nLab fully faithful (infinity,1)-functor

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 $F : C \to D$ is full and faithful if for all objects $x,y \in C$ it induces an equivalence on the hom-∞-groupoids

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

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

$$F : C \hookrightarrow D$$

to indicate this. This is definition 1.2.10 of Lurie

Properties

Lemma

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

• $F$ is fully faithful

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

$$\array{ C &\to& D \\ \downarrow && \downarrow \\ Ho(C) &\to& Ho(D) }$$

• The following square is a pullback in the $(\infty,1)$ category of $\infty$-categories:

$$\array{ Core(C^{[1]}) &\to& Core(D^{[1]}) \\ \downarrow && \downarrow \\ Core(C) \times Core(C) &\to& Core(D) \times Core(D) }$$

• The following square is a pullback in the $(\infty,1)$ category of $\infty$-categories:

$$\array{ C^{[1]} &\to& D^{[1]} \\ \downarrow && \downarrow \\ C \times C &\to& D \times D }$$

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 $X$ are the fibers of $X^{[1]} \to X \times X$ (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.

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

Assume the second point. Since $Ho(C) \to Ho(D)$ 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 $(-)^{[1]}$ 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 $(\infty,1)$-functor is a monomorphism in (∞,1)Cat, but being a full and faithful $(\infty,1)$-functor is a stronger condition. An $(\infty,1)$-functor $F$ is a monomorphism if and only if it induces a monomorphism on hom-spaces and every equivalence $F X \simeq F Y$ is in the effective image of $F$ (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.

Related concepts

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

  • homotopy final functor, final (∞,1)-functor

References

This appears as definition 1.2.10 in

• Jacob Lurie, Higher Topos Theory
• Denis-Charles Cisinski, “Higher Categories and Homotopical Algebra”