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 (,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. In other words, when CC is large, Pro(C)Pro(C) consists only of those left-exact functors which are “small cofiltered limits of representables”.

References

The large version is mentioned around def. 7.1.6.1 of

Revised on December 16, 2012 15:46:32 by Urs Schreiber (71.195.68.239)