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



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Extra stuff, structure and property

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structures in a

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    • / homotopy groups

Notions of subcategory

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


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

Then define recursively, for nn \in \mathbb{N}:

an nn-subcategory of an (,1)(\infty,1)-category DD for n1n \geq 1 is an (∞,1)-functor

F:DC F : D \to C

such that for all objects x,yDx,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 (n1)(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:DCF : D \to C a full and faithful (,1)(\infty,1)-functor, choose in each preimage F 1(c)F^{-1}(c) for each object cCc \in C a representative, and let CC' be the full sSet-enriched subcategory on these representatives.

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

F:DDC, 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:DCF : D \to C has a left adjoint (∞,1)-functor L:CDL : C \to D, then FF is full and faithful and hence exhibits a 1-subcategory precisely if the counit

LFId 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.


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 hDhCh 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 DCD \to C is on each hom-complex a full and faithful (∞,1)-functor. Hence this identifies DD as a 2-subcategory of CC.

If hDhCh D \to h C is an inclusion on objects (which is a bit evil to say) 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 CGrpdC \in \infty Grpd a 1-subobject is classified by an \infty-functor CSetC \to Set. This factors through the homotopy category of CC as ChCSetC \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 DCD \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.


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

Last revised on May 20, 2014 at 07:02:22. See the history of this page for a list of all contributions to it.