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

Idea

The notion of ind-object and ind-category in an (infinity,1)-category is the straightforward generalization of the notion of ind-object in an ordinary category. See there for idea and motivation.

Definition

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

Let in the following $C$ be a small (infinity,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-(infinity,1)-category of (infinity,1)-presheaves on those $(\mathrm{\infty }\mathrm{,1})$-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 (infinity,1)-presheaf is in ${\mathrm{ind}}_{\kappa }\text{-}C$ if there exists a $\kappa$-filtered (infinity,1)-category $D$ and an $(\mathrm{\infty }\mathrm{,1})$-functor $W:D\to C$ such that $F$ is the colimit over $Y\circ W$, where $Y$ is the Yoneda (infinity,1)-embedding?.

References

Section 5.3 and in particular 5.3.3 of

• Jacob Lurie, Higher Topos Theory