Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
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.
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…)
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 .
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. 5.3.5.3.
For any the following are equivalent:
is a -filtered colimit in of a diagram in ;
belongs to .
If, furthermore, admits -small colimits, then the above are equivalent to
This is HTT, corollary 5.3.5.4.
Every object of is a -compact object of .
This is HTT, prop. 5.3.5.5.
This makes an -category of ind-objects a compactly generated (∞,1)-category.
ind-object / ind-object in an -category
Section 5.3 and in particular 5.3.3 of
Last revised on September 1, 2019 at 20:55:31. See the history of this page for a list of all contributions to it.