on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
A model category is a context in which we can do homotopy theory or some generalization thereof; two model categories are ‘the same’ for this purpose if they are Quillen equivalent. For example, the classic version of homotopy theory can be done using either topological spaces or simplicial sets. There is a model category of topological spaces, and a model category of simplicial sets, and they are Quillen equivalent.
In short, Quillen equivalence is the right notion of equivalence for model categories — and most importantly, this notion is weaker than equivalence of categories. The work of Dwyer–Kan, Bergner and others has shown that Quillen equivalent model categories present equivalent (infinity,1)-categories.
Let $C$ and $D$ be model categories and let
be a Quillen adjunction with $L$ left adjoint to $R$.
Write $Ho C$ and $Ho D$ for the corresponding homotopy categories.
By the discussion there, $Ho C$ may be regarded as obtained by first passing to the full subcategory on cofibrant objects and then inverting weak equivalences, and $L$ (being a left Quillen adjoint) preserves weak equivalences between cofibrant objects. Thus, $L$ induces a functor
between the homotopy categories, called its (total) left derived functor. Similarly (but dually), $R$ induces a (total) right derived functor $\mathbb{R} : Ho D \to Ho C$. See at homotopy category of a model category – derived functors for more.
A Quillen adjunction $(L \dashv R)$ is a Quillen equivalence if the following equivalent conditions are satisfied.
The total left derived functor $\mathbb{L} : Ho(C) \to Ho(D)$ is an equivalence of the homotopy categories;
The total right derived functor $\mathbb{R} : Ho(D) \to Ho(C)$ is an equivalence of the homotopy categories;
For every cofibrant object $c \in C$ and every fibrant object $d \in D$, a morphism $c \to R(d)$ is a weak equivalence in $C$ precisely when the adjunct morphism $L(c) \to d$ is a weak equivalence in $D$.
For every cofibrant object $c\in C$, the composite $c \to R(L(c)) \to R(L(c)^{fib})$ (of the adjunction unit with a fibrant replacement $R(L(c) \stackrel{\simeq}{\to} L(c)^{fib})$) is a weak equivalence in $C$,
For every fibrant object $d\in D$, the composite $L(R(d)^{cof}) \to L(R(d)) \to d$ (of the adjunction counit with cofibrant replacement $L(R(d)^{cof} \stackrel{\simeq}{\to} R(d))$) is a weak equivalence in $D$.
Not every equivalence between homotopy categories of model categories lifts to a Quillen equivalence. An interesting counterexample is given for instance in (Dugger-Shipley 09).
Here are further characterizations:
If in a Quillen adjunction $\array{\mathcal{C} &\underoverset{\underset{R}{\to}}{\overset{L}{\leftarrow}}{\bot}& \mathcal{D}}$ the right adjoint $R$ “creates weak equivalences” (in that a morphism $f$ in $\mathcal{C}$ is a weak equivalence precisly if $U(f)$ is) then $(L \dashv R)$ is a Quillen equivalence precisely already if for all cofibrant objects $d \in \mathcal{D}$ the plain adjunction unit
is a weak equivalence.
Generally, $(L \dashv R)$ is a Quillen equivalence precisely if
for every cofibrant object $d\in \mathcal{D}$, the “derived adjunction unit”, hence the composite
(of the adjunction unit with image under $R$ of any fibrant replacement $L(d) \underoverset{\in W}{j_{L(d)}}{\longrightarrow} R(P(L(d)))$) is a weak equivalence;
for every fibrant object $c \in \mathcal{C}$, the “derived adjunction counit”, hence the composite
(of the adjunction counit with the image under $L$ of any cofibrant replacement $Q R(c)\underoverset{\in W}{p_{R(c)}}{\longrightarrow} R(c)$ is a weak equivalence in $D$.
Consider the first condition: Since $R$ preserves the weak equivalence $j_{L(d)}$, by two-out-of-three the composite in the first item is a weak equivalence precisely if $\eta$ is.
Hence it is now sufficient to show that in this case the second condition above is automatic.
Since $R$ also reflects weak equivalences, the composite in item two is a weak equivalence precisely if its image
under $R$ is.
Moreover, assuming, by the above, that $\eta_{Q(R(c))}$ on the cofibrant object $Q(R(c))$ is a weak equivalence, then by two-out-of-three this composite is a weak equivalence precisely if the further composite with $\eta$ is
But by the formula for adjuncts, this composite is the $(L\dashv R)$-adjunct of the original composite, which is just $p_{R(c)}$
But $p_{R(c)}$ is a weak equivalence by definition of cofibrant replacement.
Since equivalences of categories enjoy the 2-out-of-3-property, so do Quillen equivalences.
sSet-enriched Quillen equivalences between combinatorial model categories present equivalences between the corresponding locally presentable (infinity,1)-categories. And every equivalence between these is presented by a Zig-Zag of Quillen equivalences. See there for more details.
Quillen equivalence
For standard references see at model category.
An example of an equivalence of homotopy categories of model categories which does not lift to a Quillen equivalence is in