Spahn HTT, 2. fibrations of simplicial sets (Rev #5, changes)

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2.0 ~~(overview~~ (Overview and glossary of fibrations and anodyne morphisms)
2.1 Left fibrations
2.2 Simplicial categories and $\infty$ -categories
- 2.2.5 The Joyal model structure
2.3 Inner fibrations
- 2.3.3. Minimal inner fibrations
- 2.3.4 Theory of $n$ -categories
2.4 Cartesian fibrations

2.0 (overview (Overview and glossary of fibrations and anodyne morphisms)

In particular the following notions are important:

Left fibrations: These are the higher analogs of fibrations in groupoids.
Anodyne morphisms: These are morphisms possessing the left lifting property with respect to all Kan fibrations. Anodyne morphisms are acyclic cofibrations in the standard model structure on simplicial sets.
The covariant model structure: This is a model structure on the over category $sSet/S$ . It is a model structure for left fibrations. It is ~~factorial~~ functorial in $S$ . It is called covariant since by the ~~Grothendieck~~ ~~construction~~ ${Fib}_{l} ∞ (S)$ ~~Fib_l~~ \infty ~~(S)~~ -Grothendieck is construction ~~associated~~ to ~~the~~ ~~category~~ of ~~covariant~~ ~~functors~~ $Fun {Fib}_{l} (S, ∞ Grpd)$ ~~Fun(S,\infty~~ Fib_l ~~Grpd)~~ (S) . ~~There~~ is ~~also~~ associated a to ~~notion~~ the category of covariant functors $Fun(S,\infty Grpd)$ . There is also a notion of contravariant model structure where right fibrations $Fib_r (S)$ is associated to $Fun(S^{op},\infty Grpd)$ .
The Joyal model structure: Precisely $\infty$ -categories are fibrant with respect to this model structure.
inner fibration and minimal fibration: These are used to develop a theory of n-categories.
cartesian fibration: These are higher analogs to Grothendieck fibrations (not necessarily in groupoids). These are defined with respect to cartesian morphisms.
categorical ~~fibrations~~ fibration?: These are the fibrations in the Joyal model structure on $sSet$ (also called model structure for quasi-categories): Morphisms of simplicial sets possessing the right lifting properties against acyclic cofibrations in this model structure. Here the cofibrations are the monomorphisms and the weak equivalences are called weak categorical equivalences. Categorical fibrations have no intrinsic meaning in $\infty$ -category theory. Fibrant objects in this model structure are precisely $\infty$ -categories.

For a general overview see model structure on simplicial sets and fibration of quasi-categories.

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).

We have the following relations of kinds of fibrations where the arrows indicate implication (e.g. an acyclic fibration is a Kan fibration).

\array{ &acyclic\; fibration \\ &\downarrow \\ &Kan\; fibration \\ \swarrow&&\searrow \\ left \;fibration&&right \;fibration \\ \downarrow&&\downarrow \\ cocartesian\; fibration&&\cartesian\; fibration \\ \searrow&&\swarrow \\ &categorical\; fibration \\ &\downarrow \\ &inner \;fibration }

~~Remark~~
~~(anodyne morphisms)~~

Every map of simplicial sets admits a factorization into an anodyne ((left anodyne, right anodyne, inner anodyne, a coﬁbration) followed by a Kan ﬁbration (left ﬁbration, right ﬁbration, inner ﬁbration, trivial ﬁbration).

The theory of left fibrations and left anodyne morphisms is dual to that of right fibrations and right anodyne morphisms; i.e. if $S\to T$ is a left fibration (left anodyne morphisms) iff the induced map $S^{op}\to T^{op}$ right fibrations and right anodyne morphisms.

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).

Remark

(anodyne morphisms)

Every map of simplicial sets admits a factorization into an anodyne ((left anodyne, right anodyne, inner anodyne, a coﬁbration) followed by a Kan ﬁbration (left ﬁbration, right ﬁbration, inner ﬁbration, trivial ﬁbration).
The theory of left fibrations and left anodyne morphisms is dual to that of right fibrations and right anodyne morphisms; i.e. if $S\to T$ is a left fibration (left anodyne morphisms) iff the induced map $S^{op}\to T^{op}$ right fibrations and right anodyne morphisms.

2.1 Left fibrations

2.1.1 Left fibrations in classical category theory

Left fibrations are analogs to fibrations in groupoids.

Proposition 2.1.1.3 +

Let $F:C\to D$ be a functor between categories. The $F$ is a fibrations in groupoids iff the induced map $N(F):N(C)\to N(D)$ is a left fibration of simplicial sets.

Compare the following lemma with Proposition 2.1.3.1

Lemma 2.1.1.4 +

Let $q:X\to S$ be a left fibration of simplicial sets. The assignment

\begin{cases} hS\to H \\ s\mapsto X_s \\f\mapsto f_1 \end{cases}

2.1.2 Stability properties of left fibrations

The aim of this section is to show that left fibrations exist in abundance.

The following remark follows from a theorem of Joyal (not displayed here).

Remark (2.1.2.2, 2.1.2.9)

(left fibrations) 1. The projection from the over category is a left fibration.

1.The property of being a left fibration is stable under forming functor categories.

2.1.3 A characterization of Kan fibrations

Compare the following proposition with Lemma 2.1.1.4.

Proposition 2.1.3.1 +

Let $p:S\to T$ be a left fibration of simplicial sets. Then the following statements are equivalent

$p$ is a Kan fibration.
For every edge $f:t\to t^\prime$ in $T$ , the map $f_!:S_t\to S_{t^\prime}$ is an isomorphism in the homotopy category of spaces.

2.1.4 The covariant model structure

This section is a preparation for the Grothendieck consruction for $\infty$ -categories.

uses the model structure on simplicially enriched categories

Definition 2.1.4.2

left- and right cone of a morphism of simplicial sets

cone point

The covariant model structure is a ‘’relative model structure’‘

Definition 2.1.4.5

Let $S$ be a simplicial set . A morphism $f:X\to Y$ in $sSet_{/S}$ is called a

(W) a covariant weak equivalence if the induced map

X^\triangleleft\coprod_X S\to Y^\triangleleft\coprod_Y S

is a categorical weak equivalence.

(F) covariant fibration if it has the right lifting property with respect to every map wich is both a covariant cofibration and a covariant equivalence.

Lemma 2.1.4.6

every left anodyne map is a covariant equivalence

Proposition 2.1.4.7

The covariant model structure determines a left proper, combinatorial model structure on $sSet_{/S}$

Proposition 2.1.4.9

Every covariant fibration is a left fibration of simplicial sets

Proposition 2.1.4.10

The covariant model structure is functorial in $S$ .

Remark 2.1.4.12

There is also a contravariant model structure

2.2 Simplicial categories and $\infty$ -categories

2.2.5 The Joyal model structure

Requisite: Theorem 2.4.6.1: Let $X$ be a simplicial set. Then $X$ is fibrant in the Joyal model structure iff $X$ is an $\infty$ -category.

Theorem 2.2.5.1

The exists a left proper, combinatorial model structure on the category of simplicial sets such that

(W) A map $p$ is a categorical equivalence iff $S(p)$ is an equivalence of simplicial categories. Where $S:sSet\to sCat$ denotes the functor induced via Kan extension by the cosimplicial object $\mathfrak{C}:\Delta\to sCat$ , Definition 1.1.5.1, HTT.

here: give proof of Proposition 1.2.7.3

2.3 Inner fibrations

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

~~Every~~ $Fun(\Delta[2],C)\to Fun(\Lambda_1[2],C)$ ~~$\infty$~~ is a trivial fibration.~~-category is categorial equivalenct to a minimal~~ ~~$\infty$~~ ~~-category.~~

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.~~

\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 $\infty$ -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 $\infty$ -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

2.4.1 Cartesian morphisms

Definiton 2.4.1.1

Let $p:X\to S$ be an inner fibrations of simplicial sets. Let $f:x\to y$ be an edge in $X$ . Then $f$ is called $p$ -cartesian if the induced map

X_{/f}\to X_{/y}\times_{S_{p(y)}} S_{/p(f)}

is a trivial Kan fibration.

Proposition 2.4.1.3

Every edge of a simplicial set is $p$ cartesian for an isomorphism.
Let $p$ be an inner fibration, let $q$ be the pullback of $p$ (which s then also an inner fibration). Then an edge is $p$ cartesian if ‘’its pullback’‘ is $q$ -cartesian.
(…)

Corollary 2.4.1.6

Let $p:C\to D$ be an inner fibration between $\infty$ -categories. Every identity morphism of $C$ (i.e. every degenerate edge of $C$ ) is $p$ -cartesian.

Proposition 2.4.1.7

(left cancellation) Let $p:C\to D$ be an inner fibration between simplicial sets. Let

\array{ &C_1& \\ {}_f\nearrow&\searrow^g \\ C_0&\stackrel{h}{\to}&C_2 }

Let $g$ be $p$ -cartesian. Then $f$ is $p$ -cartesian iff $h$ is $p$ -cartesian.

Proposition 2.4.1.10

(…)

2.4.2 Cartesian fibrations

Definition 2.4.2.1

Let $p:X\to S$ be a map of simplicial sets. Then $p$ is called a cartesian fibration if the following coditions are satisfied.

$p$ is an inner fibration.
Every edge of has a $p$ -cartesian lift.

Proposition 2.4.2.3

Proposition 2.4.2.4

2.4.4 Mapping spaces and cartesian fibrations

Corollary 2.4.4.4

Corollary 2.4.4.6

Corollary 2.4.4.7

Corollary 2.4.4.8

2.4.6 Application: Categorical fibrations

Corollary 2.4.6.1

2.4.7 Bifibrations

Corollary 2.4.7.11

Corollary 2.4.7.12

Revision on June 21, 2012 at 17:00:35 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

Spahn HTT, 2. fibrations of simplicial sets (Rev #5, changes)

Contents

2.0 (overview (Overview and glossary of fibrations and anodyne morphisms)

Definition

Remark

Remark

Remark

Remark

2.1 Left fibrations

2.1.1 Left fibrations in classical category theory

Proposition 2.1.1.3 +

Lemma 2.1.1.4 +

2.1.2 Stability properties of left fibrations

Remark (2.1.2.2, 2.1.2.9)

2.1.3 A characterization of Kan fibrations

Proposition 2.1.3.1 +

2.1.4 The covariant model structure

Definition 2.1.4.2

Definition 2.1.4.5

Lemma 2.1.4.6

Proposition 2.1.4.7

Proposition 2.1.4.9

Proposition 2.1.4.10

Remark 2.1.4.12

2.2 Simplicial categories and ∞\infty-categories

2.2.5 The Joyal model structure

Theorem 2.2.5.1

2.3 Inner fibrations

Corollary 2.3.2.2: 2.3.2.2Fun(Δ[2],C)→Fun(Λ 1[2],C)Fun(\Delta[2],C)\to Fun(\Lambda_1[2],C) is a trivial fibration.

2.3.3. Minimal inner fibrations

Definition 2.3.3.1

Definition 2.3.3.1

2.3.4 Theory of nn-categories

Proposition 2.3.4.19

Proposition 2.3.4.19 Let CC be an ∞\infty-category. Let n≥−1n\ge -1. Then the following statements are equivalent:

Corollary 2.3.4.8

Proposition 2.3.4.12

Corollary 2.3.4.19

2.4 Cartesian fibrations

2.4.1 Cartesian morphisms

Definiton 2.4.1.1

Proposition 2.4.1.3

Corollary 2.4.1.6

Proposition 2.4.1.7

Proposition 2.4.1.10

2.4.2 Cartesian fibrations

Definition 2.4.2.1

Proposition 2.4.2.3

Proposition 2.4.2.4

2.4.4 Mapping spaces and cartesian fibrations

Corollary 2.4.4.4

Corollary 2.4.4.6

Corollary 2.4.4.7

Corollary 2.4.4.8

2.4.6 Application: Categorical fibrations

Corollary 2.4.6.1

2.4.7 Bifibrations

Corollary 2.4.7.11

Corollary 2.4.7.12

2.2 Simplicial categories and $\infty$ -categories

Corollary 2.3.2.2: 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 Let $C$ be an $\infty$ -category. Let $n\ge -1$ . Then the following statements are equivalent: