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We would like a general procedure to determine when a topos can be a category of models of homotopy types of CW-complexes, the same way simplicial sets, cubical sets, or any other category of presheaves on a test category do. In particular, given a sheaf $X$ on a site $C$, we would like that the shape of the topos $Sh(C)_{/X}$ to be strongly related with the (weak) homotopy type of $X$. Typical examples coming from geometry should be sheaves of sets on suitable categories of manifolds. Grothendieck never mentioned the concept of test topoi themselves, but the idea of `test topologies' on test categories is expressed from time to time in Pursuing Stacks. As in the case of test categories, the natural (i.e most robust) notion is in fact local: it is worth looking at models of homotopy types over a given homotopy type.
In other terms, given a topos $X$, a natural question will be to formulate how the objects of $X$ are models of homotopy types over the homotopy type of $X$ (defined through shape theory). The following is a summary of part of (Cisinski03).
A morphism of Grothendieck topoi $f\colon X\to Y$ is an Artin-Mazur weak equivalence if, for any integer $n\geq 0$ and any locally constant sheaf $F$ on $Y$, the induced map
is bijective, with $n=0$ if $F$ is a sheaf of sets, $n=1$ if $F$ is a sheaf of groups, and $n$ is arbitrary if $F$ is a sheaf of abelian groups. These are simply called weak homotopy equivalences of topoi in (Moerdijk95). In particular, any morphism inducing an equivalence of shapes of the associated hypercomplete $\infty$-topoi is Artin-Mazur weak equivalence; see (Hoyois18).
Given a Grothendieck topos $T$, a morphism $\colon x\to y$ in $T$ is an Artin-Mazur weak equivalence if the induced morphism of topoi $T_{/x}\to T_{/y}$ (whose pull-back functor is defined via the assignment $(t\to y)\mapsto (t\times_y x\to x)$) is an Artin-Mazur weak equivalence.
If $f\colon C\to D$ is a functor between small categories, it induces a morphism of topoi $f\colon PSh(C)\to PSh(D)$ whose pull-back functor is defined by precomposition with $u$. The nerve of the functor $f$ is a weak homotopy equivalence of simplicial sets if and only if the corresponding morphism of topoi is an Artin-Mazur weak equivalence. This is a reformulation of a well known theorem of Whitehead asserting that one can detect weak homotopy equivalences through cohomology with coefficients in local systems.
There is an analogue of (a relative version of) Quillen’s Theorem A which is a direct application of (hyper)descent (e.g. see Thm. 3.4.25 in (Cisinski03)):
Let
be a commutative triangle of Grothendieck topoi. Let $(S_i)_{i\in I}$ be a small family of maps in $S$ which is a covering, i.e. so that the induced map from $\coprod_i S_i$ to the terminal object of $S$ is an epimorphism. If, for each index $i$ the induced map $X_{/p^*(S_i)}\to Y_{/q^*(S_i)}$ is an Artin-Mazur weak equivalence, then the map $f:X\to Y$ is an Artin-Mazur weak equivalence as well.
A particular case is the following.
Let
be a commutative triangle in a Grothendieck topos $T$. Let $(S_i\to S)_{i\in I}$ be a small family of maps in $T$ which is a covering, i.e. so that the induced map $\coprod_i S_i\to S$ is an epimorphism. If, for each index $i$ the induced map $S_i\times_S X\to S_i\times_SY$ is an Artin-Mazur weak equivalence, then the map $f:X\to Y$ is an Artin-Mazur weak equivalence as well.
Let $T$ be a Grothendieck topos. The class of monomorphisms which are Artin-Mazur weak equivalences is saturated and stable under small filtered colimits.
This follows right away from Prop. 3.4.22 in (Cisinski03).
A Grothendieck topos $T$ is aspherical if the map $T\to pt$ to the terminal topos is an Artin-Mazur weak equivalence.
An object $x$ of a topos $T$ is aspherical if the corresponding topos $T_{/x}$ is aspherical.
A Grothendieck topos $T$ is locally aspherical there is a generating family of $T$ which consists of aspherical objects.
If $C$ is a small category, the corresponding topos of presheaves of sets $PSh(C)$ is locally aspherical. Indeed, representable presheaves generate $PSh(C)$, and, for any presheaf $x$ on $C$ represented by an object $c$ of $C$, we have a canonical equivalence
with $PSh(C_{/c})$ aspherical because the slice category $C_{/c}$ has a terminal object.
A topos $T$ is locally aspherical if and only if $T\cong Sh(C)$ where $C$ is a small site such that, for any representable presheaf $U$ on $C$, with associated sheaf $U^{\wedge}$, the topos $T_{/U^{\wedge}}$ is aspherical. Equivalently (using the toposic analogue of Quillen’s Theorem A above), for any presheaf $F$ on $C$ with associated sheaf $F^{\wedge}$, the induced morphism of topoi $T_{/F^{\wedge}}\to PSh(C)_{/F}$ is an Artin-Mazur weak equivalence.
An interval in a topos $T$ is an object $I$ with the following properties:
it has two disjoint global sections: there is a monomorphism from the disjoint union of two copies of the terminal object to $I$;
for any object $X$ in $T$, the projection $I\times X\to X$ is an Artin-Mazur weak equivalence (equivalently, there exists a small family $(S_i)_{i\in I}$ which covers $T$ such that the projection $I\times S_i\to S_i$ is an Artin-Mazur weak equivalence).
Let $T$ be a Grothendieck topos. The following conditions are equivalent:
Any morphism of $T$ with the right ligting property with respect to monomorphisms is an Artin-Mazur weak equivalence;
There exists an interval in $T$;
Since the class of Artin-Mazur weak equivalences has the 2-out-of-3 property and is saturated, this follows from Lemma 3.3 in (Cisinski02).
A local test topos is a topos which is locally aspherical and which has an interval.
A test topos is a local test topos which is aspherical.
A small category $A$ is a (local) test category if and only if the associated topos $PSh(A)=Fun(A^{op},Set)$ is a (local) test topos: this follows from the proposition right above and from the characterization of local test categories provided by Thm. 1.5.6 in (Maltsiniotis05).
Let $T$ be a topos and $C$ a small site such that $T\cong Sh(C)$. We assume that $C$ is a local test category and that, for any representable presheaf $F$ on $C$ with associated sheaf $F^\wedge$, the topos $T_{/U^{\wedge}}$ is aspherical. Then $T$ is a local test topos: any interval of $PSh(C)$ is then an interval of $T$, and $T$ is clearly locally aspherical; see Thm. 4.2.8 in (Cisinski03).
Let $T$ be a local test topos. Then there is a proper combinatorial model category structure on $T$ whose cofibrations are the monomorphisms, and whose weak equivalences are the Artin-Mazur weak equivalences. Futhermore, this model structure provides models for homotopy types over the homotopy type of $T$. More precisely, if $C$ is any small site whose underlying category is a local test category, such that $T\cong Sh(C)$, and such that, for any representable presheaf $F$ on $C$ with associated sheaf $F^\wedge$, the topos $T_{/F^{\wedge}}$ is aspherical, then the sheafification functor is a left Quillen equivalence $PSh(C)\to T$, and there is a left Quillen equivalence $PSh(C)\to SSet_{/N(C)}$ which sends any presheaf $F$ to the nerve of its category of elements $C_{/F}$.
The first assertion on the existence of a combinatorial proper model structure comes from Cor. 4.2.12 in (Cisinski03) and is an example of a Cisinski model structure provided by Thm. 3.9 in (Cisinski02). For the second part, it is obvious that the sheafification functor is a left Quillen equivalence from $PSh(C)$ to $T$ and an Artin-Mazur weak equivalence. Therefore, it is sufficient to prove the case where $T=PSh(C)$. This is then a particular case of Proposition 4.4.28 in (Cisinski06).
Let $T$ be a test topos. Then there is a proper combinatorial model category structure on $T$ whose cofibrations are the monomorphisms, and whose weak equivalences are the Artin-Mazur weak equivalences which models homotopy types of CW-complexes.
Let $T$ be a local test topos and $C$ an internal category of $T$. On can then consider the topos $PSh(C)$ of internal presheaves on $C$ (which only depends on the stack associated to $C$). Then, by virtue of Thm. 6.2.7 in (Cisinski03), $PSh(C)$ is a local test topos as well. In particular, if $G$ is a sheaf of groups on $T$, then $Rep(G)=PSh(BG)$ is the category of representations of $G$ (i.e. the objects of $T$ equipped with an action of $G$). If the Artin-Mazur weak equivalences are stable under finite products in $T$, then one can show that the Artin-Mazur weak equivalences of $Rep(G)$ simply are the $G$-equivariant maps $X\to Y$ which are Artin-Mazur weak equivalences in $T$ when we forget the action of $G$; to see this, one reduces easily to the case where $T$ is a category of presheaves on a local test category, and this follows then from Prop. 7.3.8 in (Cisinski02).
Let us consider a full subcatogry $C$ of the category of locally ringed spaces with the following properties:
for any space $X$ in $C$, there is a basis of contractible open subsets of $X$ which are $C$;
for any locally ringed space $X$, if each point of $X$ admits an open neighborhood in $C$, then $X$ belong to $X$;
the forgetful functor from $C$ to the category of topological spaces commutes with products (up to weak equivalence);
there exists a contractible space with two distinct closed points in $C$.
We consider $C$ as site with the Grothendieck topology induced by open coverings. Then, for any $X$ in $C$, the topos $Sh(C)_{/X}$ is aspherical because there is an adjunction between $Sh(X)$ and $Sh(C)_{/X}$ in the $2$-category of topoi. Therefore, since $C$ is a strict test category, we see that $Sh(C)$ is a test topos in which the Artin-Mazur weak equivalences are stable under finite products; see Thm. 6.1.8 in (Cisinski03). If $X$ is a locally ringed space which has a covering by spaces in $C$, then the sheaf on $C$ represented by $X$ corresponds in the test model structure on $Sh(C)$ to the classical weak homotopy type of $X$ (this follows by hyperdescent in shape theory). In practice, all objects of $C$ admit CW-structures, so that we even get the classical homotopy type of $X$ characterized by the sheaf represented by $X$ on $C$. Such geometric models of weak homotopy types of manifolds are used for instance by Madsen and Weiss (Madsen-Weiss07) and Kupers (Kupers19).
Examples of such ‘test sites’ $C$ are given by: euclidian spaces (seen as $C^\infty$-manifolds), or contractible Stein complex manifolds. If $G$ is a real (or complex) Lie group, then representations of $G$ in the topos of sheaves on differentiable (or complex analytic) manifolds is a local test topos. One may also replace $G$ by any orbifold, presented as an internal groupoid in the category of manifolds.
Michael Artin and Barry Mazur, Etale homotopy, Lecture Notes in Mathematics 100, Springer (1969).
Denis-Charles Cisinski, Théories homotopiques dans les topos, J. Pure Appl. Algebra 174, No. 1 (2002). (doi)
Denis-Charles Cisinski, Faisceaux localement asphériques, preprint (2003) [pdf, pdf]
Marc Hoyois, Higher Galois theory, J. Pure Appl. Algebra 222, no. 7 (2018), (arXiv)
Alexander Kupers, Three applications of delooping to h-principles, Geom. Dedicata 202 (2019), pp. 103–151. (journal, arXiv)
Ib Madsen and Michael Weiss, The stable moduli space of Riemann surfaces: Mumford’s conjecture, Ann. of Math. (2)165(2007), no. 3, 843–941. (journal, arXiv)
Georges Maltsiniotis, La théorie de l’homotopie de Grothendieck, Astérisque 301 (2005) (see
html)
Ieke Moerdijk, Classifying spaces and classifying topoi, Lecture Notes in Mathematics 1616, Springer (1995).
Last revised on November 4, 2022 at 10:04:55. See the history of this page for a list of all contributions to it.