nLab
pro-object in an (infinity,1)-category

Context

(,1)(\infty,1)-Category theory

Limits and colimits

Contents

Definition

For small (∞,1)-categories

For CC a small (∞,1)-category and κ\kappa a regular cardinal, the (,1)(\infty,1)-category of κ\kappa-pro-objects in CC is the opposite (∞,1)-category of ind-objects in the opposite of CC:

Pro κ(C):=(Ind κ(C op)) op. Pro_\kappa(C) := (Ind_\kappa(C^{op}))^{op} \,.

For κ=ω\kappa = \omega we write just Pro(C)Pro(C).

By the properties listed there, if CC has all κ\kappa-small (∞,1)-limits then this is equivalent to

Lex κ(C,Grpd) opFunc(C,Grpd) op \cdots \simeq Lex_\kappa(C, \infty Grpd)^{op} \subset Func(C,\infty Grpd)^{op}

the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that preserve these limits.

For large (∞,1)-categories

Generalizing this definition, if CC is a non-small but accessible (,1)(\infty,1)-category with finite limits, we write

Pro(C):=AccLex(C,Grpd) op. Pro(C) := AccLex(C,\infty Grpd)^{op} \,.

for the category of left exact functors CGprdC\to \infty Gprd which are moreover accessible. More generally, if CC is just locally small, then one can take Pro(C)Pro(C) to be the infinity-category of small functors whose Grothendieck construction is cofiltered?. Equivalently, Pro(C)Pro(C) consists of the functors which are “small cofiltered limits of representables”.

References

The large version is mentioned around def. 7.1.6.1 of

Last revised on March 19, 2018 at 16:08:25. See the history of this page for a list of all contributions to it.