model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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
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.
Let $\mathcal{C}$ be a model category and $X \in \mathcal{C}$ an object.
where $X\to Path(X)$ is a weak equivalence. This is called a good path object if in addition $Path(X) \to X \times X$ is a fibration.
where $Cyl(X) \to X$ is a weak equivalence. This is called a good cylinder object if in addition $X \sqcup X \to Cyl(X)$ is a cofibration.
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 $X\times [0,1]$ is a cylinder object, but not a good cylinder unless $X$ itself is cofibrant (a cell complex in this case).
More generally, the path object $Path(X)$ of def. is analogous to the powering $\pitchfork(I,X)$ with an interval object and the cyclinder object $Cyl(X)$ is analogous to the tensoring with a cylinder object $I\odot X$. In fact, if $\mathcal{C}$ is a $V$-enriched model category and $X$ 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.
Let $f,g \colon X \longrightarrow Y$ be two parallel morphisms in a model category.
If $X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{p}{\in W}{\longrightarrow} X$ is a good cylinder object for a cofibrant object $X$ def. , then both components $i_0, i_1 \colon X \to Cyl(X)$ are acyclic cofibrations.
Dually, if $X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X$
is a good path object for a fibrant object $X$, then both component $p_0,p_1 \colon Path(X)\to X$ are acyclic fibrations.
We discuss the first case, the second is formally dual. First observe that the two inclusions $X \to X \sqcup X$ are cofibrations, since they are the pushout of the cofibration $\emptyset \to X$. This implies that $i_0$ and $i_1$ are composites of two cofibrations
and hence are themselves cofibrations. That they are in addition weak equivalences follows from two-out-of-three applied to the identity
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”.
For $\eta \colon f \Rightarrow_L g \colon X \to Y$ a left homotopy in some model category, def. , such that $Y$ is a fibrant object, then for $Cyl(X)$ any choice of good cylinder object for $X$, def. , there is a commuting diagram of the form
Dually, for $\eta \colon f \Rightarrow_R g \colon X \to Y$ a right homotopy, def. , such that $X$ is cofibrant, then for $Path(X)$ any choice of good path object for $X$, def. , there is a commuting diagram of the form
We discuss the first statement, the second is formally dual. Let $\eta \colon \hat X \longrightarrow Y$ be the given left homotopy with respect to a given cylinder object $\hat X$ of $X$. Factor it as
Then find liftings $\ell$ and $k$ in the following two commuting diagrams
Now the composite $\eta \coloneqq k \circ \ell$ is of the required kind,
Let $f,g \colon X \to Y$ be two parallel morphisms in a model category.
Let $X$ be cofibrant. If there is a left homotopy $f \Rightarrow_L g$ then there is also a right homotopy $f \Rightarrow_R g$ (def. ) with respect to any chosen good path object.
Let $Y$ be fibrant. If there is a right homotopy $f \Rightarrow_R g$ then there is also a left homotopy $f \Rightarrow_L g$ with respect to any chosen good cylinder object.
We discuss the first case, the second is formally dual. Let $\eta \colon Cyl(X) \longrightarrow Y$ be the given left homotopy. By lemma we may assume without restriction that $Cyl(X)$ is good in the sense of def. , for otherwise replace it by one that is. With this, lemma implies that we have a lift $h$ in the following commuting diagram
where on the right we have the chosen path space object. Now the composite $\tilde \eta \coloneqq h \circ i_1$ is a right homotopy as required.
For $X$ a cofibrant object in a model category and $Y$ a fibrant object, then the relations of left homotopy $f \Rightarrow_L g$ and of right homotopy $f \Rightarrow_R g$ (def. ) on the hom set $Hom(X,Y)$ coincide and are both equivalence relations.
That both relations coincide under the (co-)fibrancy assumption follows directly from lemma .
To see that left homotopy with domain $X$ is a transitive relation first use lemma to obtain that every left homotopy is exhibited by a good cylinder object $Cyl(X)$ and then lemma to see that the cofiber coproduct $Cyl(X)\underset{X}{\sqcup} Cyl(X)$ in
is again a cylinder object, def. . The symmetry and reflexivity of the relation is obvious.
See the references at model category.