nLab coshape of an (infinity,1)-topos

Idea

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

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

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

Proposition

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

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

Proof

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

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

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

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

Proposition

Coshape Yoneda-extends to a pair of adjoint functorss

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

Proof

By HTT, lemma 6.3.5.21 we have a functor

$$\Uparrow Topos \to \Uparrow Grpd = GRPD$$

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

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

to the composite

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

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 $F = Topos(-,\mathbf{H})$ this composite is hence $A \mapsto \Gamma(\mathbf{H})(A)$.

Related concepts

• shape of an (∞,1)-topos

• coshape of an $(\infty,1)$-topos