nLab
minimal inner fibration

Idea

The condition that aa 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). 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

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

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

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

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

such that

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

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

$p\circ F=v\circ {\pi }_B$

$F\circ (i\times {\mathrm{id}}_{\Delta [1]})=u\circ {\pi }_A$

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

and $F\mid \{b\}\times \Delta [1]$ is an equivalence in the $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-category $C$ is called minimal $(\mathrm{\infty },1)$-category if $C\to *$ is minimal.

Proposition 2.3.4.18

Let $C$ be an $(\mathrm{\infty },1)$-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 ${\mathrm{Map}}_C(X,Y)\in H$ 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,