nLab
coshape of an (infinity,1)-topos

Idea

Just as the shape of an (∞,1)-topos is the functor $\mathrm{\infty }\mathrm{Gpd}\to \mathrm{\infty }\mathrm{Gpd}$ which it corepresents (after identifying ∞-groupoids with their presheaf (∞,1)-topoi), so the coshape of an (∞,1)-topos is the functor $\mathrm{\infty }{\mathrm{Gpd}}^{\mathrm{op}}\to \mathrm{\infty }\mathrm{Gpd}$ 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 $H$ a topos, we say its co-shape $\Gamma H$ is the functor $\Gamma (H):{\mathrm{Gpd}}^{\mathrm{op}}\to \mathrm{Gpd}$ defined by

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

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

Proposition

The coshape $\Gamma (H)$ is represented by $\mathrm{Pt}(H)$, i.e. for any (small) groupoid $A$ we have

$$\Gamma (H)(A)\simeq \mathrm{GPD}(A,\mathrm{Pt}(H)).$$\Gamma(\mathbf{H})(A) \simeq GPD(A, Pt(\mathbf{H})).

Proof

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

Thus, in even more particular, the functor $\mathrm{Psh}$ preserves copowers, since each groupoid is the copower of the terminal object by itself. Therefore, we have $\mathrm{Topos}(\mathrm{Psh}(A),H)\simeq \mathrm{GPD}(A,\mathrm{Topos}(*,H))=\mathrm{GPD}(A,\mathrm{Pt}(H))$, 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 $\mathrm{Psh}$ preserves all small colimits. Therefore, since $\Gamma (H):{\mathrm{Gpd}}^{\mathrm{op}}\to \mathrm{Gpd}$ is the composite of this embedding with the representable functor $\mathrm{Topos}(-,H)$, it must also preserves all small limits in ${\mathrm{Gpd}}^{\mathrm{op}}$ (i.e. small colimits in $\mathrm{Gpd}$).

Therefore, we can regard it as an object of the category $\mathrm{Cts}({\mathrm{Gpd}}^{\mathrm{op}},\mathrm{Gpd})$ of small-limit-preserving functors, also known as the very large (∞,1)-sheaf (∞,1)-topos on Gpd (and also the $\kappa$-ind-objects of $\mathrm{Gpd}$, for $\kappa$ the cardinality of the universe). However, by the general theory of universe enlargement (generalized to $(\mathrm{\infty },1)$-categories), this category is equivalent to $\mathrm{GPD}$, 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 $\mathrm{Psh}$. For this to be possible, we would need to enlarge $\mathrm{Gpd}$ to $\mathrm{GPD}$, but if we also enlarged $\mathrm{Topos}$ to its naive enlargement $\mathrm{TOPOS}$, we would face the same problem “one universe higher.”

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

Proposition

Coshape Yoneda-extends to a pair of adjoint functorss

$$\mathrm{GPD}\stackrel{\stackrel{\Gamma }{\leftarrow }}{\underset{\mathrm{Codisc}}{\to }}\Uparrow \mathrm{Topos}.$$GPD \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\to}} \Uparrow Topos.

Proof

By HTT, lemma 6.3.5.21 we have a functor

$$\Uparrow \mathrm{Topos}\to \Uparrow \mathrm{Grpd}=\mathrm{GRPD}$$\Uparrow Topos \to \Uparrow Grpd = GRPD

that preserves $𝒰$-small colimits and finite limits and is given by sending

$$F:{\mathrm{Topos}}^{\mathrm{op}}\to \mathrm{Grpd}$$F : Topos^{op} \to Grpd

to the composite

$$\mathrm{Grpd}\stackrel{\mathrm{PSh}(-)}{\to }(\mathrm{Topos}/\mathrm{Grpd}{)}_{\mathrm{et}}^{\mathrm{op}}\stackrel{}{\to }{\mathrm{Topos}}^{\mathrm{op}}\stackrel{F}{\to }\mathrm{Grpd},$$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 $F=\mathrm{Topos}(-,H)$ this composite is hence $A\mapsto \Gamma (H)(A)$.

Related concepts

• shape of an (∞,1)-topos

• coshape of an $(\mathrm{\infty },1)$-topos