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

Idea

In the (∞,1)-topos Top to every object – every topological space – $X$ is associated the set ${\pi }_0(X)$ of connected components and the homotopy groups ${\pi }_n(X,x)$ for $x\in X$ and $n\in \mathbb{N}$, $n>0$.

By the general logic of space, we may think of the objects in an arbitrary ∞-stack (∞,1)-topos as generalized spaces of sorts. Accordingly, there is a notion of homotopy groups of an $\mathrm{\infty }$-stack .

But care has to be taken. It turns out that there are actually two different notions of homotopy groups in an arbitrary $(\mathrm{\infty }\mathrm{,1})$-topos, two notions that accidentally coincide for Top:

• there is a notion of categorical homotopy group:

  every $(\mathrm{\infty }\mathrm{,1})$-topos $H$ is powered over ∞Grpd usually modeled as SSet, hence for every object $X\in H$ there is the categorical $n$-sphere object $X^{S_c^n}$, where $S_c^n={\Delta }^n/\partial {\Delta }^n$.

• there should be a notion of geometric homotopy group, induced from the monodromy of locally constant ∞-stacks on objects $X\in H$.

For instance let $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{Diff})$ be the $(\mathrm{\infty }\mathrm{,1})$-topos of Lie ∞-groupoids. An ordinary smooth manifold $X$ is represented in $H$ by a sheaf of sets on Diff. This has no higher nontrivial categorical homotopy groups – ${\pi }_{n>0}^{\mathrm{cat}}(X)=0$ – reflecting the fact regarded as a smooth ∞-groupoid, $X$ is a categorically discrete groupoid.

But of course the manifold $X$ may have nontrivial homotopy groups in terms of its underlying topological space. For instance if $X=S^1$ is the circle, then the geometric first homotopy group is nontrivial, ${\pi }_1^{\mathrm{geom}}(X)=\mathbb{Z}$.

We discuss below both cases. The case of categorical homotopy groups is fully understood, for the case of geometric homotopy groups at the moment only a few aspects are in the literature, more is in the making. Some authors of this page (U.S.) thank Richard Williamson for pointing this out.

Categorical homotopy groups

Definition

Every ∞-stack (∞,1)-topos $H$ is naturally powered over ∞Grpd (HTT, remark 5.5.2.6):

for every ∞-groupoid $K$ and every object $A\in H$ there is an object $A^K\in H$ such that for every object $X\in H$ there is an equivalence

$$H(X,A^K)\simeq \mathrm{\infty }\mathrm{Grpd}(K,H(X,A)).$$\mathbf{H}(X,A^K) \simeq \infty Grpd(K, \mathbf{H}(X,A)) \,.

Let $S^n:={\mathrm{Ex}}^{\mathrm{\infty }}\partial \Delta [n+1]$ be the Kan fibrant replacement of the boundary of the (n+1)-simplex, i.e. the model in ∞Grpd of the pointed $n$-sphere.

For every $X\in H$, evaluation at the base point induces a morphism

$$s:X^{S^n}\to X.$$s : X^{S^n} \to X \,.

This morphism may be regarded as an object of the over category (∞,1)-topos $H/X$.

For $n\in \mathbb{N}$ define

$${\pi }_n(X):={\tau }_{\le 0}s\in H/X$$\pi_n(X) := \tau_{\leq 0} s \in \mathbf{H}/X

to be the discrete (i.e. 0-truncated) object obtained from $s$.

As in classical homotopy theory, for $n\ge 1$ the object ${\pi }_n(X)$ is equipped with the structure of a group object in $H/X$, which is an abelian group object for $n\ge 2$.

Remark The 0-truncated objects in $H/X$ have the interpretation of sheaves on $X$. So in the world of ∞-stacks a homotopy group object is a sheaf of groups.

References

section 6.5.1 of

• Jacob Lurie, Higher Topos Theory

Geometric homotopy groups

Idea

An ordinary topos $E$ is a locally connected topos of the global sections geometric morphism $(\mathrm{LConst}⊣\Gamma ):E\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\stackrel{\mathrm{Gamma}}{\to }}\mathrm{Set}$ is in fact an essential geometric morphism in that $\mathrm{LConst}$ has also a left adjoint $({\Pi }_0⊣\mathrm{LConst})$:

$$({\Pi }_0⊣\mathrm{LConst}⊣\Gamma ):E\stackrel{\stackrel{{\Pi }_0}{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\stackrel{\Gamma }{\to }}}\mathrm{Set}.$$(\Pi_0 \dashv LConst \dashv \Gamma) : E \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set \,.

This left adjoint ${\Pi }_0$ sends each object $X$ of $A$ to its set ${\Pi }_0$ of connected components. In other words this left adjoint produces the degree 0-part of the homotopy groups of objects of $E$.

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

Definition

The obvious generalization of the notion of ${\Pi }_0$ for a locally connected topos is to say that for $n\in \mathbb{N}$ an (n,1)-topos $H$ 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 $\mathrm{LConst}$ has a left adjoint ${\Pi }_n$

$$({\Pi }_n⊣\mathrm{LConst}⊣\Gamma ):H\stackrel{\stackrel{{\Pi }_0}{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\stackrel{\Gamma }{\to }}}n\mathrm{Grpd}.$$(\Pi_n \dashv LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} n Grpd \,.

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

$$(\Pi ⊣\mathrm{LConst}⊣\Gamma ):H\stackrel{\stackrel{{\Pi }_0}{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\stackrel{\Gamma }{\to }}}\mathrm{\infty }\mathrm{Grpd}$$(\Pi \dashv LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} \infty Grpd

to ∞ Grpd, with $\Pi$ the left adjoint (∞,1)-functor to the constant ∞-stack (∞,1)-functor $\mathrm{LConst}$. For $X\in H$ any object, the ∞-groupoid $\Pi (X)$ deserves to be called the homtopy $\mathrm{\infty }$-groupoid of X Its ordinary homotopy groups should be the homotopy groups of $X$.

This perspective on homotopy groups in an $(\mathrm{\infty }\mathrm{,1})$-topos is made explicit at

• homotopy ∞-groupoid.

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 $\Pi (-)$.

1. by constructing the left adjoint $\Pi (-)$ 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 $\Pi ()$ by Tannaka duality – this is described in the section In terms of monodromy and Galois theory;

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

In terms of local contractions

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

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

Proposition The (∞,1)-adjunction $(\Pi ⊣\mathrm{LConst}):{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)\overleftarrow{\to }\mathrm{\infty }\mathrm{Grpd}$ is presented by an SSet-enriched Quillen adjunction

$$(\Pi ⊣\mathrm{LConst}):\mathrm{sPSh}(C{)}_{\mathrm{proj}}^{\mathrm{loc}}\stackrel{\stackrel{\Pi }{\to }}{\stackrel{\mathrm{LConst}}{\leftarrow }}{\mathrm{sSet}}_{\mathrm{Quillen}},$$(\Pi \dashv LConst) : sPSh(C)_{proj}^{loc} \stackrel{\overset{\Pi}{\to}}{\overset{LConst}{\leftarrow}} sSet_{Quillen} \,,

where

• for $S\in \mathrm{sSet}$ the presheaf ${\mathrm{LConst}}_S$ sends all $U\mapsto S$, for all $U$;

• the functor $\Pi$ acts by $\mathrm{Pi}(X)={\int }^{U\in C}X={\lim }_{\to }X$.

The total left derived functor of $\Pi$ first takes an object $X$ 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:

$$𝕃\Pi :X\mapsto QX={\int }^{[n]\in \Delta }\Delta [n]\cdot \left(\coprod _{i_n}{U}_i\right)\mapsto \Pi (QX)={\int }^{[n]\in \Delta }\Delta [n]\cdot (\coprod _{i_n}*).$$\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 discuseed at homotopy ∞-groupoid.

References

Essentially the constrution of $𝕃\Pi$ 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 $E=\mathrm{Sh}(C)$ be a locally connected topos

$$({\Pi }_0⊣\mathrm{LConst}):\mathrm{Sh}(C)\overleftarrow{\to }\mathrm{Set}$$(\Pi_0 \dashv LConst) : Sh(C) \stackrel{\leftarrow}{\to} Set

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

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

The definition of the ∞-groupoid $\Pi (X)$ 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

$$\Pi (X)=\underrightarrow{\lim }{\Pi }_0(Y_{•}),$$\Pi(X) = \lim_\to \Pi_0( Y_\bullet) \,,

where

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

• the connected components functor ${\Pi }_0:\mathrm{Sh}(C)\to \mathrm{Set}$ is applied degreewise to the simplicial sheaf $Y=(Y_{•})$ to produce a simplicial set.

In Artin-Mazur it is discussed that this prescription does produce the right homotopy groups for $X$ 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 ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ is a locally contractible (∞,1)-topos, then this prescription can be seen to model implicitly the left Quillen functor $\Pi (-)$ that we described above:

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

$$Y=QX={\int }^{[n]\in \Delta }\Delta [n]\cdot \left(\coprod _{i_n}{U}_i\right)$$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 ${\Pi }_0(U_i)=*$. 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

$$\Pi (QX)={\Pi }_0((QX{)}_{•})={\int }^{[n]\in \Delta }\Delta [n]\cdot {\Pi }_0\left(\coprod _{i_n}{U}_{i_n}\right)={\int }^{[n]\in \Delta }\Delta [n]\cdot {\Pi }_0(\coprod _{i_n}*).$$\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.

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

• Daniel Dugger, Universal homotopy theories

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

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

* Carlos Simpson, Constantin Teleman, de Rham’s theorem for $\mathrm{\infty }$-stacks (pdf)

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

In terms of monodromy and Galois theory

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

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

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

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

$$\mathrm{\infty }\mathrm{CovBund}(X)\simeq \mathrm{Func}(\Pi (X),\mathrm{\infty }\mathrm{Grpd})$$\infty CovBund(X) \simeq Func(\Pi(X), \infty Grpd)

with locally constant ∞-stacks/$\mathrm{\infty }$-covering spaces on the one hand and (∞,1)-functors from $\Pi (X)$ 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 $\Pi (X)$ from just knowing $\mathrm{\infty }\mathrm{CovBund}(X)$ in terms of the automorphism ∞-group of a fiber functor

$$F_x:\mathrm{\infty }\mathrm{CovBund}(X)\to \mathrm{\infty }\mathrm{Grpd}$$F_x : \infty CovBund(X) \to \infty Grpd

from $\mathrm{\infty }$-coverin bundles/locally constant ∞-stacks over $X$ to ∞-groupoid.

– these automorphism are called the monodromy of $X$.

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

$$\mathrm{Aut}(F_x)={\Omega }_x^{\mathrm{geom}}X=:{\Omega }_x\Pi (X).$$Aut (F_x) = \Omega^{geom}_x X =: \Omega_x \Pi(X) \,.

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

Proof

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

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

$$\mathrm{CovBund}(X)\simeq \mathrm{\infty }\mathrm{Func}(\Pi (X),\mathrm{\infty }\mathrm{Grpd}).$$CovBund(X) \simeq \infty Func(\Pi(X), \infty Grpd) \,.

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

$$F_x:=\mathrm{\infty }\mathrm{Func}(i,\mathrm{\infty }\mathrm{Grpd}):\mathrm{Func}(\Pi (X),\mathrm{\infty }\mathrm{Grpd})\to \mathrm{\infty }\mathrm{Grpd}$$F_x := \infty Func(i, \infty Grpd) : Func(\Pi(X), \infty Grpd) \to \infty Grpd

which takes a local system $\rho :\Pi (X)\to \mathrm{\infty }\mathrm{Grpd}$ and evaluates it on $x$. By the (∞,1)-Yoneda lemma this means that $F_x$ is given by homming out of the local system $Y_{\Pi (X{)}^{\mathrm{op}}}x$ represented by $x$:

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

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

$$\mathrm{\infty }\mathrm{Func}(i,\mathrm{\infty }\mathrm{Grpd})\simeq Y_{({\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(\Pi (X{)}^{\mathrm{op}}){)}^{\mathrm{op}}}{Y}_{\Pi (X{)}^{\mathrm{op}}}x.$$\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

$$\begin{array}{rl}{\mathrm{Aut}}_{{\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(({\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(\Pi (X{)}^{\mathrm{op}}){)}^{\mathrm{op}})}\mathrm{\infty }\mathrm{Func}(i,\mathrm{\infty }\mathrm{Grod})& ={\mathrm{Aut}}_{{\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(({\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(\Pi (X{)}^{\mathrm{op}}){)}^{\mathrm{op}})}{Y}_{({\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(\Pi (X{)}^{\mathrm{op}}){)}^{\mathrm{op}}}{Y}_{\Pi (X{)}^{\mathrm{op}}}x\\ & \simeq {\mathrm{Aut}}_{({\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(\Pi (X{)}^{\mathrm{op}}){)}^{\mathrm{op}}}{Y}_{\Pi (X{)}^{\mathrm{op}}}x\\ & \simeq {\mathrm{Aut}}_{\Pi (X{)}^{\mathrm{op}}}x\\ & \simeq {\Omega }_x\Pi (X)\\ & =:{\Omega }_x^{\mathrm{geom}}X.\end{array}$$\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 $\Pi (X)$ in hands yet, but only know the ∞-groupoid $\mathrm{CovBund}(X)$ of covering $\mathrm{\infty }$-bundles / locally constant ∞-stacks in $X$: in terms of this we may reconstruct the automorphism ∞-groups of $\Pi (X)$ as

$$\mathrm{Aut}(\mathrm{CovBund}(X)\stackrel{F_x}{\to }\mathrm{\infty }\mathrm{Grpd})\simeq {\Omega }_x\Pi (X)=:{\Omega }_x^{\mathrm{geom}}X.$$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, $\mathrm{\infty }$-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 $X$ may be recovered from the behaviour of locally constant ∞-stacks on $X$ was

• Bertrand Toen, Toward a Galoisian interpretation of homotopy theory (arXiv:0007157)

Very similar constructions and statement then appeared in

• Pietro Polesello and Ingo Waschkies, Higher monodromy , Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150 (pdf)

• Mike Shulman, Parametrized spaces model locally constant homotopy sheaves (arXiv:0706.2874)

and, building on that, in example 1.8 of

• Denis-Charles Cisinski, Locally constant functors , Math. Proc. Camb. Phil. Soc. , 147 (pdf)

Notably the article by Pietro Polesello and Ingo Waschkies makes fully explicit the observation that locally constant $n$-stacks are precisely the sections of the constant $(n+1)$-stack on the $(n+1)$-groupoid $n\mathrm{Grpd}$. This is a key observation for bringing the full power of the adjunction $(\Pi ⊣\mathrm{LConst})$ 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 $(\mathrm{\infty }\mathrm{,1})$-toposes $H$ comes canonically equipped with a notion of homotopy ∞-groupoid $\Pi (X)$ of every object $X\in H$.

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 ${\Pi }_0$ of a sheaf on a locally connected topological space

Here we discuss the 0-th geometric homotopy group ${\Pi }_0:\mathrm{Sh}(X)\to \mathrm{Set}$ of objects in a sheaf topos in terms of a left adjoint ${\Pi }_0$ 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 $X$ be a sufficiently nice topological space.

Observation

There is a triple of adjoint functors

$$({\Pi }_0⊣\mathrm{LConst}⊣\Gamma ):\mathrm{Sh}(X)\stackrel{\stackrel{{\Pi }_0}{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\stackrel{\Gamma }{\to }}}\mathrm{Set}$$(\Pi_0 \dashv LConst \dashv \Gamma) \;\;\; : \;\;\; Sh(X) \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\overset{\Gamma}{\to}}} Set

where

• $(\mathrm{LConst}⊣\Gamma )$ is the usual global section geometric morphism with ${\mathrm{LConst}}_S$ the constant sheaf of locally constant functions with values in $S\in \mathrm{Set}$ and

• $\Pi :\mathrm{Sh}(X)\to \mathrm{Set}$ is left adjoint to $\mathrm{LConst}$ and sends each sheaf $A$ to the set of connected components of the corresponding etale space $p_A:\mathrm{Et}(A)\to X$:

  $${\Pi }_0(A)={\pi }_0\mathrm{Et}(A).$$\Pi_0(A) = \pi_0 Et(A) \,.

Proof

The etale space of ${\mathrm{LConst}}_S$ is $E({\mathrm{LConst}}_S)=X\times S$. By the relation of sheaves on $X$ with etale spaces over $X$ we have

$${\mathrm{Hom}}_{\mathrm{Sh}(X)}(A,{\mathrm{LConst}}_S)\simeq {\mathrm{Hom}}_{\mathrm{Et}/X}(E(A),X\times S)$$Hom_{Sh(X)}(A, LConst_S) \simeq Hom_{Et/X}(E(A), X \times S)

For $\gamma :I\to E(A)$ any continuous path in $E(A)$, and for $f:E(A)\to X\times S$ a morphism in $\mathrm{Et}/X$, the image of $\gamma$ in $X\times I$ is fixed by, say, the image $f(\gamma (0))=(p_A({\gamma }_0),s)$ to be $f(\gamma ):t\mapsto (p_A(\gamma (t)),s)$. This means that the value of $f$ on any path component of $E(A)$ is uniquely fixed by its value on any point in that path component.

Choosing a basepoint in each path component therefore induces bijection

$$\simeq {\mathrm{Hom}}_{\mathrm{Set}}({\pi }_0(\mathrm{Et}(A)),S)={\mathrm{Hom}}_{\mathrm{Set}}({\Pi }_0(A),S).$$\simeq Hom_{Set}(\pi_0(Et(A)), S) = Hom_{Set}(\Pi_0(A),S) \,.

Geometric ${\Pi }_0$ of a general object in a locally connected topos

More generally, if $E$ is a locally connected topos then the global sections geometric morphism $(\mathrm{LConst}⊣\Gamma ):E\overleftarrow{\to }\mathrm{Set}$ has also a left adjoint ${\Pi }_0$ to $\mathrm{LConst}$:

$$({\Pi }_0⊣\mathrm{LConst}⊣\Gamma ):E\stackrel{\stackrel{{\Pi }_0}{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\stackrel{\Gamma }{\to }}}\mathrm{Set}.$$(\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 ${\pi }_1$ of objects in a 1-topos

The general idea is that of

• Grothendieck's Galois theory.

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

• fundamental group of a topos.

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

• Peter Johnstone, Topos theory .

Geometric ${\Pi }_2$ of a topological space

This case is discussed in

• Pietro Polesello and Ingo Waschkies, Higher monodromy , Homology, Homotopy and Applications, Vol. 7(2005), No. 1, pp. 109-150 (pdf)

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

$$(\mathrm{LConst}⊣\Gamma ):{\mathrm{Sh}}_{(\mathrm{2,1})}(X)\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}\mathrm{Grpd}$$(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

$$({\Pi }_1⊣\mathrm{LConst}):\mathrm{Op}(X)\hookrightarrow {\mathrm{Sh}}_{(\mathrm{2,1})}(X)\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{{\Pi }_1}{\to }}\mathrm{Grpd}$$(\Pi_1\dashv LConst) : Op(X) \hookrightarrow Sh_{(2,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Pi_1}{\to}} Grpd

with the right adjoint to $\mathrm{LConst}$ 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 $\Gamma (X,C_X)$ we think of this here equivalently as ${\mathrm{Sh}}_{(\mathrm{2,1})}(X)(X,\mathrm{LConst}(C))$, so that the theorem then gives the adjunction equivalence $\cdots \simeq \mathrm{Grpd}({\Pi }_1(X),C)$).

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

$$({\Pi }_2⊣\mathrm{LConst}):\mathrm{Op}(X)\hookrightarrow {\mathrm{Sh}}_{(\mathrm{3,1})}(X)\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{{\Pi }_3}{\to }}\mathrm{Grpd}$$(\Pi_2\dashv LConst) : Op(X) \hookrightarrow Sh_{(3,1)}(X) \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Pi_3}{\to}} Grpd

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

Geometric ${\Pi }_{\mathrm{\infty }}$ of a topological space

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

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

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

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

for the $\mathrm{\infty }$-groupoid of locally constant $\mathrm{\infty }$-stacks on $X$.

Write $\Pi (X):=\mathrm{Sing}X$ for the fundamental ∞-groupoid of $X$.

Claim

There is an equivalence of $\mathrm{\infty }$-groupoids

$$\mathrm{LConst}(X)\simeq \mathrm{\infty }\mathrm{Grpd}(\Pi (X),\mathrm{\infty }\mathrm{Grpd}).$$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

• Bertrand Toen, Toward a Galoisian interpretation of homotopy theory (arXiv:0007157)

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

$$\mathrm{LConst}(X)\simeq \mathrm{Fib}(\Pi (X))$$LConst(X) \simeq Fib(\Pi(X))

of locally constant ∞-stacks on $X$ and Kan fibrations over the fundamental ∞-groupoid $\Pi (X)=\mathrm{Sing}(X)$.

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

$$\mathrm{LConst}(X)\simeq \mathrm{Func}(\Pi (X),\mathrm{\infty }\mathrm{Grpd}).$$LConst(X) \simeq Func(\Pi(X), \infty Grpd) \,.

Very similar statements are discussed in

• Mike Shulman, Parametrized spaces model locally constant homotopy sheaves (arXiv:0706.2874)

and, building on that, in example 1.8 of

• Denis-Charles Cisinski, Locally constant functors , Math. Proc. Camb. Phil. Soc. , 147 (pdf)

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

• Jacob Lurie, Higher Topos Theory

as theorem 7.1.0.1.

There it is shown that for any $K\in \mathrm{\infty }\mathrm{Grpd}$ there is a bijection of homotopy sets of morphisms

$${\pi }_0\mathrm{Top}(X,\mid K\mid )\simeq {\pi }_0(p_{*}{p}^{*}K),$$\pi_0 Top(X, |K|) \simeq \pi_0(p_* p^* K) \,,

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

If we also rewrite the left using the equivalence of $\mathrm{Top}$ with $\mathrm{sSet}$, this reads

$${\pi }_0\mathrm{\infty }\mathrm{Grpd}(\Pi (X),K)\simeq {\pi }_0(\Gamma {\mathrm{LConst}}_K)={\pi }_0{\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(X,{\mathrm{LConst}}_K),$$\pi_0 \infty Grpd(\Pi(X), K) \simeq \pi_0(\Gamma LConst_K) = \pi_0 Sh_{(\infty,1)}(X,LConst_K) \,,

For $K=\mathrm{Core}(\mathrm{\infty }\mathrm{Grpd})$ this is the ${\pi }_0$-decategorification of the above statement.

Geometric ${\Pi }_{\mathrm{\infty }}$ of the terminal object in a locally connected $(\mathrm{\infty }\mathrm{,1})$-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 $\Pi (X)\in \mathrm{\infty }\mathrm{Grpd}$ for $X$ the terminal object in ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ on an ordinary site $C$ with $\Pi (X)$ as described above in Geometric fundamental oo-groupoid.

This reviews in particular (slightly implicitly)

Theorem

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

$${\pi }_n\Pi (X)\simeq {\pi }_n(X).$$\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 $\Pi (X)$ to make sense.

Examples

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

• $S^1$, the $\mathrm{\infty }$-stack represented by the standard circle in $\mathrm{Diff}$;

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

Then

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

  $${\pi }_n^{\mathrm{cat}}(S^1)=*$$\pi_n^{cat}(S^1) = {*}

• the geometric homotopy groups of $S^1$ are the usual ones obtained from regarding $S^1$ as an object in Top:

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

  etc.

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

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

  $${\pi }_n^{\mathrm{cat}}(B\mathbb{Z})=\{\begin{array}{cc}\mathbb{Z}& \mid \mathrm{if}n=1\\ *& \mid \mathrm{otherwise}\end{array}.$$\pi_n^{cat}(\mathbf{B}\mathbb{Z}) = \left\{ \array{ \mathbb{Z} & | if\; n=1 \\ * & | otherwise } \right. \,.