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coshape of 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

Contents

Idea

Just as the shape of an (∞,1)-topos is the functor GpdGpd\infty Gpd\to \infty Gpd which it corepresents (after identifying ∞-groupoids with their presheaf (∞,1)-topoi), so the coshape of an (∞,1)-topos is the functor Gpd opGpd\infty Gpd^{op}\to \infty Gpd which it represents.

Unlike the shape, which is only (co)representable (by an ∞-groupoid) when the topos is locally ∞-connected, the coshape is always representable, albeit possibly by a large ∞-groupoid—specifically the ∞-groupoid of points of the (∞,1)-topos in question.

From here on, this page uses the implicit ∞-category theory convention.

Definition

Definition

For H\mathbf{H} a topos, we say its co-shape ΓH\Gamma \mathbf{H} is the functor Γ(H):Gpd opGpd\Gamma(\mathbf{H}) \colon Gpd^{op}\to Gpd defined by

Γ(H)(A)=Topos(PSh(A),H) \Gamma(\mathbf{H})(A) = Topos(PSh(A), \mathbf{H})

Let Pt(H)=Topos(*,H)Pt(\mathbf{H}) = Topos(*,\mathbf{H}) denote the (possibly large) groupoid of points of H\mathbf{H}, where ** denotes the terminal topos GpdGpd.

Proposition

The coshape Γ(H)\Gamma(\mathbf{H}) is represented by Pt(H)Pt(\mathbf{H}), i.e. for any (small) groupoid AA we have

Γ(H)(A)GPD(A,Pt(H)). \Gamma(\mathbf{H})(A) \simeq GPD(A, Pt(\mathbf{H})).
Proof

Recall that colimits in ToposTopos are calculated via limits on the level of underlying categories. In particular, the copower of K\mathbf{K} by a groupoid AA is the topos K A\mathbf{K}^A.

Thus, in even more particular, the functor PshPsh preserves copowers, since each groupoid is the copower of the terminal object by itself. Therefore, we have Topos(Psh(A),H)GPD(A,Topos(*,H))=GPD(A,Pt(H))Topos( Psh(A), \mathbf{H} ) \simeq GPD(A, Topos( *, \mathbf{H} )) = GPD(A,Pt(\mathbf{H})), as desired.

In terms of Universe Enlargements

Another way of phrasing the above argument is as follows. For the same reason cited in the proof, the embedding PshPsh preserves all small colimits. Therefore, since Γ(H):Gpd opGpd\Gamma(\mathbf{H}) \colon Gpd^{op}\to Gpd is the composite of this embedding with the representable functor Topos(,H)Topos(-,\mathbf{H}), it must also preserves all small limits in Gpd opGpd^{op} (i.e. small colimits in GpdGpd).

Therefore, we can regard it as an object of the category Cts(Gpd op,Gpd)Cts(Gpd^{op},Gpd) of small-limit-preserving functors, also known as the very large (∞,1)-sheaf (∞,1)-topos on Gpd (and also the κ\kappa-ind-objects of GpdGpd, for κ\kappa the cardinality of the universe). However, by the general theory of universe enlargement (generalized to (,1)(\infty,1)-categories), this category is equivalent to GPDGPD, and the equivalence gives the representability theorem above.

Enlarging the category of toposes

Instead of being content with a “large-representability” result as above, we might wish that the coshape would actually give us a right adjoint to the embedding PshPsh. For this to be possible, we would need to enlarge GpdGpd to GPDGPD, but if we also enlarged ToposTopos to its naive enlargement TOPOSTOPOS, we would face the same problem “one universe higher.”

Thus, to get “better behavior” we can instead replace ToposTopos by its locally presentable enlargement Topos\Uparrow Topos, also called the very large (∞,1)-sheaf (∞,1)-topos on ToposTopos. We can then say:

Proposition

Coshape Yoneda-extends to a pair of adjoint functorss

GPDCodiscΓTopos. GPD \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\to}} \Uparrow Topos.
Proof

By HTT, lemma 6.3.5.21 we have a functor

ToposGrpd=GRPD \Uparrow Topos \to \Uparrow Grpd = GRPD

that preserves 𝒰\mathcal{U}-small colimits and finite limits and is given by sending

F:Topos opGrpd F : Topos^{op} \to Grpd

to the composite

GrpdPSh()(Topos/Grpd) et opTopos opFGrpd, Grpd \stackrel{PSh(-)}{\to} (Topos/Grpd)_{et}^{op} \stackrel{}{\to} Topos^{op} \stackrel{F}{\to} Grpd \,,

where the first step is forming presheaf toposes which sit by their terminal global section geometric morphisms over Grpd, and the second step is the evident projection.

Applied to a representable F=Topos(,H)F = Topos(-,\mathbf{H}) this composite is hence AΓ(H)(A)A \mapsto \Gamma(\mathbf{H})(A).

Revised on December 13, 2010 22:39:53 by Mike Shulman (71.137.3.108)