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
The notion of ind-object and ind-category in an (∞,1)-category is the straightforward generalization of the notion of ind-object in an ordinary category. See there for idea and motivation.
We describe -ind-objects for a regular cardinal.
The different equivalent definitions of ordinary ind-objects have their analog for (∞,1)-categories.
Let in the following be a small (∞,1)-category.
In terms of formal colimits
The definition in terms of formal colimits is precisely analogous to the one for ordinary ind-objects, with colimits and limits replaced by the corresponding -notion (compare homotopy limit and limit in quasi-categories)
So the objects of are small filtered diagrams in , and the morphisms are given by
(… should be made more precise…)
In terms of filtered fibrations
Write for a regular cardinal and write for the full sub-(∞,1)-category of (∞,1)-presheaves on those -presheaves
which classify right fibrations such that is -filtered.
In the case write .
In terms of filtered colimits
Equivalently, an (∞,1)-presheaf is in if there exists a -filtered (∞,1)-category and an -functor such that is the colimit over , where is the (∞,1)-Yoneda embedding.
Let a small -category and a regular cardinal.
is closed in under -filtered (∞,1)-colimits.
This is HTT, prop. 22.214.171.124.
For any the following are equivalent:
is a -filtered colimit in of a diagram in ;
belongs to ;
preserves -small limits.
This is HTT, corollary 126.96.36.199.
Every object of is a -compact object of .
This is HTT, prop. 188.8.131.52.
This makes an -category of ind-objects a compactly generated (∞,1)-category.
Section 5.3 and in particular 5.3.3 of