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test topos

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

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 XX on a site CC, we would like that the shape of the topos Sh(C) /XSh(C)_{/X} to be strongly related with the (weak) homotopy type of XX. 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 XX, a natural question will be to formulate how the objects of XX are models of homotopy types over the homotopy type of XX (defined through shape theory). The following is a summary of part of (Cisinski03).

Artin-Mazur weak equivalences

A morphism of Grothendieck topoi f:XYf\colon X\to Y is an Artin-Mazur weak equivalence if, for any integer n0n\geq 0 and any locally constant sheaf FF on YY, the induced map

f *:H n(Y,F)H n(X,F) f^*\colon H^n(Y,F)\to H^n(X,F)

is bijective, with n=0n=0 if FF is a sheaf of sets, n=1n=1 if FF is a sheaf of groups, and nn is arbitrary if FF 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 TT, a morphism :xy\colon x\to y in TT is an Artin-Mazur weak equivalence if the induced morphism of topoi T /xT /yT_{/x}\to T_{/y} (whose pull-back functor is defined via the assignment (ty)(t× yxx)(t\to y)\mapsto (t\times_y x\to x)) is an Artin-Mazur weak equivalence.

Remark

If f:CDf\colon C\to D is a functor between small categories, it induces a morphism of topoi f:PSh(C)PSh(D)f\colon PSh(C)\to PSh(D) whose pull-back functor is defined by precomposition with uu. The nerve of the functor ff 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)):

Proposition

Let

X f Y p q S \array{ X & & \overset{f}{\to} & & Y\\ & _p \searrow & & \swarrow_q \\ & & S }

be a commutative triangle of Grothendieck topoi. Let (S i) iI(S_i)_{i\in I} be a small family of maps in SS which is a covering, i.e. so that the induced map from iS i\coprod_i S_i to the terminal object of SS is an epimorphism. If, for each index ii the induced map X /p *(S i)Y /q *(S i)X_{/p^*(S_i)}\to Y_{/q^*(S_i)} is an Artin-Mazur weak equivalence, then the map f:XYf:X\to Y is an Artin-Mazur weak equivalence as well.

A particular case is the following.

Corollary

Let

X f Y p q S \array{ X & & \overset{f}{\to} & & Y\\ & _p \searrow & & \swarrow_q \\ & & S }

be a commutative triangle in a Grothendieck topos TT. Let (S iS) iI(S_i\to S)_{i\in I} be a small family of maps in TT which is a covering, i.e. so that the induced map iS iS\coprod_i S_i\to S is an epimorphism. If, for each index ii the induced map S i× SXS i× SYS_i\times_S X\to S_i\times_SY is an Artin-Mazur weak equivalence, then the map f:XYf:X\to Y is an Artin-Mazur weak equivalence as well.

Proposition

Let TT be a Grothendieck topos. The class of monomorphisms which are Artin-Mazur weak equivalences is saturated and stable under small filtered colimits.

Proof

This follows right away from Prop. 3.4.22 in (Cisinski03).

Definition

A Grothendieck topos TT is aspherical if the map TptT\to pt to the terminal topos is an Artin-Mazur weak equivalence.

An object xx of a topos TT is aspherical if the corresponding topos T /xT_{/x} is aspherical.

A Grothendieck topos TT is locally aspherical there is a generating family of TT which consists of aspherical objects.

Example

If CC is a small category, the corresponding topos of presheaves of sets PSh(C)PSh(C) is locally aspherical. Indeed, representable presheaves generate PSh(C)PSh(C), and, for any presheaf xx on CC represented by an object cc of CC, we have a canonical equivalence

PSh(C) /xPSh(C /c)PSh(C)_{/x}\cong PSh(C_{/c})

with PSh(C /c)PSh(C_{/c}) aspherical because the slice category C /cC_{/c} has a terminal object.

Example

A topos TT is locally aspherical if and only if TSh(C)T\cong Sh(C) where CC is a small site such that, for any representable presheaf UU on CC, with associated sheaf U U^{\wedge}, the topos T /U T_{/U^{\wedge}} is aspherical. Equivalently (using the toposic analogue of Quillen’s Theorem A above), for any presheaf FF on CC with associated sheaf F F^{\wedge}, the induced morphism of topoi T /F PSh(C) /FT_{/F^{\wedge}}\to PSh(C)_{/F} is an Artin-Mazur weak equivalence.

Characterization and examples

Definition

An interval in a topos TT is an object II 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 II;

  • for any object XX in TT, the projection I×XXI\times X\to X is an Artin-Mazur weak equivalence (equivalently, there exists a small family (S i) iI(S_i)_{i\in I} which covers TT such that the projection I×S iS iI\times S_i\to S_i is an Artin-Mazur weak equivalence).

Proposition

Let TT be a Grothendieck topos. The following conditions are equivalent:

  • Any morphism of TT with the right ligting property with respect to monomorphisms is an Artin-Mazur weak equivalence;

  • There exists an interval in TT;

Proof

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).

Definition

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.

Example

A small category AA is a (local) test category if and only if the associated topos PSh(A)=Fun(A op,Set)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).

Example

Let TT be a topos and CC a small site such that TSh(C)T\cong Sh(C). We assume that CC is a local test category and that, for any representable presheaf FF on CC with associated sheaf F F^\wedge, the topos T /U T_{/U^{\wedge}} is aspherical. Then TT is a local test topos: any interval of PSh(C)PSh(C) is then an interval of TT, and TT is clearly locally aspherical; see Thm. 4.2.8 in (Cisinski03).

Theorem

Let TT be a local test topos. Then there is a proper combinatorial model category structure on TT 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 TT. More precisely, if CC is any small site whose underlying category is a local test category, such that TSh(C)T\cong Sh(C), and such that, for any representable presheaf FF on CC with associated sheaf F F^\wedge, the topos T /F T_{/F^{\wedge}} is aspherical, then the sheafification functor is a left Quillen equivalence PSh(C)TPSh(C)\to T, and there is a left Quillen equivalence PSh(C)SSet /N(C)PSh(C)\to SSet_{/N(C)} which sends any presheaf FF to the nerve of its category of elements C /FC_{/F}.

Proof

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)PSh(C) to TT and an Artin-Mazur weak equivalence. Therefore, it is sufficient to prove the case where T=PSh(C)T=PSh(C). This is then a particular case of Proposition 4.4.28 in (Cisinski06).

Corollary

Let TT be a test topos. Then there is a proper combinatorial model category structure on TT whose cofibrations are the monomorphisms, and whose weak equivalences are the Artin-Mazur weak equivalences which models homotopy types of CW-complexes.

Example

Let TT be a local test topos and CC an internal category of TT. On can then consider the topos PSh(C)PSh(C) of internal presheaves on CC (which only depends on the stack associated to CC). Then, by virtue of Thm. 6.2.7 in (Cisinski03), PSh(C)PSh(C) is a local test topos as well. In particular, if GG is a sheaf of groups on TT, then Rep(G)=PSh(BG)Rep(G)=PSh(BG) is the category of representations of GG (i.e. the objects of TT equipped with an action of GG). If the Artin-Mazur weak equivalences are stable under finite products in TT, then one can show that the Artin-Mazur weak equivalences of Rep(G)Rep(G) simply are the GG-equivariant maps XYX\to Y which are Artin-Mazur weak equivalences in TT when we forget the action of GG; to see this, one reduces easily to the case where TT is a category of presheaves on a local test category, and this follows then from Prop. 7.3.8 in (Cisinski02).

Example

Let us consider a full subcatogry CC of the category of locally ringed spaces with the following properties:

  • for any space XX in CC, there is a basis of contractible open subsets of XX which are CC;

  • for any locally ringed space XX, if each point of XX admits an open neighborhood in CC, then XX belong to XX;

  • the forgetful functor from CC to the category of topological spaces commutes with products (up to weak equivalence);

  • there exists a contractible space with two distinct closed points in CC.

We consider CC as site with the Grothendieck topology induced by open coverings. Then, for any XX in CC, the topos Sh(C) /XSh(C)_{/X} is aspherical because there is an adjunction between Sh(X)Sh(X) and Sh(C) /XSh(C)_{/X} in the 22-category of topoi. Therefore, since CC is a strict test category, we see that Sh(C)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 XX is a locally ringed space which has a covering by spaces in CC, then the sheaf on CC represented by XX corresponds in the test model structure on Sh(C)Sh(C) to the classical weak homotopy type of XX (this follows by hyperdescent in shape theory). In practice, all objects of CC admit CW-structures, so that we even get the classical homotopy type of XX characterized by the sheaf represented by XX on CC. 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’ CC are given by: euclidian spaces (seen as C C^\infty-manifolds), or contractible Stein complex manifolds. If GG is a real (or complex) Lie group, then representations of GG in the topos of sheaves on differentiable (or complex analytic) manifolds is a local test topos. One may also replace GG by any orbifold, presented as an internal groupoid in the category of manifolds.

References

  • 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)

  • 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 June 8, 2021 at 15:57:23. See the history of this page for a list of all contributions to it.