Background
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equivalences in/of $(\infty,1)$-categories
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The generalization to the context of (∞,1)-category-theory of the notion of a full and faithful functor in ordinary category theory.
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
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
to indicate this. This is definition 1.2.10 of Lurie
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:
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.
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.
basic properties of…
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.