nLab
pro-object in an (infinity,1)-category
Context
$(\infty,1)$ -Category theory
(∞,1)-category theory

Background
Basic concepts
Universal constructions
Local presentation
Theorems
Models
Limits and colimits
limits and colimits

1-Categorical
limit and colimit

limits and colimits by example

commutativity of limits and colimits

small limit

filtered colimit

sifted colimit

connected limit , wide pullback

preserved limit , reflected limit , created limit

product , fiber product , base change , coproduct , pullback , pushout , cobase change , equalizer , coequalizer , join , meet , terminal object , initial object , direct product , direct sum

finite limit

Kan extension

weighted limit

end and coend

2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Definition
For small (∞,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 $C$ :

$Pro_\kappa(C) := (Ind_\kappa(C^{op}))^{op}
\,.$

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

By the properties listed there, if $C$ has all $\kappa$ -small (∞,1)-limit s then this is equivalent to

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

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 $C$ is a non-small but accessible $(\infty,1)$ -category with finite limits, we write

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

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