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 $\mathrm{\infty }$-notion (compare homotopy limit and limit in quasi-categories)

So the objects of $\mathrm{Ind}C$ are small filtered diagrams $X:D_X\to C$ in $C$, and the morphisms are given by

$${\mathrm{Hom}}_{\mathrm{Ind}C}(X,Y):={\lim }_{d\in D_X}{\mathrm{colim}}_{d\prime \in D_Y}{\mathrm{Hom}}_C(X(d),Y(d\prime )).$$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 ${\mathrm{ind}}_{\kappa }\text{-}C$ for the full sub-(∞,1)-category of (∞,1)-presheaves on those $(\mathrm{\infty },1)$-presheaves

$$F:C^{\mathrm{op}}\to \mathrm{Top}$$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 ${\mathrm{ind}}_{\kappa }\text{-}C=\mathrm{ind}\text{-}C$.

In terms of filtered colimits

Equivalently, an (∞,1)-presheaf is in ${\mathrm{ind}}_{\kappa }\text{-}C$ if there exists a $\kappa$-filtered (∞,1)-category $D$ and an $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-category and $\kappa$ a regular cardinal.

Proposition

${\mathrm{Ind}}_{\kappa }(C)$ is closed in $\mathrm{PSh}(C)$ under $\kappa$-filtered (∞,1)-colimits.

This is HTT, prop. 5.3.5.3.

Proposition

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

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

2. $F$ belongs to ${\mathrm{Ind}}_{\kappa }(C)$;

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

This is HTT, corollary 5.3.5.4.

Proposition

Every object of $C$ is a $\kappa$-compact object of ${\mathrm{Ind}}_{\kappa }(C)$.

This is HTT, prop. 5.3.5.5.

Related concepts

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

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

• accessible (∞,1)-category

References

Section 5.3 and in particular 5.3.3 of

• Jacob Lurie, Higher Topos Theory