equivalences in/of $(\infty,1)$-categories
For $C$ a small (∞,1)-category and $\kappa$ a regular cardinal, the $(\infty,1)$-category of $\kappa-$pro-objects in $C$ is the opposite (∞,1)-category of ind-objects in the opposite of $C$:
For $\kappa = \omega$ we write just $Pro(C)$.
By the properties listed there, if $C$ has all $\kappa$-small (∞,1)-limits then this is equivalent to
the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that preserve these limits.
Generalizing this definition:
If $\mathcal{C}$ is a possibly non-small but accessible $(\infty,1)$-category with finite $(\infty,1)$-limits, we write :
for the opposite $\infty$-category of $\infty$-functors $\mathcal{C} \to$ ∞Grpd which are
left exact in that they preserve finite $(\infty,1)$-limits;
Yet more generally, if $C$ is just locally small, then one can take $Pro(\mathcal{C})$ to be the $\infty$-category of small functors whose Grothendieck construction is cofiltered?. Equivalently, $Pro(\mathca;{C})$ consists of the functors which are “small cofiltered limits of representables”.
For $\mathcal{C}$ a possibly large but accessible $(\infty,1)$-category which is tensored over ∞Grpd, in that there is a natural equivalence
then it is still a full sub-$(\infty,1)$-category of its pro-objects, in the sense of Def. , via the usual $(\infty,1)$-Yoneda embedding:
This is because the above tensoring means that $\mathcal{C}(c,-)$ is a right$\,$adjoint $(\infty,1)$-functor and these preserve limits and are accessible (by this Prop.)
The large version is mentioned in:
See also:
Last revised on October 4, 2021 at 04:59:34. See the history of this page for a list of all contributions to it.