fundamental infinity-groupoid of a locally infinity-connected (infinity,1)-topos


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

(∞,1)-topos theory





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structures in a cohesive (∞,1)-topos

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

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Basic facts




Every (∞,1)-topos EE has a shape Shape(E)ProGrpdShape(E) \in Pro\infty Grpd. When EE is locally ∞-connected then this is a genuine ∞-groupoid Π(E)\Pi(E) \in ∞Grpd. We may think of this as the fundamental ∞-groupoid of the (,1)(\infty,1)-topos regarded as a generalized space.

But also every locally ∞-connected (∞,1)-topos has an internal notion of fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos for objects of EE, denoted Π E:EGrpd\Pi_E : E \to \infty Grpd. (If in addition Δ:GrpdE\Delta \colon \infty Grpd \to E is fully faithful then we call Π\Pi the shape modality.) Applied to its terminal object this does agree with the fundamental ∞-groupoid of the topos:

Π(E)Π E(*). \Pi(E) \simeq \Pi_E(*) \,.

Conversely, for an object XEX\in E, the fundamental ∞-groupoid Π E(X)\Pi_E(X) internal to EE can be identified with the fundamental ∞-groupoid of the locally ∞-connected (∞,1)-topos E/XE/X.



For (Π EΓ ELConst E):EGrpd(\Pi_E \dashv \Gamma_E \dashv LConst_E) : E \to \infty Grpd a locally ∞-connected (∞,1)-topos we say its fundamental \infty-groupoid is

Π(E):=Π E(*), \Pi(E) := \Pi_E(*) \,,

where ** is the terminal object of EE.

In other words, it is the internal fundamental ∞-groupoid of the terminal object of EE.


Let H\mathbf{H} be a locally \infty-connected (,1)(\infty,1)-topos and XHX \in \mathbf{H} an object. Then also the over-(∞,1)-topos H/X\mathbf{H}/X is locally \infty-connected (as discussed there).

We have then two different definitions of the fundamental \infty-groupoid of XX: once as Π H(X)\Pi_{\mathbf{H}}(X) – the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos – and once as Π(H/X)\Pi(\mathbf{H}/X).


These agree:

Π H(X)Π(H/X). \Pi_{\mathbf{H}}(X) \simeq \Pi(\mathbf{H}/X) \,.

Since XIdXX \stackrel{Id}{\to} X is the terminal object in H/X\mathbf{H}/X we have by definition

Π(H/X)=Π H/X(Id X). \Pi(\mathbf{H}/X) = \Pi_{\mathbf{H}/X}(Id_X) \,.

Now observe that Π H/X=Π HX !\Pi_{\mathbf{H}/X} = \Pi_{\mathbf{H}} \circ X_! since the terminal global section geometric morphism of the over-topos is

H/XX *X *X !HΓ HLConst HΠ HGrpd \mathbf{H}/X \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}} \mathbf{H} \stackrel{\overset{\Pi_{\mathbf{H}}}{\to}}{\stackrel{\overset{LConst_{\mathbf{H}}}{\leftarrow}}{\underset{\Gamma_{\mathbf{H}}}{\to}}} \infty Grpd

and that X !X_! in the etale geometric morphism is the projection map that sends YXY \stackrel{}{\to} X to YY.


Let LC(,1)ToposLC(\infty,1)Topos denote the full sub-(∞,1)-category of (∞,1)Topos determined by the locally ∞-connected objects.


The (,1)(\infty,1)-category Gpd\infty Gpd (as the category of local homeomorphisms over Gpd\infty Gpd) is reflective in LC(,1)ToposLC(\infty,1)Topos,

GrpdΠLC(,1)Topos \infty Grpd \stackrel{\overset{\Pi}{\leftarrow}}{\hookrightarrow} LC(\infty,1)Topos

with the reflector given by forming the fundamental \infty-groupoid.


Any ∞-groupoid GG gives rise to an (∞,1)-presheaf (∞,1)-topos PSh(G)=[G op,Gpd]PSh(G) = [G^{op},\infty Gpd], which by the (∞,1)-Grothendieck construction is equivalent to the over-(∞,1)-topos Gpd/G\infty Gpd / G. The (,1)(\infty,1)-toposes of this form are, by definition, those for which the unique (∞,1)-geometric morphism to Gpd\infty Gpd is a local homeomorphism of toposes. This construction embeds Gpd\infty Gpd as a full sub-(∞,1)-category of (∞,1)Topos:

PSh():GrpdLC(,1)Topos. PSh(-) : \infty Grpd \hookrightarrow LC(\infty,1)Topos \,.

since in particular the (,1)(\infty,1)-toposes PSh(G)PSh(G) are locally ∞-connected.

To show that Π\Pi is a left adjoint (∞,1)-functor to PSh()PSh(-) we demonstrate a natural hom-equivalence

LC(,1)Topos(E,(,1)PSh(A))Grpd(Π E(*),A) LC(\infty,1)Topos(E,(\infty,1)PSh(A)) \simeq \infty Grpd(\Pi_E(*), A)

for ELC(,1)ToposE\in LC(\infty,1)Topos and AGrpdA \in \infty Grpd.

At shape of an (∞,1)-topos it is shown that we have a natural equivalence

(,1)Topos(E,PSh(A))Γ ELConst EG=:Shape(E)(A). (\infty,1)Topos(E, PSh(A)) \simeq \Gamma_E LConst_E G =: Shape(E)(A) \,.

Now observe that furthermore we have a sequence of natural equivalences

Shape(E)(A) =Γ(LConst(A)) Grpd(*,Γ(LConst(A))) E(LConst(*),LConst(A)) E(*,LConst(A)) Grpd(Π E(*),A).. \begin{aligned} Shape(E)(A) &= \Gamma(LConst(A))\\ &\simeq \infty Grpd(*, \Gamma(LConst(A)))\\ &\simeq E(LConst(*), LConst(A)) \\ &\simeq E(*, LConst(A)) \\ &\simeq \infty Grpd(\Pi_E(*),A). \end{aligned} \,.

So equivalently, one may say that a locally ∞-connected (∞,1)-topos EE has a shape which is representable, and its fundamental ∞-groupoid Π(H)\Pi(\mathbf{H}) is the representing object.



For XX a locally contractible topological space, we have an equivalence

Π((,1)Sh(X))SingX \Pi ((\infty,1)Sh(X)) \simeq Sing X

between the ordinary fundamental ∞-groupoid of XX defined by the singular simplicial complex and the topos-theoretic fundamental \infty-groupoid of the (∞,1)-sheaf (∞,1)-topos (,1)Sh(X)(\infty,1)Sh(X) over XX.

More generally the shape of an (∞,1)-topos of (,1)Sh(X)(\infty,1)Sh(X) reproduces the shape theory of XX.

Revised on September 2, 2014 14:30:58 by Urs Schreiber (