nLab
geometric homotopy groups in an (infinity,1)-topos

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Homotopy theory

This is a sub-entry of homotopy groups in an (∞,1)-topos. It discusses the general notions of étale homotopy in the context of locally ∞-connected (∞,1)-toposes.

For the other notion of homotopy groups in an (,1)(\infty,1)-topos see categorical homotopy groups in an (∞,1)-topos.

Contents

Idea

An ordinary topos EE is a locally connected topos if the global sections geometric morphism (LConstΓ):EΓLConstSet(LConst \dashv \Gamma) : E \stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}} Set is in fact an essential geometric morphism in that LConstLConst has also a left adjoint (Π 0LConst)(\Pi_0 \dashv LConst):

(Π 0LConstΓ):EΓLConstΠ 0Set. (\Pi_0 \dashv LConst \dashv \Gamma) : E \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set \,.

This left adjoint Π 0\Pi_0 sends each object XX of AA to its set Π 0\Pi_0 of connected components. In other words this left adjoint produces the degree 0-part of the homotopy groups of objects of EE.

This has an obvious generalization of (∞,1)-toposes.

Definition

The obvious generalization of the notion of Π 0\Pi_0 for a locally connected topos is to say that for nn \in \mathbb{N} an (n,1)-topos H\mathbf{H} is a locally n-connected (n,1)-topos if again the terminal geometric morphism is an essential geometric morphism in that the constant n-stack functor LConstLConst has a left adjoint Π n\Pi_n

(Π nLConstΓ):HΓLConstΠ nnGrpd. (\Pi_n \dashv LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi_n}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} n Grpd \,.

Here we may take n=n = \infty and say that an (∞,1)-topos is locally contractible if we have an essential geometric morphism

(ΠLConstΓ):HΓLConstΠGrpd (\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} \infty Grpd

to ∞ Grpd, with Π\Pi the left adjoint (∞,1)-functor to the constant ∞-stack (∞,1)-functor LConstLConst. For XHX \in \mathbf{H} any object, the ∞-groupoid Π(X)\Pi(X) deserves to be called the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos of XX Its ordinary homotopy groups are the homotopy groups of XX.

While an obvious slight generalization or refinement of what is considered in previous literature, it seems that the simple picture of a left adjoint (∞,1)-functor to the constant ∞-stack functor has not been made explicit in the existing literature (though possibly in the thesis by Richard Williamson).

However, up to some straightforward translations of concepts and notation, it turns out that essentially all aspects of this picture are present and well known – if somewhat implicitly – in existing literature. A detailed commented account of what is in the literature is in the following subsection and in particular in the section Examples below.

There are essentially three different methods concretely constructing the abstractly defined homotopy ∞-groupoid?-functor Π()\Pi(-).

  1. by constructing the left adjoint Π()\Pi(-) as the functor that takes an object to its local contraction – this is described in the section In terms of local contractions;

  2. by using monodromy/Galois theory of locally constant ∞-stacks to reproduce Π()\Pi() by Tannaka duality – this is described in the section In terms of monodromy and Galois theory;

  3. by constructing Π()\Pi(-) explicitly as a path \infty-groupoid in terms of paths modeled on an interval object in H\mathbf{H} – this is described in the section In terms of concrete paths .

In terms of local contractions

If the locally contractible (∞,1)-topos H\mathbf{H} has a site CC with HSh (,1)(C)\mathbf{H} \simeq Sh_{(\infty,1)}(C) such that the objects of the site are geometrically contractible in that constant (∞,1)-presheaves already satisfy descent over objects in CC, then the left adjoint Π:HGrpd\Pi : \mathbf{H} \to \infty Grpd to LConstLConst may be constructed explicitly as follows.

Following the discussion at models for ∞-stack (∞,1)-toposes there is a model structure on simplicial presheaves sPSh(C) proj locsPSh(C)_{proj}^{loc} wich presents H\mathbf{H}.

Proposition The (∞,1)-adjunction (ΠLConst):Sh (,1)(C)Grpd(\Pi \dashv LConst) : Sh_{(\infty,1)}(C) \stackrel{\leftarrow}{\to} \infty Grpd is presented by an SSet-enriched Quillen adjunction

(ΠLConst):sPSh(C) proj locLConstΠsSet Quillen, (\Pi \dashv LConst) : sPSh(C)_{proj}^{loc} \stackrel{\overset{\Pi}{\to}}{\overset{LConst}{\leftarrow}} sSet_{Quillen} \,,

where

  • for SsSetS \in sSet the presheaf LConst SLConst_S sends all USU \mapsto S, for all UU;

  • the functor Π\Pi acts by Pi(X)= UCX=lim XPi(X) = \int^{U \in C} X = \lim_\to X.

The total left derived functor of Π\Pi first takes an object XX to a simplicial presheaf that is degreewise a coproduct of representables and then contracts all these representables to the terminal object, regarding the resulting constant simplicial presheaf as a simplicial set:

𝕃Π:XQX= [n]ΔΔ[n]( i nU i)Π(QX)= [n]ΔΔ[n]( i n*). \mathbb{L} \Pi : X \mapsto Q X = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} U_i \right) \mapsto \Pi(Q X) = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} * \right) \,.

Proof

This is discussed at fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos and cohesive (∞,1)-topos.

References

Essentially the construction of 𝕃Π\mathbb{L} \Pi as above is an old construction in terms of – somewhat implicitly – the structure of a category of fibrant objects on simplicial objects in a topos:

the discussion on page 18 of

  • Ieke Moerdijk, Classifying Spaces and Classifying Topoi Lecture Notes in Mathematics 1616, Springer (1995) .

which goes back to

  • Artin, Mazur, Etale Homotopy Springer Lecture Notes in Mathematics 100, Berlin (1969)

goes as follows:

Let E=Sh(C)E = Sh(C) be a locally connected topos

(Π 0LConst):Sh(C)Set (\Pi_0 \dashv LConst) : Sh(C) \stackrel{\leftarrow}{\to} Set

that here we think of as a petit over-topos over a given object XX in some ambient gros topos. Accordingly we write X=*X = * for the terminal object in Sh(C)Sh(C).

Assume that EE has enough point. Then stalkwise Kan-fibrant simplicial objects in EE, i.e. stalk-wise Kan-fibrant simplicial sheaves on CC form a category of fibrant objects. In particular a fibrant simplicial object Y[Δ op,Sh(C)]Y \in [\Delta^{op}, Sh(C)] equipped with an acyclic fibration YXY \to X to the terminal object X=*X = * is a hypercover of XX.

The definition of the ∞-groupoid Π(X)\Pi(X) as defined in the above references (notice that only its homotopy groups are written down explicitly there, but it’s immediate to equivalently write it as we do now) is

Π(X)=lim Π 0(Y ), \Pi(X) = \lim_\to \Pi_0( Y_\bullet) \,,

where

  • the colimit is taken over the category of acyclic fibrations/hypercovers YXY \to X;

  • the connected components functor Π 0:Sh(C)Set\Pi_0 : Sh(C) \to Set is applied degreewise to the simplicial sheaf Y=(Y )Y = (Y_\bullet) to produce a simplicial set.

In Artin-Mazur it is discussed that this prescription does produce the right homotopy groups for XX a topological space if one assumes that this space is locally contractible space.

If we therefore interpret this as saying that for the above prescription to yield the correct result we generally ought to assume that Sh (,1)(C)Sh_{(\infty,1)}(C) is a locally contractible (∞,1)-topos, then this prescription can be seen to model implicitly the left Quillen functor Π()\Pi(-) that we described above:

In terms of the full model category structure on sPSh(C) proj locsPSh(C)_{proj}^{loc} among all these hypercovers is one that is the cofibrant object

Y=QX= [n]ΔΔ[n]( i nU i) Y = Q X = \int^{[n] \in \Delta} \Delta[n] \cdot \left( \coprod_{i_n} U_i \right)

mentioned above, consisting degreewise of coproducts of representables with Π 0(U i)=*\Pi_0(U_i) = *. For instance if XX admits a good open cover, we can take YY to be the Cech nerve of that good cover. (For more on this see ∞-Lie groupoid.) Due to the lifting property of cofibrant objects, any colimit over all hypercovers can be computed by evaluating just at that hypercover.

There the Artin-Mazur-Moerdijk-prescription yields

Π(QX)=Π 0((QX) )= [n]ΔΔ[n]Π 0( i nU i n)= [n]ΔΔ[n]Π 0( i n*). \Pi(Q X) = \Pi_0((Q X)_\bullet) = \int^{[n] \in \Delta} \Delta[n] \cdot \Pi_0\left( \coprod_{i_n} U_{i_n} \right) = \int^{[n] \in \Delta} \Delta[n] \cdot \Pi_0\left( \coprod_{i_n} * \right) \,.

This is indeed the action of the left Quillen functor from above.

It is the nerve theorem that asserts that for YY the Cech nerve of a good open cover, this simplicial set is homotopy equivalent to the original paracompact space.

A closely related, implicitly slightly more general statement is in on p. 25 of

which describes this construction for the case H=Sh (,1)(Diff)\mathbf{H} = Sh_{(\infty,1)}(Diff) (the gros topos of \infty-stacks on Diff).

With even more general sites allowed, but working only at the level of homotopy categories the left adjoint Π\Pi and its construction is described in Proposition 2.18 of

See also the discussion at locally contractible (∞,1)-topos.

In terms of monodromy and Galois theory

Given an (∞,1)-topos H=Sh (,1)(C)\mathbf{H} = Sh_{(\infty,1)}(C) we define the ∞-groupoid of locally constant ∞-stacks on an object XHX \in \mathbf{H} to be

CovBund(X):=H(X,LConst Core(Grpd)), \infty CovBund(X) := \mathbf{H}(X, LConst_{Core(\infty Grpd)}) \,,

where LConst Core(Grpd)LConst_{Core(\infty Grpd)} is the constant ∞-stack on the core ∞-groupoid of ∞ Grpd.

If H\mathbf{H} is a locally contractible (∞,1)-topos in that LConstLConst has the left adjoint (∞,1)-functor Π()\Pi(-), then by definition of adjunction we have the equivalence

CovBund(X)Func(Π(X),Grpd) \infty CovBund(X) \simeq Func(\Pi(X), \infty Grpd)

with locally constant ∞-stacks/\infty-covering spaces on the one hand and (∞,1)-functors from Π(X)\Pi(X) to ∞Grpd on the other.

Concrete realizations of this equivalence are discussed in the Examples-section below. Here we describe how one may reconstruct in terms Tannaka duality Π(X)\Pi(X) from just knowing CovBund(X)\infty CovBund(X) in terms of the automorphism ∞-group of a fiber functor

F x:CovBund(X)Grpd F_x : \infty CovBund(X) \to \infty Grpd

from \infty-coverin bundles/locally constant ∞-stacks over XX to ∞-groupoid.

– these automorphism are called the monodromy of XX.

We want to show that these automorphism ∞-groups are the loop space objects of Π(X)\Pi(X), hence the geometric homotopy \infty-groups.

Aut(F x)=Ω x geomX=:Ω xΠ(X). Aut (F_x) = \Omega^{geom}_x X =: \Omega_x \Pi(X) \,.

This is the reconstruction of the geometric homotopy ∞-groups of an ∞-stack XX from its monodromy or Galois theory.

Proof

The underlying mechanism is just (,1)(\infty,1)-Tannaka duality, i.e. essentially the (∞,1)-Yoneda lemma applied a few times in a row:

suppose we knew Π(X)\Pi(X), so that by adjunction we have

CovBund(X)Func(Π(X),Grpd). CovBund(X) \simeq \infty Func(\Pi(X), \infty Grpd) \,.

Then for each point xΠ(X)x \in \Pi(X) given by a morphism i:*Π(X)i : {*} \to \Pi(X) we get a fiber functor

F x:=Func(i,Grpd):Func(Π(X),Grpd)Grpd F_x := \infty Func(i, \infty Grpd) : Func(\Pi(X), \infty Grpd) \to \infty Grpd

which takes a local system ρ:Π(X)Grpd\rho : \Pi(X) \to \infty Grpd and evaluates it on xx. By the (∞,1)-Yoneda lemma this means that F xF_x is given by homming out of the local system Y Π(X) opxY_{\Pi(X)^{op}} x represented by xx:

Func(i,Grpd)Hom PSh (,1)(Π(X) op)(Y Π(X) op)x,). \infty Func(i, \infty Grpd) \simeq Hom_{PSh_{(\infty,1)}(\Pi(X)^{op})}(Y_{\Pi(X)^{op}}) x, -) \,.

But this in turn means that Func(i,Grpd):Func(Π(X),Grod)Grpd\infty Func(i,\infty Grpd) : \infty Func(\Pi(X),\infty Grod) \to \infty Grpd is itself a representable functor, in the (∞,1)-category of (∞,1)-presheaves PSh (,1)(PSh (,1)(Π(X) op) op)PSh_{(\infty,1)}(PSh_{(\infty,1)}(\Pi(X)^{op})^{op}):

Func(i,Grpd)Y (PSh (,1)(Π(X) op)) opY Π(X) opx. \infty Func(i, \infty Grpd) \simeq Y_{(PSh_{(\infty,1)}(\Pi(X)^{op}))^{op}} Y_{\Pi(X)^{op}} x \,.

This way we find, by applying the (∞,1)-Yoneda lemma two more times, that the automorphism ∞-group of the fiber functor is

Aut PSh (,1)((PSh (,1)(Π(X) op)) op)Func(i,Grod) =Aut PSh (,1)((PSh (,1)(Π(X) op)) op)Y (PSh (,1)(Π(X) op)) opY Π(X) opx Aut (PSh (,1)(Π(X) op)) opY Π(X) opx Aut Π(X) opx Ω xΠ(X) =:Ω x geomX. \begin{aligned} Aut_{PSh_{(\infty,1)}((PSh_{(\infty,1)}(\Pi(X)^{op}))^{op})} \infty Func(i, \infty Grod) & = Aut_{PSh_{(\infty,1)}((PSh_{(\infty,1)}(\Pi(X)^{op}))^{op})} Y_{(PSh_{(\infty,1)}(\Pi(X)^{op}))^{op}} Y_{\Pi(X)^{op}} x \\ & \simeq Aut_{(PSh_{(\infty,1)}(\Pi(X)^{op}))^{op}} Y_{\Pi(X)^{op}} x \\ & \simeq Aut_{\Pi(X)^{op}} x \\ & \simeq \Omega_x \Pi(X) \\ & =: \Omega_x^{geom} X \,. \end{aligned}

Now, the same is of course true even if we don’t have Π(X)\Pi(X) in hands yet, but only know the ∞-groupoid CovBund(X)CovBund(X) of covering \infty-bundles / locally constant ∞-stacks in XX: in terms of this we may reconstruct the automorphism ∞-groups of Π(X)\Pi(X) as

Aut(CovBund(X)F xGrpd)Ω xΠ(X)=:Ω x geomX. Aut( CovBund(X) \stackrel{F_x}{\to} \infty Grpd ) \simeq \Omega_x \Pi(X) =: \Omega^{geom}_x X \,.

References

The idea that geometric homotopy groups of generalized spaces given by sheaves, stacks, ∞-stacks is detected and definable by the behaviour of locally constant sheaves, stacks, \infty-stacks on these objects goes back to Grothendieck's Galois theory and the notion of fundamental group of a topos. The state of the art treatment of the Galois theory of coverings in a topos is in

  • Marta Bunge, Galois groupoids and covering morphisms in topos theory, Galois theory, Hopf algebras, and semiabelian categories, 131–161, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004, links.

In Pursuing Stacks Grothendieck talked about how this 1-categorical situation generalizes to ∞-stacks.

After Pursuing Stacks, apparently the first to publish a detailed formalization and proof of how the homotopy groups of a topological space XX may be recovered from the behaviour of locally constant ∞-stacks on XX was

This has a followup construction in

Very similar constructions and statement then appeared in

and, building on that, in example 1.8 of

Notably the article by Pietro Polesello and Ingo Waschkies makes fully explicit the observation that locally constant nn-stacks are precisely the sections of the constant (n+1)(n+1)-stack on the (n+1)(n+1)-groupoid nGrpdn Grpd. This is a key observation for bringing the full power of the adjunction (ΠLConst)(\Pi \dashv LConst) into the picture, as we do here.

It was pointed out to Urs Schreiber by Richard Williamson that these constructions should generalize from topological spaces to objects in any (∞,1)-topos, maybe along the lines outlined above, and that this way suitable (,1)(\infty,1)-toposes H\mathbf{H} comes canonically equipped with a notion of homotopy ∞-groupoid? Π(X)\Pi(X) of every object XHX \in \mathbf{H}.

In terms of concrete paths

References

The following references discuss fundamental groupoids of an entire topos constructed from concrete interval objects. In the context of the above discussion these toposes are to be thought of as petit over-toposes over a given object in an ambient gros topos, and as such are concerned with the fundamental groupoid of that object, in our sense.

The construction of the fundamental groupoid of a topos from interval objects is in

  • Ieke Moerdijk, Gavin Wraith, Connected locally connected toposes are path-connected , Transactions of the AMS, volume 295, number 2, (1986)

The comparison of this construction with the one by monodromy/Galois theory is in

  • Marta Bunge, Ieke Moerdijk, On the construction of the Grothendieck fundamental group of a topos by paths , J. Pure and Applied Algebra, 116 (1997)

Examples

Geometric Π 0\Pi_0 of a sheaf on a locally connected topological space

Here we discuss the 0-th geometric homotopy group Π 0:Sh(X)Set\Pi_0 : Sh(X) \to Set of objects in a sheaf topos in terms of a left adjoint Π 0\Pi_0 of the constant sheaf functor. This is a special case of the more general situation discussed in Pi0 of a general object in a locally connected topos below.

Let XX be a sufficiently nice topological space.

Observation

There is a triple of adjoint functors

(Π 0LConstΓ):Sh(X)ΓLConstΠ 0Set (\Pi_0 \dashv LConst \dashv \Gamma) \;\;\; : \;\;\; Sh(X) \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set

where

Proof

The etale space of LConst SLConst_S is E(LConst S)=X×SE(LConst_S) = X \times S. By the relation of sheaves on XX with etale spaces over XX we have

Hom Sh(X)(A,LConst S)Hom Et/X(E(A),X×S) Hom_{Sh(X)}(A, LConst_S) \simeq Hom_{Et/X}(E(A), X \times S)

For γ:IE(A)\gamma : I \to E(A) any continuous path in E(A)E(A), and for f:E(A)X×Sf : E(A) \to X \times S a morphism in Et/XEt/X, the image of γ\gamma in X×IX \times I is fixed by, say, the image f(γ(0))=(p A(γ 0),s)f(\gamma(0)) = (p_A(\gamma_0),s) to be f(γ):t(p A(γ(t)),s)f(\gamma) : t \mapsto (p_A(\gamma(t)),s). This means that the value of ff on any path component of E(A)E(A) is uniquely fixed by its value on any point in that path component.

Choosing a basepoint in each path component therefore induces bijection

Hom Set(π 0(Et(A)),S)=Hom Set(Π 0(A),S). \simeq Hom_{Set}(\pi_0(Et(A)), S) = Hom_{Set}(\Pi_0(A),S) \,.

Geometric Π 0\Pi_0 of a general object in a locally connected topos

More generally, if EE is a locally connected topos then the global sections geometric morphism (LConstΓ):ESet(LConst \dashv \Gamma) : E \stackrel{\leftarrow}{\to} Set has also a left adjoint Π 0\Pi_0 to LConstLConst:

(Π 0LConstΓ):EΓLConstΠ 0Set. (\Pi_0 \dashv LConst \dashv \Gamma) : E \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set \,.

For instance page 17 of

  • Ieke Moerdijk, Classifying Spaces and Classifying Topoi Lecture Notes in Mathematics 1616, Springer (1995)

Geometric π 1\pi_1 of objects in a 1-topos

The general idea is that of

A discussion of of how this produces first homotopy groups of a 1-topos is at

The general construction of the first geometric homotopy group of objects in a Grothendieck topos is for instance in section 8.4 of

Geometric Π 2\Pi_2 of a topological space

This case is discussed in

We indicate briefly how the results stated in this article fit into the general abstract picture as indicated above:

The authors consider locally constant 1-stacks and 2-stacks on sites of open subsets of sufficiently nice topological spaces.

Prop. 1.1.9 gives the adjunction

(LConstΓ):Sh (2,1)(X)ΓLConstGrpd (LConst \dashv \Gamma) : Sh_{(2,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} Grpd

between forming constant stacks and taking global sections.

Then prop 1.2.5, 1.2.6, culminating in theorem 1.2.9, p. 121 gives (somewhat implicitly) the other adjunction

(Π 1LConst):Op(X)Sh (2,1)(X)LConstΠ 1Grpd (\Pi_1\dashv LConst) : Op(X) \hookrightarrow Sh_{(2,1)}(X) \stackrel{\overset{\Pi_1}{\to}}{\underset{LConst}{\leftarrow}} Grpd

with the right adjoint to LConstLConst being the fundamental groupoid functor on representables. (Where we change a bit the perspective on the results as presented there, to amplify the pattern indicated above. For instance where the authors write Γ(X,C X)\Gamma(X,C_X) we think of this here equivalently as Sh (2,1)(X)(X,LConst(C))Sh_{(2,1)}(X)(X,LConst(C)), so that the theorem then gives the adjunction equivalence Grpd(Π 1(X),C)\cdots \simeq Grpd(\Pi_1(X),C)).

Then in essentially verbatim analogy, these results are lifted from stacks to 2-stacks in section 2, where now prop 2.2.2, 2.2.3, culminating in theorem 2.2.5, p. 132 gives (somewhat implicitly) the adjunction

(Π 2LConst):Op(X)Sh (3,1)(X)LConstΠ 2Grpd (\Pi_2\dashv LConst) : Op(X) \hookrightarrow Sh_{(3,1)}(X) \stackrel{\overset{\Pi_2}{\to}}{\underset{LConst}{\leftarrow}} Grpd

now with the path 2-groupoid operation (locally) left adjoint to forming constant 2-stacks.

Geometric Π \Pi_\infty of a topological space

Let XX be a sufficiently nice (I think this should be locally (relatively) contractible. -DR) (paracompact) topological space. The canonical map X*X \to {*} induces the geometric morphism

Sh (,1)(X)ΓLConstGrpd Sh_{(\infty,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

where the right adjoint Γ\Gamma is taking global sections and the left adjoint is forming the constant ∞-stack on an \infty-groupoid KK. If K=Core(Grpd)K = Core (\infty Grpd) then LConst KLConst_K is the constant ∞-stack of locally constant ∞-stacks and we write

LConst(X):=Sh (,1)(X,LConst Grpd)=ΓLConst Grpd LConst(X) := Sh_{(\infty,1)}(X, LConst_{\infty Grpd})= \Gamma LConst_{\infty Grpd}

for the \infty-groupoid of locally constant \infty-stacks on XX.

Write Π(X):=SingX \Pi(X) := Sing X for the fundamental ∞-groupoid of XX.

Claim

There is an equivalence of \infty-groupoids

LConst(X)Grpd(Π(X),Grpd). LConst(X) \simeq \infty Grpd(\Pi(X), \infty Grpd) \,.

Urs Schreiber: I think this is proven in the literature, if maybe slightly implicitly so. I’ll now go through the available references to discuss this.

After old ideas by Alexander Grothendieck from Pursuing Stacks, it seems that the first explicit formalization and proof of this statement is given in

In theorem 2.13, p. 25 the author proves an equivalence of (∞,1)-categories (modeled there as Segal categories)

LConst(X)Fib(Π(X)) LConst(X) \simeq Fib(\Pi(X))

of locally constant ∞-stacks on XX and Kan fibrations over the fundamental ∞-groupoid Π(X)=Sing(X)\Pi(X) = Sing(X).

But Kan fibrations over a Kan complex such as Π(X)\Pi(X) are equivalently left fibrations (as discussed there) and by by the (∞,1)-Grothendieck construction these are equivalent to (∞,1)-functors Π(X)Grpd\Pi(X) \to \infty Grpd. So under the (∞,1)-Grothendieck construction Toën’s result does actually produce an equivalence

LConst(X)Func(Π(X),Grpd). LConst(X) \simeq Func(\Pi(X), \infty Grpd) \,.

In

this is discussed in the context of Segal-toposes.

Very similar statements are discussed in

and, building on that, in example 1.8 of

A variant of this statement – more general in one respect, less general in another – appears in

as theorem 7.1.0.1.

There it is shown that for any KGrpdK \in \infty Grpd there is a bijection of homotopy sets of morphisms

π 0Top(X,K)π 0(p *p *K), \pi_0 Top(X, |K|) \simeq \pi_0(p_* p^* K) \,,

where (p *p *):Sh (,1)(X)Grpd(p^* \dashv p_*) : Sh_{(\infty,1)}(X) \to \infty Grpd is the geometric morphism we denoted (LConstΓ)(LConst \dashv \Gamma) above.

If we also rewrite the left using the equivalence of TopTop with sSetsSet, this reads

π 0Grpd(Π(X),K)π 0(ΓLConst K)=π 0Sh (,1)(X,LConst K), \pi_0 \infty Grpd(\Pi(X), K) \simeq \pi_0(\Gamma LConst_K) = \pi_0 Sh_{(\infty,1)}(X,LConst_K) \,,

For K=Core(Grpd)K = Core(\infty Grpd) this is the π 0\pi_0-decategorification of the above statement.

Geometric Π \Pi_\infty of the terminal object in a locally \infty-connected (,1)(\infty,1)-topos

The geometric Π \Pi_\infty of the terminal object in a locally ∞-connected (∞,1)-topos can be called the fundamental ∞-groupoid of the topos. It represents the shape of the topos.

On page 18-19 of

  • Ieke Moerdijk, Classifying Spaces and Classifying Topoi Lecture Notes in Mathematics 1616, Springer (1995)

is described the construction of Π(X)Grpd\Pi(X) \in \infty Grpd for XX the terminal object in Sh (,1)(C)Sh_{(\infty,1)}(C) on an ordinary site CC with Π(X)\Pi(X) as described above in Geometric fundamental oo-groupoid.

This reviews in particular (slightly implicitly)

Theorem

Let XX be a topological space that has a basis of contractible open subsets. Write XX also for XX regarded as the terminal object in Sh (,1)(X)Sh_{(\infty,1)}(X). Then the image of XX under Π:Sh (,1)(X)Grpd\Pi : Sh_{(\infty,1)}(X) \to \infty Grpd has the same homtopy groups as XX regarded as an object in Top:

π nΠ(X)π n(X). \pi_n \Pi(X) \simeq \pi_n(X) \,.
Proof

This is a slight reformulation of the statement in

M. Artin, B. Mazur, Etale homotopy , Springer lecture notes in mathematics 100, Berlin 1969

Notice the local contractibility assumption. This is necessary in general for Π(X)\Pi(X) to make sense.

Examples

Let C=C = Diff and consider in Sh (,1)(Diff)Sh_{(\infty,1)}(Diff) the two objects

  • S 1S^1, the \infty-stack represented by the standard circle in DiffDiff;

  • B\mathbf{B}\mathbb{Z} – the \infty-stack constant on the delooping groupoid of the additive group \mathbb{Z}.

Then

  • the categorical homotopy groups of S 1S^1 are all trivial

    π n cat(S 1)=* \pi_n^{cat}(S^1) = {*}
  • the geometric homotopy groups of S 1S^1 are the usual ones obtained from regarding S 1S^1 as an object in Top:

    π 0 geom(S 1)=* \pi^{geom}_0(S^1) = *
    π 1 geom(S 1)= \pi^{geom}_1(S^1) = \mathbb{Z}

    etc.

For B\mathbf{B}\mathbb{Z} it is the other way round:

  • the categorical homotopy groups of B\mathbf{B}\mathbb{Z} are

    π n cat(B)={ ifn=1 * otherwise. \pi_n^{cat}(\mathbf{B}\mathbb{Z}) = \left\{ \array{ \mathbb{Z} & | if\; n=1 \\ * & | otherwise } \right. \,.

Revised on April 4, 2014 23:15:36 by Tim Porter (2.26.36.99)