Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
For a small (∞,1)-category and a regular cardinal, the -category of pro-objects in is the opposite (∞,1)-category of ind-objects in the opposite of :
For we write just .
By the properties listed there, if has all -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 is a possibly non-small but accessible -category with finite -limits, we write :
for the opposite -category of -functors ∞Grpd which are
left exact in that they preserve finite -limits;
Yet more generally, if is just locally small, then one can take to be the -category of small functors whose Grothendieck construction is cofiltered?. Equivalently, consists of the functors which are “small cofiltered limits of representables”.
For a possibly large but accessible -category which is tensored over ∞Grpd, in that there is a natural equivalence
then it is still a full sub--category of its pro-objects, in the sense of Def. , via the usual -Yoneda embedding:
This is because the above tensoring means that is a rightadjoint -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 08:59:34. See the history of this page for a list of all contributions to it.