nLab
homotopy in a model category

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

The extra structure of a model category over a category with weak equivalences induces concrete constructions for expressing homotopy between morphisms. These lead in particular to an explicit construction of the homotopy category of a model category.

Definition

Definition

Let 𝒞\mathcal{C} be a model category and X𝒞X \in \mathcal{C} an object.

  • A path object Path(X)Path(X) for XX is a factorization of the diagonal X:XX×X\nabla_X \colon X \to X \times X as
X:XWiPath(X)(p 0,p 1)X×X. \nabla_X \;\colon\; X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X \,.

where XPath(X)X\to Path(X) is a weak equivalence. This is called a good path object if in addition Path(X)X×XPath(X) \to X \times X is a fibration.

  • A cylinder object Cyl(X)Cyl(X) for XX is a factorization of the codiagonal (or “fold map”) Δ XXXX\Delta_X X \sqcup X \to X as
Δ X:XX(i 0,i 1)Cyl(X)pWX. \Delta_X \;\colon\; X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X \,.

where Cyl(X)XCyl(X) \to X is a weak equivalence. This is called a good cylinder object if in addition XXCyl(X)X \sqcup X \to Cyl(X) is a cofibration.

Remark

By the factorization axioms every object in a model category has both a good path object and as well as a good cylinder object according to def. . But in some situations one is genuinely interested in using non-good such objects.

For instance in the classical model structure on topological spaces, the obvious object X×[0,1]X\times [0,1] is a cylinder object, but not a good cylinder unless XX itself is cofibrant (a cell complex in this case).

More generally, the path object Path(X)Path(X) of def. is analogous to the powering (I,X)\pitchfork(I,X) with an interval object and the cyclinder object Cyl(X)Cyl(X) is analogous to the tensoring with a cylinder object IXI\odot X. In fact, if 𝒞\mathcal{C} is a VV-enriched model category and XX is fibrant/cofibrant, then these powers and copowers are in fact examples of (good) path and cylinder objects if the interval object is sufficiently good.

Definition

Let f,g:XYf,g \colon X \longrightarrow Y be two parallel morphisms in a model category.

  • A left homotopy η:f Lg\eta \colon f \Rightarrow_L g is a morphism η:Cyl(X)Y\eta \colon Cyl(X) \longrightarrow Y from a cylinder object of XX, def. , such that it makes this diagram commute:
X Cyl(X) X f η g Y. \array{ X &\longrightarrow& Cyl(X) &\longleftarrow& X \\ & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}} \\ && Y } \,.
  • A right homotopy η:f Rg\eta \colon f \Rightarrow_R g is a morphism η:XPath(Y)\eta \colon X \to Path(Y) to some path object of XX, def. , such that this diagram commutes:
X f η g Y Path(Y) Y. \array{ && X \\ & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}} \\ Y &\longleftarrow& Path(Y) &\longrightarrow& Y } \,.

Properties

Basic lemmas

Lemma

If XX(i 0,i 1)Cyl(X)pWX X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X is a good cylinder object for a cofibrant object XX def. , then both components i 0,i 1:XCyl(X)i_0, i_1 \colon X \to Cyl(X) are acyclic cofibrations.

Dually, if XWiPath(X)(p 0,p 1)X×X X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X

is a good path object for a fibrant object XX, then both component p 0,p 1:Path(X)Xp_0,p_1 \colon Path(X)\to X are acyclic fibrations.

Proof

We discuss the first case, the second is formally dual. First observe that the two inclusions XXXX \to X \sqcup X are cofibrations, since they are the pushout of the cofibration X\emptyset \to X. This implies that i 0i_0 and i 1i_1 are composites of two cofibrations

i 0,i 1:XCofXXCof i_0, i_1 \;\colon\; X \overset{\in Cof}{\longrightarrow} X\sqcup X \overset{\in Cof}{\longrightarrow}

and hence are themselves cofibrations. That they are in addition weak equivalences follows from two-out-of-three applied to the identity

id X:XWCyl(X)i 0X. id_X \;\colon\; X \overset{\in W}{\longrightarrow} Cyl(X) \overset{i_0}{\longrightarrow} X \,.

implied by the fact that the cylinder by definition factors the codiagonal.

The following says that the choice of cylinder/path objects in def. is irrelevant as long it is “good”.

Lemma

For η:f Lg:XY\eta \colon f \Rightarrow_L g \colon X \to Y a left homotopy in some model category, def. , such that YY is a fibrant object, then for Cyl(X)Cyl(X) any choice of good cylinder object for XX, def. , there is a commuting diagram of the form

X Cyl(X) X f η˜ g Y. \array{ X &\longrightarrow& Cyl(X) &\longleftarrow& X \\ & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\tilde \eta}}& \swarrow_{\mathrlap{g}} \\ && Y } \,.

Dually, for η:f Rg:XY\eta \colon f \Rightarrow_R g \colon X \to Y a right homotopy, def. , such that XX is cofibrant, then for Path(X)Path(X) any choice of good path object for XX, def. , there is a commuting diagram of the form

X f η˜ g Y Path(Y) Y. \array{ && X \\ & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\tilde \eta}} & \searrow^{\mathrlap{g}} \\ Y &\longleftarrow& Path(Y) &\longrightarrow& Y } \,.
Proof

We discuss the first statement, the second is formally dual. Let η:X^Y\eta \colon \hat X \longrightarrow Y be the given left homotopy with respect to a given cylinder object X^\hat X of XX. Factor it as

η:X^CofZWFibY. \eta \;\colon\; \hat X \overset{\in Cof}{\longrightarrow} Z \overset{\in W \cap Fib}{\longrightarrow} Y \,.

Then find liftings \ell and kk in the following two commuting diagrams

XX X^ Z Cyl(X) Y,X^ η Y k Z *. \array{ X \sqcup X &\overset{}{\longrightarrow}& \hat X &\longrightarrow& Z \\ \downarrow && & {}^{\mathllap{\ell}}\nearrow & \downarrow \\ Cyl(X) &\longrightarrow& &\longrightarrow& Y } \;\;\;\;\; \,, \;\;\;\;\; \array{ \hat X &\overset{\eta}{\longrightarrow}& Y \\ \downarrow &{}^{\mathllap{k}}\nearrow& \downarrow \\ Z &\longrightarrow& \ast } \,.

Now the composite ηk\eta \coloneqq k \circ \ell is of the required kind,

XX X^ Z k Y Cyl(X) . \array{ X \sqcup X &\overset{}{\longrightarrow}& \hat X &\longrightarrow& Z &\overset{k}{\longrightarrow}& Y \\ \downarrow &&& {}^{\mathllap{\ell}}\nearrow & \\ Cyl(X) &\longrightarrow& } \,.
Lemma

Let f,g:XYf,g \colon X \to Y be two parallel morphisms in a model category.

  • Let XX be cofibrant. If there is a left homotopy f Lgf \Rightarrow_L g then there is also a right homotopy f Rgf \Rightarrow_R g (def. ) with respect to any chosen path object.

  • Let YY be fibrant. If there is a right homotopy f Rgf \Rightarrow_R g then there is also a left homotopy f Lgf \Rightarrow_L g with respect to any chosen cylinder object.

Proof

We discuss the first case, the second is formally dual. Let η:Cyl(X)Y\eta \colon Cyl(X) \longrightarrow Y be the given left homotopy. By lemma we may assume without restriction that Cyl(X)Cyl(X) is good in the sense of def. , for otherwise replace it by one that is. With this, lemma implies that we have a lift hh in the following commuting diagram

X if Path(Y) WCof i 0 h Fib p 0,p 1 Cyl(X) (fp,η) Y×Y, \array{ X &\overset{i \circ f}{\longrightarrow}& Path(Y) \\ {}^{\mathllap{i_0}}_{\mathllap{\in W \cap Cof}}\downarrow &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}} \\ Cyl(X) &\underset{(f \circ p,\eta)}{\longrightarrow}& Y \times Y } \,,

where on the right we have the chosen path space object. Now the composite η˜hi 1\tilde \eta \coloneqq h \circ i_1 is a right homotopy as required.

Path(Y) h Fib p 0,p 1 X i 1 Cyl(X) (fp,η) Y×Y. \array{ && && Path(Y) \\ && &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}} \\ X &\overset{i_1}{\longrightarrow}& Cyl(X) &\underset{(f \circ p,\eta)}{\longrightarrow}& Y \times Y } \,.

Equivalence relation

Proposition

For XX a cofibrant object in a model category and YY a fibrant object, then the relations of left homotopy f Lgf \Rightarrow_L g and of right homotopy f Rgf \Rightarrow_R g (def. ) on the hom set Hom(X,Y)Hom(X,Y) coincide and are both equivalence relations.

Proof

That both relations coincide under the (co-)fibrancy assumption follows directly from lemma .

To see that left homotopy with domain XX is a transitive relation first use lemma to obtain that every left homotopy is exhibited by a good cylinder object Cyl(X)Cyl(X) and then lemma to see that the cofiber coproduct Cyl(X)XCyl(X)Cyl(X)\underset{X}{\sqcup} Cyl(X) in

X i 0 X Wi 1 Cyl(X) i 0 (po) X i 1 Cyl(X) W Cyl(X)XCyl(X) W X \array{ && && X \\ && && \downarrow^{\mathrlap{i_0}} \\ && X &\underoverset{\in W}{i_1}{\longrightarrow}& Cyl(X) \\ && {}^{\mathllap{i_0}}\downarrow &(po)& \downarrow \\ X &\underset{i_1}{\longrightarrow}& Cyl(X) &\underset{\in W}{\longrightarrow}& Cyl(X) \underset{X}{\sqcup} Cyl(X) \\ && &{}_{\mathllap{}}\searrow& & \searrow \\ && && \underset{\in W}{\longrightarrow} &\longrightarrow& X }

is again a cylinder object, def. . The symmetry and reflexivity of the relation is obvious.

References

See the references at model category.

Last revised on February 22, 2019 at 12:51:34. See the history of this page for a list of all contributions to it.