Spahn HTT, 2. fibrations of simplicial sets (Rev #1)

2.0

Definition

A morphism of simplicial sets is called

a Kan fibration if it has the right lifting property with respect to every horn inclusion.
a trivial fibration if it has the right lifting property with respect to every boundary inclusion $\partial \Delta[n]\hookrightarrow \Delta[n]$ .
a left fibration if it has the right lifting property with respect to every horn inclusion except the right outer one.
a left fibration if it has the right lifting property with respect to every horn inclusion except the left outer one.
a left fibration if it has the right lifting property with respect to every inner horn inclusion.
left anodyne if it has the left lifing property with respect to every left fibration.
right anodyne if it has the left lifing property with respect to every right fibration.
inner anodyne if it has the left lifing property with respect to every inner fibration.
minimal fibration roughly said, when the morphism is determined by its values on the boundaries.
cartesian fibration
cocartesian fibration
categorical fibration

Remark

We have the following intuition in regard to these types of fibrations

Right fibrations are the $\infty$ -categorical analog of fibrations in groupoids.
Left fibrations are the $\infty$ -categorical analog of cofibrations in groupoids.
Cartesian fibrations are the $\infty$ -categorical analog of fibrations (not necessarily in groupoids).

2.2.5 Joyal model structure

2.3 Inner fibrations and minimal inner fibrations

Corollary 2.3.2.2: $Fun(\Delta[2],C)\to Fun(\Lambda_1[2],C)$ is a trivial fibration.

Every $\infty$ -category is categorial equivalenct to a minimal $\infty$ -category.

2.3.3. Minimal inner fibrations

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 $\infty$ -category $C$ is called minimal $\infty$ -category if $C\to *$ is minimal.

(…)

Every $\infty$ -category is equivalent to a minimal $\infty$ -category.

2.3.4 Theory of $n$ -categories

Proposition 2.3.4.19

Proposition 2.3.4.5: For a simplicial set $X$ the following statements are equivalent:

the unit $u:X\to N(hX)$ is an isomorphism of simplicial sets.
There is small category $C$ and an isomorphism of simpliial sets $X\simeq N(C)$ .
$S$ is a 1-category.

Proposition 2.3.4.19 Let $C$ be an $\infty$ -category. Let $n\ge -1$ . Then the following statements are equivalent:

$C$ is an $n$ -category.
For every simplicial set $K$ and every pair of maps $f,g:K\to C$ such that $f| sk^n K$ and $g|sk^n K$ are homotopic relative to $sk^{n-1}K$ , we have $f=g$ .

Corollary 2.3.4.8

Let $C$ be an $n$ -category and let $X$ be a simplicial set. Then $Fun(X,C)$ is an $n$ -category.

Proposition 2.3.4.12

Let $C$ be an $\infty$ -category. Let $n\ge 1$ .

There exists a simplicial set $h_n C$ with the following universal mapping property: $Fun(K,h_n C)=[K,C]/\sim$ .
$h_n C$ is an $n$ -category.
If $C$ is an $n$ -category, then the natural map $\Theta:C\to h_n C$ is an isomorphism.
For every $n$ -category $D$ , composition with $\Theta$ is an isomorphism of simplicial sets $Fun(h_n C,D)\to Fun(C,D)$ .

Corollary 2.3.4.19

Let $X$ be a Kan complex. Then is is equivalent to an $n$ -category iff it is $n$ -truncated.

2.4 cartesian fibrations

1.1 (definitions of $\infty$ -categories)

$\infty$ -categories as simplicial sets

$\infty$ -categories as categories enriched in

$sSet$
$Top_CG$

Revision on June 20, 2012 at 18:21:32 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.