Spahn a reading guide to HTT (Rev #3)

Appendix A.2 (model categories and their homotopy categories)

2. Fibrations of simplicial sets

2.2

2.2.5 Joyal model structure

2.3 inner fibrations and minimal inner fibrations

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

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

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

$sSet$
$Top_CG$

1.2 (basic $\infty$ -category theory)

1.2.3 (the homotopy category of a simplicial set)

1.2.4 (objects and morphisms in an $\infty$ -category)

1.2.5 ( $\infty$ -groupoids)

1.2.6 (homotopy commutativity and homotopy coherence)

1.2.7 (functors between $\infty$ -categories)

Proposition 1.2.7.3

1.2.10, 1.2.11, 1.2.16

4 Limits and colimits

4.1

Definition 4.1: cofinal arrow Proposition 4.1.3.1: Cofinal arrows preserve colimits

4.2

Theorem 4.2.4.1: relation of $\infty$ -categorial colimits and homotopy colimits in simplicially enriched categories.

Proposition 4.2.4.4 (and Corollary 4.2.4.7)in a simplicial model category every homotopy coherent diagram is equivalent to a commutative diagram

4.3 (Kan extensions)

4.4 Examples for limits and colimits

construction of colimits from basic diagrams

5

5.1 Presheaves

5.2

Definition 5.2.2.1

Proposition 5.2.2.6

Proposition 5.2.2.8

Proposition 5.2.2.9

Proposition 5.2.2.12

Proposition 5.2.3.5 Adjoint functors preserve (co)limits

Revision on June 20, 2012 at 16:55:09 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.