equivalences in/of $(\infty,1)$-categories
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 $\kappa$-ind-objects for $\kappa$ a regular cardinal.
The different equivalent definitions of ordinary ind-objects have their analog for (∞,1)-categories.
Let in the following $C$ 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 $\infty$-notion (compare homotopy limit and limit in quasi-categories)
So the objects of $Ind C$ are small filtered diagrams $X : D_X \to C$ in $C$, and the morphisms are given by
(… should be made more precise…)
Write $\kappa$ for a regular cardinal and write $ind_\kappa \text{-}C$ for the full sub-(∞,1)-category of (∞,1)-presheaves on those $(\infty,1)$-presheaves
which classify right fibrations $\tilde C \to C$ such that $\tilde C$ is $\kappa$-filtered.
In the case $\kappa = \omega$ write $ind_\kappa\text{-}C = ind\text{-}C$.
Equivalently, an (∞,1)-presheaf is in $ind_\kappa\text{-}C$ if there exists a $\kappa$-filtered (∞,1)-category $D$ and an $(\infty,1)$-functor $W: D \to C$ such that $F$ is the colimit over $Y \circ W$, where $Y$ is the (∞,1)-Yoneda embedding.
Let $C$ a small $(\infty,1)$-category and $\kappa$ a regular cardinal.
$Ind_\kappa(C)$ is closed in $PSh(C)$ under $\kappa$-filtered (∞,1)-colimits.
This is HTT, prop. 5.3.5.3.
For any $F \in PSh(C)$ the following are equivalent:
$F$ is a $\kappa$-filtered colimit in $PSh(C)$ of a diagram in $C$;
$F$ belongs to $Ind_\kappa(C)$.
If, furthermore, $C$ admits $\kappa$-small colimits, then the above are equivalent to
This is HTT, corollary 5.3.5.4.
Every object of $C$ is a $\kappa$-compact object of $Ind_\kappa(C)$.
This is HTT, prop. 5.3.5.5.
This makes an $\infty$-category of ind-objects a compactly generated (∞,1)-category.
ind-object / ind-object in an $(\infty,1)$-category
Section 5.3 and in particular 5.3.3 of
Last revised on September 1, 2019 at 16:55:31. See the history of this page for a list of all contributions to it.