nLab minimal inner fibration

Idea

The condition that a quasicategory has “no non-trivial cells” above degree $n$ (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 “$n$-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 $(n+1)$-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, see also at minimal fibration). 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

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

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

$$X^B\to X^A\times_{S^A}S^B$$

Equivalently $f,g$ are homotopic relative $A$ over $B$ if there is a map

$$F:B\times \Delta[1]\to X$$

such that

$F|B\times\{0\}=f$

$F|B\times\{1\}=g$

$p\circ F=v\circ \pi_B$

$F\circ(i\times id_{\Delta[1]})=u\circ\pi_A$

$F|\{b\}\times \Delta[1]$

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

Definition 2.3.3.1

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

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

Proposition 2.3.4.18

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

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

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

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

Corollary 2.3.4.19

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

Related concepts

• minimal Kan fibration

• inner fibration, left fibration, right fibration

References

Section 2.3.3 and section 2.3.4 of

• Jacob Lurie, Higher Topos Theory,