structures in a cohesive (∞,1)-topos
Just as the shape of an (∞,1)-topos is the functor which it corepresents (after identifying ∞-groupoids with their presheaf (∞,1)-topoi), so the coshape of an (∞,1)-topos is the functor 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 a topos, we say its co-shape is the functor defined by
Let denote the (possibly large) groupoid of points of , where denotes the terminal topos .
The coshape is represented by , i.e. for any (small) groupoid we have
Recall that colimits in are calculated via limits on the level of underlying categories. In particular, the copower of by a groupoid is the topos .
Thus, in even more particular, the functor preserves copowers, since each groupoid is the copower of the terminal object by itself. Therefore, we have , as desired.
Another way of phrasing the above argument is as follows. For the same reason cited in the proof, the embedding preserves all small colimits. Therefore, since is the composite of this embedding with the representable functor , it must also preserves all small limits in (i.e. small colimits in ).
Therefore, we can regard it as an object of the category of small-limit-preserving functors, also known as the very large (∞,1)-sheaf (∞,1)-topos on Gpd (and also the -ind-objects of , for the cardinality of the universe). However, by the general theory of universe enlargement (generalized to -categories), this category is equivalent to , 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 . For this to be possible, we would need to enlarge to , but if we also enlarged to its naive enlargement , we would face the same problem “one universe higher.”
Thus, to get “better behavior” we can instead replace by its locally presentable enlargement , also called the very large (∞,1)-sheaf (∞,1)-topos on . We can then say:
that preserves -small colimits and finite limits and is given by sending
to the composite
Applied to a representable this composite is hence .
coshape of an -topos