pro-object in an (infinity,1)-category
Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
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
end and coend
For small (∞,1)-categories
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.
For large (∞,1)-categories
Generalizing this definition, if is a non-small -category with finite limits, we write
for the category of left exact functors which are moreover accessible. In other words, when is large, consists only of those left-exact functors which are ”small cofiltered limits of representables”.
The large version is mentioned around def. 220.127.116.11 of
Revised on December 16, 2012 15:46:32
by Urs Schreiber