(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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.
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
Let $Pt(\mathbf{H}) = Topos(*,\mathbf{H})$ denote the (possibly large) groupoid of points of $\mathbf{H}$, where $*$ denotes the terminal topos $Gpd$.
The coshape $\Gamma(\mathbf{H})$ is represented by $Pt(\mathbf{H})$, i.e. for any (small) groupoid $A$ we have
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.
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.
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:
Coshape Yoneda-extends to a pair of adjoint functorss
By HTT, lemma 6.3.5.21 we have a functor
that preserves $\mathcal{U}$-small colimits and finite limits and is given by sending
to the composite
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 $A \mapsto \Gamma(\mathbf{H})(A)$.
coshape of an $(\infty,1)$-topos