nLab
fully faithful (infinity,1)-functor

Contents

Contents

Idea

The generalization to the context of (∞,1)-category-theory of the notion of a full and faithful functor in ordinary category theory.

Definition

An (∞,1)-functor F:CDF : C \to D is full and faithful if for all objects x,yCx,y \in C it induces an equivalence on the hom-∞-groupoids

F x,y:Hom C(x,y)Hom D(F(x),F(y)). F_{x,y} : Hom_C(x,y) \stackrel{\simeq}{\to} Hom_D(F(x), F(y)) \,.

A full and faithful (,1)(\infty,1)-functor F:CDF : C \to D exhibits CC as a full sub-(∞,1)-category of DD and one tends to write

F:CD F : C \hookrightarrow D

to indicate this. This is definition 1.2.10 of Lurie

Properties

Lemma

For an (,1)(\infty,1)-functor F:CDF : C \to D, the following are equivalent:

  • FF is fully faithful

  • Ho(F):Ho(C)Ho(D)Ho(F) : Ho(C) \to Ho(D) is a fully faithful functor of (1-)categories and the following square is a pullback in the (,1)(\infty,1) category of \infty-categories:

C D Ho(C) Ho(D) \array{ C &\to& D \\ \downarrow && \downarrow \\ Ho(C) &\to& Ho(D) }
  • The following square is a pullback in the (,1)(\infty,1) category of \infty-categories:
Core(C [1]) Core(D [1]) Core(C)×Core(C) Core(D)×Core(D) \array{ Core(C^{[1]}) &\to& Core(D^{[1]}) \\ \downarrow && \downarrow \\ Core(C) \times Core(C) &\to& Core(D) \times Core(D) }
  • The following square is a pullback in the (,1)(\infty,1) category of \infty-categories:
C [1] D [1] C×C D×D \array{ C^{[1]} &\to& D^{[1]} \\ \downarrow && \downarrow \\ C \times C &\to& D \times D }
Proof

The equivalence of the first two points is basically remark 1.2.11.1 of Lurie that a fully faithful functor factors as an equivalence onto a full subcategory.

The equivalence of the first and third points is proposition 3.9.6 of Cisinski. The idea is to exploit the fact that the hom-spaces of an \infty-category XX are the fibers of X [1]X×XX^{[1]} \to X \times X (and similarly for the map on cores).

The fourth point implies the third point since Core preserves pullbacks. The second point implies the fourth point by the following arugment.

C [1] D [1] C [1] D [1] Ho(C) [1] Ho(D) [1] C×C D×D Ho(C)×Ho(C) Ho(D)×Ho(D) Ho(C)×Ho(C) Ho(D)×Ho(D) \array{ C^{[1]} &\to& D^{[1]} &\qquad \qquad& C^{[1]} &\to& D^{[1]} \\ \downarrow && \downarrow && \downarrow && \downarrow \\ Ho(C)^{[1]} &\to& Ho(D)^{[1]} && C \times C &\to& D \times D \\ \downarrow && \downarrow && \downarrow && \downarrow \\ Ho(C) \times Ho(C) &\to& Ho(D) \times Ho(D) && Ho(C) \times Ho(C) &\to& Ho(D) \times Ho(D) }

Assume the second point. Since Ho(C)Ho(D)Ho(C) \to Ho(D) is a fully faithful functor of 1-categories, the bottom square is a pullback (and a homotopy pullback). The top square is a pullback since () [1](-)^{[1]} preserves limits. Thus the outer square is a pullback.

On the right, we’ve seen the outer square is a pullback and the bottom square is a pullback since limits commute with limits. Thus, the upper square is a pullback.

Every full and faithful (,1)(\infty,1)-functor is a monomorphism in (∞,1)Cat, but being a full and faithful (,1)(\infty,1)-functor is a stronger condition. An (,1)(\infty,1)-functor FF is a monomorphism if and only if it induces a monomorphism on hom-spaces and every equivalence FXFYF X \simeq F Y is in the effective image of FF (see this MathOverflow question).

An (∞,1)-functor which is both full and faithful as well as an essentially surjective (∞,1)-functor is an equivalence of (∞,1)-categories.

basic properties of…

References

This appears as definition 1.2.10 in

Last revised on June 1, 2020 at 14:45:34. See the history of this page for a list of all contributions to it.