nLab ind-object in an (infinity,1)-category

Idea

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.

Definition

The different equivalent definitions of ordinary ind-objects have their analog for (∞,1)-categories.

Let in the following $C$ 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 $\infty$-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

$$Hom_{Ind C}(X,Y) := lim_{d\in D_X} colim_{d' \in D_Y} Hom_C(X(d), Y(d')) \,.$$

(… should be made more precise…)

In terms of filtered fibrations

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

$$F : C^{op} \to Top$$

which classify right fibrations $\tilde C \to C$ such that $\tilde C$ is $\kappa$-filtered.

In the case $\kappa = \omega$ write $ind_\kappa\text{-}C = ind\text{-}C$.

In terms of filtered colimits

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.

Properties

Let $C$ a small $(\infty,1)$-category and $\kappa$ a regular cardinal.

Proposition

$Ind_\kappa(C)$ is closed in $PSh(C)$ under $\kappa$-filtered (∞,1)-colimits.

This is HTT, prop. 5.3.5.3.

Proposition

For any $F \in PSh(C)$ the following are equivalent:

1. $F$ is a $\kappa$-filtered colimit in $PSh(C)$ of a diagram in $C$;

2. $F$ belongs to $Ind_\kappa(C)$.

If, furthermore, $C$ admits $\kappa$-small colimits, then the above are equivalent to

1. $F : C^{op} \to \infty Grpd$ preserves $\kappa$-small limits.

This is HTT, corollary 5.3.5.4.

Proposition

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.

Related concepts

• ind-object / ind-object in an $(\infty,1)$-category

• pro-object / pro-object in an (∞,1)-category

• compactly generated (∞,1)-category

• accessible (∞,1)-category

References

Section 5.3 and in particular 5.3.3 of

• Jacob Lurie, Higher Topos Theory