For i = 1, 2, let _i: Y ! Z 2 C be a morphism such that _1 O ' = _2 O ' :
X ! Z. Then for all points p in Shv(C) , p*ay(_1O ') = p*ay(_2O '). >From this
it follows that
p*ay(_1) O p*ay(') = p*ay(_2) O p*ay(').
But by Lemma 7.1 p*ay(') is a set isomorphism, so in particular it is epi. Ther*
*efore
p*ay_1 = p*ay_2 for all points p in Shv (C). Since C has enough points, ay_1 =
ay_2. By Lemma 7.2 a O y is faithful, thus _1 = _2. Therefore ' is epi.
Proposition 7.5. Let ' : X ! Y 2 C and assume that ~y(') is a stalkwise
equivalence. Then ' is mono.
Proof.For i = 1, 2, let _i: W ! X 2 C be a morphism such that ' O _1 = ' O _2 :
W ! Y . As in the proof of the previous proposition, for all points p in Shv(C)*
* ,
p*ay(') O p*ay(_1) = p*ay(') O p*ay(_2).
Again by Lemma 7.1, p*ay(') is mono. Therefore p*ay_1 = p*ay_2 for all points
p in Shv (C) . Since C has enough points, ay_1 = ay_2. By Lemma 7.2 a O y is
faithful, thus _1 = _2. Therefore ' is mono.
Let C = Spaces or LPS with the open cover topology. By Example 4.3 and
Proposition 4.4 this topology is subcanonical.
Recall from Section 6 that if Z 2 C and x 2 Z, then
op
p*x: SetC ! Set
(3) F 7! colim F (L)
x2LopenZ
descends to a point in Shv (C) (where the colimit is taken over open subspaces *
*of
Z which contain x).
Theorem 7.6. Let ' : X ! Y 2 C. Then ~y(') is a stalkwise equivalence if and
only if ' is an isomorphism in C.
20 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
Proof.(() If ' is an isomorphism, then for all points p in Shv (C) p*ay(') is an
isomorphism. Hence by Lemma 7.1 ~y(') is a stalkwise equivalence.
()) Assume that ~y(') is a stalkwise equivalence. Then by Theorem 7.3, ' is a
bijection.
Let x 2 Y . Let px be the corresponding point defined in (3). Then
p*xay(') : colimx2LCY(L, X) '*--!colimx2LCY(L, Y ) 2 Set
is a bijection. Let f : Y ! Y be given by f = IdY. Let ~f= [f] 2 colimx2L YC(L,*
* Y ).
Let ~g= ('*)-1(f~). Then there is some x 2 W Y such that ~ghas a representati*
*ve
g 2 C(W, X).
Let f0 = '*g = ' O g. Then [f0] = '*[g] = [f]. Therefore there exists x 2 S *
* Y
such that S Y \ W and f0|S = f|S = IdY|S.
Let _ = g|S. Therefore '_ = IdS. Let T = im(_). Then '|T O _ = IdSand '|T
is a bijection. Hence '|T : T ! S is an isomorphism, where x 2 S.
Finally this construction can be repeated for all x 2 Y . For each x 2 Y ther*
*e is
a x 2 Sx Y and there is a map
_x : Sx ! X such that_x = ('|im(_x))-1.
Since ' is a bijection, all local inverses must agree. That is, {_x : Sx ! X} i*
*s a
matching family on the open cover {Sx} of Y . Since the topology is subcanonica*
*l,
there is a unique amalgamation _ : Y ! X. It remains to show that _ is an inver*
*se
for '.
For allSx, ' O _|Sx = ' O _x = IdSx.
Therefore ' is an isomorphism in C .
8.Model categories for local po-spaces
8.1. A model category for local po-spaces. Using our results on LPS , Theo-
rem 1.1 will now follow directly from Jardine's model structure (Theorem 2.20).
Proof of Theorem 1.1.The open dicovers induce a Grothendieck topology on the
small category LPS . Applying Theorem 6.4, the Grothendieck topos Shv (LPS )
has enough points. So by Jardine's Theorem (Theorem 2.20), sPre(LPS ) has a
proper, simplicial, cellular model structure in which
o the cofibrations are the monomorphisms, i.e. the levelwise monomorphisms
of presheaves,
o the weak equivalences are the stalkwise equivalences, and
o the fibrations are the morphisms which have the right lifting property w*
*ith
respect to all trivial cofibrations.
Finally by Theorem 7.6 the weak equivalences coming from LPS (via the Yoneda
embedding) are precisely the isomorphisms.
8.2. Localization. Our main motivation for constructing a model category for
local po-spaces was to model concurrent systems. In particular we would like to
be able to define and understand equivalences of concurrent systems using such a
model category. However our model structure on sPre(LPS ) does not have any
non-trivial equivalences among the morphisms coming from LPS . To obtain a
model category more directly useful for studying concurrency, we need to locali*
*ze
with respect to a set of morphisms. In particular we want morphisms which prese*
*rve
certain computer-scientific information.
A MODEL CATEGORY FOR LOCAL PO-SPACES 21
How to best choose such morphisms is an important question and has been
studied in [Bub04 ]. For the sake of simplicity that paper studied only the cat*
*egory
PoSpaces of po-spaces (a subcategory of LPS ). There it was shown that the set *
*of
morphisms which should be equivalences depends on the context. That is, instead
of choosing equivalences for PoSpaces one should be choosing equivalences for t*
*he
coslice category or undercategory A # PoSpaces of po-spaces under a po-space A,
where A is called the context.
This result can be easily extended to our setting. First we remark that if
we choose a local po-space A then the undercategory A # LPS is the category
whose objects are dimaps 'M : A ! (M, ~U) and whose morphisms are dimaps
f : (M, ~U) ! (N, ~V) such that the following diagram commutes:
xA EE
'Mxxxxx EEE'NEE
__xxx f E""E
(M, ~U)____________//(N, ~V)
Next, ~y(A) 2 sPre(LPS ) and the undercategory ~y(A) # sPre(LPS )is the cat-
egory whose objects are morphisms of simplicial presheaves 'ff: ~y(A) ! ff and
whose morphisms are morphisms of simplicial presheaves f : ff ! fi such that the
following diagram commutes:
~y(A)B
'ff___ BB'fiB
__ BBB
""___ f B__
ff______________//fi
Since ~y: LPS ! sPre(LPS ) is a functor
~y('M ) : ~y(A) ! ~y(M, ~U) and ~y('N ) = ~y(f O 'M ) = ~y(f) O ~y('M ).
Hence A # LPS embeds as a subcategory of ~y(A) # sPre(LPS.)
Define morphisms in ~y(A) # sPre(LPS )to be weak equivalences, cofibrations
and fibrations if and only if they are weak equivalence, cofibrations and fibra-
tions in sPre(LPS ). Then this makes ~y(A) # sPre(LPS )into a model category
(see [Hir03, Theorem 7.6.5]).
We will show that this model category is again proper and cellular. We will n*
*eed
the following definitions and a theorem of Kan.
Definition 8.1. oLet C be a category and I be a set of maps in C. A relati*
*ve
I-cell complex is a map that can be constructed by a transfinite composi*
*tion
of pushouts of elements of I.
o An object A 2 C is small relative to a collection of morphisms D in C if
there exists a cardinal ~ such that for all regular cardinals ~ ~ and *
*for
all ~-sequences
X0 ! X1 ! X2 ! . .!.Xfi! . . .
with Xfi! Xfi+1in D for fi + 1 < ~, the set map
colimfi<~C(A, Xfi) ! C(A, colimfi<~Xfi)
is an isomorphism.
22 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
Definition 8.2. A model category M is cofibrantly generated if there are sets I
and J such that
o the domains of I are small relative to the relative I-cell complexes,
o the domains of J are small relative to the relative J-cell complexes,
o the fibrations have the right lifting property with respect to J, and
o the trivial fibrations have the right lifting property with respect to I.
We say that M is cofibrantly generated by I and J.
Definition 8.3. oLet M be a model category cofibrantly generated by I
and J. An object A 2 M is compact if there is a cardinal fl such that
for all relative I-cell complexes f : X ! Y with a particular presentati*
*on,
every map A ! Y factors through a subcomplex of size at most fl.
o f : A ! B is an effective monomorphism if f is the equalizer of the incl*
*u-
sions B ' B qA B.
Definition 8.4. A cellular model category is a model category cofibrantly gener-
ated by I and J such that
o the domains and codomains of elements of I and J are compact,
o the domains of elements of J are small relative to relative I-cell compl*
*exes,
and
o the cofibrations are effective monomorphisms.
Theorem 8.5 ([Hir03, Theorem 11.3.2]). Let M be a model category cofibrantly
generated by the sets I and J, and let N be a bicomplete category such that the*
*re
exists a pair of adjoint functors F : M o N : U. Define F I = {F u | u 2 I} and
F J = {F v | v 2 J}. If
(1)the domains of F I and F J are small relative to F I-cell and F J-cell, *
*re-
spectively, and
(2)U maps relative F J-cell complexes to weak equivalences,
then N has a model category structure cofibrantly generated by F I and F J such
that f is a weak equivalence in N if and only if Uf is a weak equivalence in M,
and (F, U) is a Quillen pair.
Theorem 8.6. Let M be a model category and let A 2 M. Then A # M has a
model structure where a morphism A @ is a weak equivalence, cofibration
"""""@__@f
B ______//C
or fibration in A # M if and only if f is a weak equivalence, cofibration or fi*
*bration,
respectively, in M. If M is proper, cofibrantly generated or cellular, then so*
* is
A # M .
Remark 8.7. For a more detailed proof we invite the reader to regard Hirschhorn*
*'s
note [Hir05].
Proof.That A # M has the stated model structure follows from the definitions (s*
*ee
[Hir03, Theorem 7.6.5]).
Pushouts and pullbacks in A # M can be formed by taking pushouts and pull-
backs of the underlying morphisms in M, and then taking the induced maps from
A. It thus follows that if M is proper so is A # M .
Assume M is cofibrantly generated by I and J. The method for showing that
A # M is cofibrantly generated will be to apply Theorem 8.5 to the following a*
*djoint
A MODEL CATEGORY FOR LOCAL PO-SPACES 23
functors:
F : M o (A # M ) : U
where for B 2 M and f : B ! C 2 M,
A i vA HHi
F (B) = fi1flffl||, F (f) = z1zvvvvvHH1$$HHIdqf
A q B A q B ________//_A q C
and U is the forgetful functor
0 1 0 1
A ' A @@'
U @ f'Bflffl||A= B, U @ "B""""""@COO@@fA= B f-!C.
B B ________//_C
Define F I = {F u | u 2 I} and F J = {F v | v 2 J}.
The main observation for the proof is that for a morphism u in M, the pushout
of F u is obtained from the pushout of u in M. That is,
A4TTTTTFF
4FFFTTTT44 u
44FFF TTTTT B _____//C
44 F## TTT))T// | |
44A q B Idqu_A q C where P is defined byf| __ |
44 | | fflffl|fflffl||
44|'Xqf __| X _____//P
4ssssfflffl|fflffl||
X ________//_P
From this it follows that for a set of morphisms S in M, the underlying morphis*
*ms
of a relative F S-complex are a relative S-complex.
Hence the conditions on A # M in Theorem 8.5 and the definition of a cellular
model category (Definition 8.4) are all inherited from the corresponding condit*
*ions
in M.
Finally one can check that the model category structure given by Theorem 8.5
coincides with the one in the statement of the theorem.
Let M denote the model structure above on ~y(A) # sPre(LPS ). Since M
is cellular we can apply left Bousfield localization [Hir03] to this model stru*
*c-
ture M with respect to a set of morphisms which will preserve the computer-
scientific properties we are interested in. In [Bub04 ], one inverted the set *
*of di-
homotopy equivalences in A # PoSpaces . So in our setting we will let I be the
set of dihomotopy equivalences in A # LPS defined below. We will invert the set
I = {~y(f) | f 2 I} ~y(A) # sPre(LPS.)
Definition 8.8. oLet "Ibe the po-space ([0, 1], ) where is the usual to*
*tal
order on [0, 1]. Given dimaps f, g : (M, ~U) ! (N, ~V) 2 A # LPS , OE is*
* a
dihomotopy from f to g if OE : (M, ~U) x "I! (N, ~V), OE|(M,U~)x{0}= f,
OE|(M,U~)x{1}= g, and for all a 2 A, OE('M (a), t) = 'N (a). In this cas*
*e write
OE : f ! g.
o The symmetric, transitive closure of dihomotopy is an equivalence relati*
*on.
Write f ' g if there is a chain of dihomotopies f ! f1 f2 ! . . .fn !
g.
o A dimap f : (M, ~U) ! (N, ~V) is a dihomotopy equivalence if there is a
dimap g : (N, ~V) ! (M, ~U) such that g O f ' IdM and f O g ' IdN.
24 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
The left Bousfield localization of M with respect to I provides a model struc*
*ture
on ~y(A) # sPre(LPS )in which the weak equivalences are the I-local equivalences
(see [Hir03]), the cofibrations are the cofibrations in M and the fibrations are
morphisms which have the right lifting property with respect to morphisms which
are both cofibrations and I-local equivalences.
Theorem 8.9 (Theorem 1.2). Let I = {~y(f) | f is a directed homotopy equivalence
rel A}. The category ~y(A) # sPre(LPS )has a left proper, cellular model struct*
*ure
in which
o the cofibrations are the monomorphisms,
o the weak equivalences are the I-local equivalences, and
o the fibrations are those morphisms which have the right lifting property*
* with
respect to monomorphisms which are I-local equivalences.
We claim that this model category provides a good model for studying concur-
rency. An analysis of this model category will be the subject of future researc*
*h.
Appendix A. Hypercovers
Suppose now C is small and equipped withoapGrothendieck topology, i.e. we ha*
*ve
a site (C, o). The ~Cechstructure sSetC~c(o)is obtained from the projective str*
*ucture
by homotopically localizing the comparison morphisms given by the ~Cechcovers
with respect to o or, up-to homotopy, from the injective structure by localizin*
*g at
the same set of morphisms.
n u o
Definition A.1. Let U = Ui-!i X 2 J (X) be a cover. Let ip 2 I for each
i2I
0 p n and Ui0...inbe the wide pullback of the uip's, i.e. the limiting obje*
*ct of
the diagram
Ui0QQ . . . Uip . . . mUin
QQQQui0QQ ui| uinmmmmmm
QQQQ | p mmmmm
QQ((Qfflffl|vvmmmX
The ~Cech nerve ~Uof U is the simplicial presheaf given by
a
~Undef= y (Ui0...in)
i0,...,in2I
Remark A.2. For any n 2 N, X 2 C and U 2 J (X) there is a morphism
ui0...in: Ui0...in! X
and a diagram of presheaves
~Un_____EU,X,n__//y (X)
OO ll66l
iny(Ui | lllll
0,...,in)||ly(ui0,...,in)lllll
y (Ui0,...,in)
where EU,X,nis given by universal property. The EU,X,nassemble to a morphism
of simplicial presheaves
EU,X : ~U! ~y(X)
A MODEL CATEGORY FOR LOCAL PO-SPACES 25
Remark A.3. Given U 2 J (X) seen as a subcategory of the slice C=X, there is the
evident functor op
ffiU :U ! sSetC
ui 7! ~y(Ui)
op
Proposition A.4. Localizing sSetCinjat the sets
(i){EU,X | X 2 C, U 2 J (X)};
(ii) hocolim (ffiU!)~y(X) | X 2 C, U 2 J (X);
(iii){~ ('U )| X 2 C, U 2 J (X)}where, given X 2 C and R a sieve on X, 'R :
Rn,! y (X) is the corresponding inclusion of presheaves;
opo Cop Cop
(iv) jF : F ! j (F )| F 2 sSetC where j : sSet ! sSet is the ob-
jectwise sheafification functor;
op
yields the same model structure sSetC~c(o). The same holds for the projective v*
*ersion.
op
Finally, there is a model structure sSetChyp(o)obtained from the projective *
*struc-
ture by homotopically localizing at the set of the comparison morphisms given by
hypercovers with respect to o. This model structureoispQuillen equivalent to Ja*
*r-
dine's model structure (Theorem 2.20) on sSetC [DHI04 , Theorem 1.2]. As with
the ~Cechstructure, there is also an injective version. Since ~Cechcovers are p*
*artic-
ular hypercovers, there is the series of inclusions
Wprj W~c(o) Whyp(o)
and a similar series for the injective version. It is in general the case that *
*W~c(o)$
Whyp(o), yet equality holds in some important particular cases like the smooth
Nisnevitch site (c.f. [DHI04 , Example A10]). It is an interesting question whe*
*ther
or not W~c(o)= Whyp(o)for local po-spaces.
References
[AGV72] M. Artin, A. Grothendieck, and J. L. Verdier. Th'eorie des Topos et Coh*
*omologie Etale
des Sch'emas, volume 269 and 270 of Lecture Notes in Math. Springer-Ver*
*lag, 1972.
[BK72] A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localiz*
*ations.
Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.
[Bub04] Peter Bubenik. Context for models of concurrency. In Prelimary Proceedi*
*ngs of the
Workshop on Geometry and Topology in Concurrency and Distributed Comput*
*ing
GETCO 2004, volume NS-04-2 of BRICS Notes, pages 33-49. BRICS, Amsterda*
*m,
The Netherlands, 2004 (also to appear in Elec. Notes in Th. Comp. Sci.).
[DHI04] Daniel Dugger, Sharon Hollander, and Daniel C. Isaksen. Hypercovers and*
* simplicial
presheaves. Math. Proc. Cambridge Philos. Soc., 136(1):9-51, 2004.
[Dij68] E.W. Dijkstra. Cooperating sequential processes. Academic press, 1968.
[Dug01] Daniel Dugger. Universal homotopy theories. Adv. Math., 164(1):144-176,*
* 2001.
[FGR99] Lisbeth Fajstrup, Eric Goubault, and Martin Raussen. Algebraic topology*
* and concur-
rency. to appear in Theoretical Computer Science, 1999. Also preprint R*
*-99-2008, Dept.
of Mathematical Sciences, Aalborg University, Aalborg, Denmark.
[Gau03] Philippe Gaucher. A model category for the homotopy theory of concurren*
*cy. Homology,
homotopy and applications, 5(1):549-599, 2003.
[Gou96] Eric Goubault. Durations for truly-concurrent transitions. Lecture Note*
*s in Computer
Science, 1058, 1996.
[Gou02] Eric Goubault. Labelled cubical sets and asynchronous transitions syste*
*ms: an adjunc-
tion. In Alexander Kurz, editor, Electronic Notes in Theoretical Comput*
*er Science,
volume 68. Elsevier Science Publishers, 2002.
[Gou03] Eric Goubault. Some geometric perspectives in concurrency theory. Homol*
*ogy Homo-
topy Appl., 5(2):95-136 (electronic), 2003. Algebraic topological metho*
*ds in computer
science (Stanford, CA, 2001).
26 PETER BUBENIK AND KRZYSZTOF WORYTKIEWICZ
[Gra03]Marco Grandis. Directed homotopy theory. I. Cah. Topol. G'eom. Diff'er. *
*Cat'eg.,
44(4):281-316, 2003.
[Hir03]Philip S. Hirschhorn. Model categories and their localizations, volume 9*
*9 of Mathemat-
ical Surveys and Monographs. American Mathematical Society, Providence, *
*RI, 2003.
[Hir05]Philip S. Hirschhorn. Overcategories and undercategories of model*
* categories.
http://www-math.mit.edu/~psh/, 2005.
[Hov99]Mark Hovey. Model categories, volume 63 of Mathematical Surveys and Mono*
*graphs.
American Mathematical Society, Providence, RI, 1999.
[Jar87]J. F. Jardine. Simplicial presheaves. J. Pure Appl. Algebra, 47(1):35-87*
*, 1987.
[Jar96]J. F. Jardine. Boolean localization, in practice. Doc. Math., 1:No. 13, *
*245-275 (elec-
tronic), 1996.
[Joh77]Peter T. Johnstone. Topos Theory. London Mathemathical Society Monograph*
*s. Aca-
demic Press, 1977.
[Joy84]Andr'e Joyal. Homotopy theory of simplicial sheaves. unpublished (circul*
*ated as a letter
to Grothendieck), 1984.
[Law63]F. William Lawvere. Functorial Semantics of Algebraic Theories. PhD thes*
*is, Columbia
University, 1963.
[Law64]F. William Lawvere. An elementary theory of the category of sets. Porc. *
*Nat. Acad.
Sci. USA, 52:1506-1511, 1964.
[Law73]F.W. Lawvere. Metric spaces, generalized logic, and closed categories. R*
*end. Sem. Mat.
Fis. di Milano, 43:135-166, 1973.
[Mil80]R. Milner. A Calculus of Communicating Systems, volume 92 of Lecture Not*
*es in Com-
puter Science. Springer-Verlag, 1980.
[MLM92]Saunders Mac Lane and Ieke Moerdijk. Sheaves in geometry and logic. Univ*
*ersitext.
Springer-Verlag, New York, 1992. A first introduction to topos theory.
[Qui67]Daniel G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No*
*. 43.
Springer-Verlag, Berlin, 1967.
E-mail address: p.bubenik@csuohio.edu
Department of Mathematics, Cleveland State University, 2121 Euclid Ave. RT 15*
*15,
Cleveland OH, 44115-221, USA
E-mail address: kworytki@uwo.ca
Department of Mathematics, University of Western Ontario, Middlesex College,
London, Ontario N6A 5B7, Canada