sub-(infinity,1)-category in nLab

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Idea

The analog of the notion of subcategory for (∞,1)-categories.

Definition

Say that an equivalence of (∞,1)-categories D→≃CD \stackrel{\simeq}{\to} C exhibits DD as a 0-subcategory of CC.

Then define recursively, for n∈ℕn \in \mathbb{N}:

an nn-subcategory of an (∞,1)(\infty,1)-category DD for n≥1n \geq 1 is an (∞,1)-functor

F:D→C F : D \to C

such that for all objects x,y∈Dx,y \in D the component-(∞,1)(\infty,1)-functor on the hom-objects

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

exhibits an (n−1)(n-1)-subcategory.

Special cases

1-Subcategory / full subcategory

A full subcategory is a 1-subcategory, exhibited by a full and faithful (∞,1)-functor.

For sSetsSet-enriched and quasi-categories

Let CC and DD be incarnated specifically as fibrant simplicially enriched categories. Then for F:D→CF : D \to C a full and faithful (∞,1)(\infty,1)-functor, choose in each preimage F −1(c)F^{-1}(c) for each object c∈Cc \in C a representative, and let C′C' be the full sSet-enriched subcategory on these representatives.

Then the evident projection functor D→≃D′D \stackrel{\simeq}{\to} D' is manifestly an equivalence and the original F:D→CF : D \to C factors as

F:D→≃D′↪C, F : D \stackrel{\simeq}{\to} D' \hookrightarrow C \,,

where the second morphism is an ordinary inclusion of objects and hom-complexes.

Reflective sub-(∞,1)(\infty,1)-categories

If the (∞,1)(\infty,1)-functor F:D→CF : D \to C has a left adjoint (∞,1)-functor L:C→DL : C \to D, then FF is full and faithful and hence exhibits a 1-subcategory precisely if the counit

L∘F→Id D L \circ F \stackrel{}{\to} Id_D

is an equivalence of (∞,1)-functors. (See also HTT, p. 308).

In this case DD is a reflective (∞,1)-subcategory.

2-Subcategory

For sSetsSet-enriched and quasi-categories

Let the (∞,1)(\infty,1)-categories CC and DD concretely be incarnated as fibrant simplicially enriched categories.

Write hC:=Ho(C)h C := Ho(C) and hD:=Ho(D)h D := Ho(D) for the corresponding homotopy category of an (∞,1)-category (hom-wise the connected components of the corresponding simplicially enriched category).

Let hD→hCh D \to h C be a faithful functor. Then if we have a pullback in sSet-Cat

D → C ↓ ↓ hD → hC \array{ D &\to& C \\ \downarrow && \downarrow \\ h D &\to& h C }

DD is a 2-sub-(∞,1)(\infty,1)-category of CC. This pullback manifestly produces the simplicially enriched category whose

• objects are those of hDh D;

• hom-complexes are precisely the unions of those connected components of the hom-complexes of CC whose equivalence class is present in hDh D.

Therefore the inclusion functor D→CD \to C is on each hom-complex a full and faithful (∞,1)-functor. Hence this identifies DD as a 2-subcategory of CC.

If hD→hCh D \to h C is an inclusion on equivalence classes of objects then this is the definition of subcategory of an (∞,1)(\infty,1)-category that appears in HTT, section 1.2.11.

As 2-subobjects

Let core(Set *)→core(Set)core(Set_*) \to core(Set) be the 2-subobject classifier in the (∞,1)-topos ∞Grpd. Then for C∈∞GrpdC \in \infty Grpd a 1-subobject is classified by an ∞\infty-functor C→SetC \to Set. This factors through the homotopy category of CC as C→hC→SetC \to h C \to Set. Since Set *→SetSet_* \to Set is the universal faithful functor, the pullback

Q → Set * ↓ ↓ hC → Set \array{ Q &\to& Set_* \\ \downarrow && \downarrow \\ h C &\to& Set }

gives an ordinary subcategory of hCh C. This means that the total pullback D→CD \to C

D → hD → Set * ↓ ↓ ↓ C → hC → Set \array{ D &\to& h D &\to& Set_* \\ \downarrow && \downarrow && \downarrow \\ C &\to& h C &\to& Set }

gives a 2-sub-(∞,1)(\infty,1)-category KK of XX (where both happen to be ∞\infty-groupoids) here.

References

What we call a 2-subcategory of an (∞,1)(\infty,1)-category appears in section 1.2.11 of

• Jacob Lurie, Higher Topos Theory