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

2.0

Definition

A morphism of simplicial sets is called

  1. a Kan fibration if it has the right lifting property with respect to every horn inclusion.

  2. a trivial fibration if it has the right lifting property with respect to every boundary inclusion Δ[n]Δ[n]\partial \Delta[n]\hookrightarrow \Delta[n].

  3. a left fibration if it has the right lifting property with respect to every horn inclusion except the right outer one.

  4. a left fibration if it has the right lifting property with respect to every horn inclusion except the left outer one.

  5. a left fibration if it has the right lifting property with respect to every inner horn inclusion.

  6. left anodyne if it has the left lifing property with respect to every left fibration.

  7. right anodyne if it has the left lifing property with respect to every right fibration.

  8. inner anodyne if it has the left lifing property with respect to every inner fibration.

  9. minimal fibration roughly said, when the morphism is determined by its values on the boundaries.

  10. cartesian fibration

  11. cocartesian fibration

  12. categorical fibration

Remark

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

  1. Right fibrations are the \infty-categorical analog of fibrations in groupoids.

  2. Left fibrations are the \infty-categorical analog of cofibrations in groupoids.

  3. 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(Δ[2],C)Fun(Λ 1[2],C)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: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 \infty-category CC is called minimal \infty-category if C*C\to * is minimal.

(…)

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

2.3.4 Theory of nn-categories

Proposition 2.3.4.19

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

  1. the unit u:XN(hX)u:X\to N(hX) is an isomorphism of simplicial sets.

  2. There is small category CC and an isomorphism of simpliial sets XN(C)X\simeq N(C).

  3. SS is a 1-category.

Proposition 2.3.4.19 Let CC be an \infty-category. Let n1n\ge -1. Then the following statements are equivalent:
  1. CC is an nn-category.

  2. For every simplicial set KK and every pair of maps f,g:KCf,g:K\to C such that f|sk nKf| sk^n K and g|sk nKg|sk^n K are homotopic relative to sk n1Ksk^{n-1}K, we have f=gf=g.

Corollary 2.3.4.8

Let CC be an nn-category and let XX be a simplicial set. Then Fun(X,C)Fun(X,C) is an nn-category.

Proposition 2.3.4.12

Let CC be an \infty-category. Let n1n\ge 1.

  1. There exists a simplicial set h nCh_n C with the following universal mapping property: Fun(K,h nC)=[K,C]/Fun(K,h_n C)=[K,C]/\sim.

  2. h nCh_n C is an nn-category.

  3. If CC is an nn-category, then the natural map Θ:Ch nC\Theta:C\to h_n C is an isomorphism.

  4. For every nn-category DD, composition with Θ\Theta is an isomorphism of simplicial sets Fun(h nC,D)Fun(C,D)Fun(h_n C,D)\to Fun(C,D).

Corollary 2.3.4.19

Let XX be a Kan complex. Then is is equivalent to an nn-category iff it is nn-truncated.

2.4 cartesian fibrations

1.1 (definitions of \infty-categories)

\infty-categories as simplicial sets

\infty-categories as categories enriched in

  1. sSetsSet

  2. Top CGTop_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.