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



(,1)(\infty,1)-Category theory

Limits and colimits



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 κ\kappa-ind-objects for κ\kappa a regular cardinal.


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

Let in the following CC 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 IndCInd C are small filtered diagrams X:D XCX : D_X \to C in CC, and the morphisms are given by

Hom IndC(X,Y):=lim dD Xcolim dD YHom C(X(d),Y(d)). 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 κ\kappa for a regular cardinal and write ind κ-Cind_\kappa \text{-}C for the full sub-(∞,1)-category of (∞,1)-presheaves on those (,1)(\infty,1)-presheaves

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

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

In the case κ=ω\kappa = \omega write ind κ-C=ind-Cind_\kappa\text{-}C = ind\text{-}C.

In terms of filtered colimits

Equivalently, an (∞,1)-presheaf is in ind κ-Cind_\kappa\text{-}C if there exists a κ\kappa-filtered (∞,1)-category DD and an (,1)(\infty,1)-functor W:DCW: D \to C such that FF is the colimit over YWY \circ W, where YY is the (∞,1)-Yoneda embedding.


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


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

This is HTT, prop.


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

  1. FF is a κ\kappa-filtered colimit in PSh(C)PSh(C) of a diagram in CC;

  2. FF belongs to Ind κ(C)Ind_\kappa(C).

If, furthermore, CC admits κ\kappa-small colimits, then the above are equivalent to

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

This is HTT, corollary


Every object of CC is a κ\kappa-compact object of Ind κ(C)Ind_\kappa(C).

This is HTT, prop.

This makes an \infty-category of ind-objects a compactly generated (∞,1)-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.