nLab
minimal inner fibration

Contents

Idea

The condition that aa quasicategory has “no non-trivial cells” above degree nn (which makes it a particularly strict model of an (n,1)-category) is not invariant under categorical equivalence. Hence there is no intrinsic characterization of the class of the simplicial sets which are “(n+1)-coskeletal” in this sense.

(Warning: in Lurie, Def. 2.3.4.1 such an “(n+1)-coskeletal” quasi-category is called an ”nn-category”, but this is not the intrinsic notion of (n,1)-category.)

However there is such a description of the class of quasi-categories which are equivalent to such (n+1)(n+1)-coskeletal quasicategories. To make this more concrete the notion of a minimal inner fibration can be used (a quasi-categorical analog of minimal Kan fibrations). THis is an inner fibration of simplicial sets satisfying a relative homotopy condition and that of a minimal quasi-category .

Every quasi-category is equivalent to a minimal quasi-category.

Definition

Definition

Let

A u X i p B v S\array{ A&\stackrel{u}{\to}&X \\ \downarrow^i&&\downarrow^p \\ B&\stackrel{v}{\to}&S}

denote a lifting problem. Then putative solutions f,gf,g of this lifting problem are called homotopic relative AA over SS if they are equivalent as objects in the fiber of the map

X BX A× S AS BX^B\to X^A\times_{S^A}S^B

Equivalently f,gf,g are homotopic relative AA over BB if there is a map

F:B×Δ[1]XF:B\times \Delta[1]\to X

such that

FB×{0}=fF|B\times\{0\}=f

FB×{1}=gF|B\times\{1\}=g

pF=vπ Bp\circ F=v\circ \pi_B

F(i×id Δ[1])=uπ AF\circ(i\times id_{\Delta[1]})=u\circ\pi_A

F{b}×Δ[1]F|\{b\}\times \Delta[1]

and F{b}×Δ[1]F|\{b\}\times\Delta[1] is an equivalence in the (,1)(\infty,1)-category X v(b)X_{v(b)} for every vertex bb of BB.

Definition 2.3.3.1

Let p:XSp : X \to S be an inner fibration of simplicial sets. pp is called minimal inner fibration if f=f f = f^\prime for every pair of maps f,f :Δ[n]Xf , f ^\prime : \Delta[n] \to X which are homotopic relative to Δ[n]\partial \Delta[n] over SS .

An (,1)(\infty,1)-category CC is called minimal (,1)(\infty,1)-category if C*C\to * is minimal.

Proposition 2.3.4.18

Let CC be an (,1)(\infty,1)-category and let n1n\ge -1. The the following statements are equivalent:

  1. There exists a minimal model C CC^\prime\subseteq C such that C C^\prime is an (n+1)(n+1)-coskeletal quasi-category.

  2. There exists a categorical equivalence DCD\to C, where DD is an (n+1)(n+1)-coskeletal quasi-category.

  3. For every pair of objects X,YCX,Y\in C, the mapping space Map C(X,Y)HMap_C(X,Y)\in H is (n1)(n-1)-truncated.

Corollary 2.3.4.19

Let XX be a Kan complex. Then is is equivalent to an (n+1)(n+1)-coskeletal quasi-category iff it is nn-truncated.

References

Section 2.3.3 and section 2.3.4 of

Revised on June 21, 2012 23:24:33 by Urs Schreiber (89.204.137.104)