Model categories and Quillen adjunctions

A model category is a category $\mathcal{M}$ with all small limits and colimits (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), together with three classes $\mathcal{W},\mathrm{F}\mathrm{i}\mathrm{b},\mathrm{C}\mathrm{o}\mathrm{f}$ of maps, each closed under retracts (that is, if a map is a retract of a map in the class, in the arrow category, then it lies in the class), subject to the following axioms.

1. $\mathcal{W}$ satisfies two-out-of-three: if two of $f$, $g$ and $g\circ f$ lie in $\mathcal{W}$, then so does the third.
2. Lifting. For a cofibration $i\text{\hspace{0.17em}⁣}:A\to B$, a fibration $p\text{\hspace{0.17em}⁣}:X\to Y$ and morphisms $a\text{\hspace{0.17em}⁣}:A\to X$, $b\text{\hspace{0.17em}⁣}:B\to Y$ with $p\circ a=b\circ i$, if $i$ or $p$ lies in $\mathcal{W}$, then there is a lift $\mathrm{\ell }\text{\hspace{0.17em}⁣}:B\to X$ with $\mathrm{\ell }\circ i=a$ and $p\circ \mathrm{\ell }=b$.
3. Factorization. Every map $f$ factors both as $f=p\circ i$ with $i$ in $\mathrm{C}\mathrm{o}\mathrm{f}$ and $p$ in $\mathrm{F}\mathrm{i}\mathrm{b}\cap \mathcal{W}$, and as $f=q\circ j$ with $j$ in $\mathrm{C}\mathrm{o}\mathrm{f}\cap \mathcal{W}$ and $q$ in $\mathrm{F}\mathrm{i}\mathrm{b}$.

A trivial fibration is a map in $\mathrm{F}\mathrm{i}\mathrm{b}\cap \mathcal{W}$ and a trivial cofibration is a map in $\mathrm{C}\mathrm{o}\mathrm{f}\cap \mathcal{W}$. An object $X$ is cofibrant when the structure map $\mathrm{\varnothing }\to X$ from an initial object lies in $\mathrm{C}\mathrm{o}\mathrm{f}$, and fibrant when the structure map $X\to \ast$ to a terminal object lies in $\mathrm{F}\mathrm{i}\mathrm{b}$; these properties are independent of the chosen initial and terminal objects, since any two are canonically isomorphic. A map in $\mathcal{W}$ is a weak equivalence.

A Quillen adjunction $L⊣R$ between model categories $\mathcal{M}$ and $\mathcal{N}$ is an adjunction (in the sense of The unit-counit, hom-set, unit-universal, and counit-universal encodings of an adjunction are equivalent) whose right adjoint $R$ sends fibrations to fibrations and trivial fibrations to trivial fibrations. It is equivalent to require that the left adjoint $L$ sends cofibrations to cofibrations and trivial cofibrations to trivial cofibrations: transposing each lifting square across the adjunction bijection identifies a lift on the left with a lift on the right, so the two conditions are exchanged by the adjunction. A Quillen equivalence is a Quillen adjunction such that for every cofibrant $X$ and fibrant $Y$ a map $LX\to Y$ lies in $\mathcal{W}$ exactly when its adjoint $X\to RY$ does.

When source and target carry simplicial mapping objects, an enriched Quillen adjunction is a Quillen adjunction together with natural isomorphisms $\mathrm{M}\mathrm{a}\mathrm{p}(LX,Y)\cong \mathrm{M}\mathrm{a}\mathrm{p}(X,RY)$ compatible with the simplicial operators and natural with respect to the enriched mapping objects. A simplicial Quillen adjunction between simplicial model categories is an enriched adjunction in this sense whose underlying adjunction is Quillen.

These definitions do not assert the existence of any model structure: being a model category is structure on a category, and a functor between model categories is not required to preserve anything. Two objects require care throughout: the initial and terminal objects. In the category of unital commutative $A$-algebras the initial object is $A$ and the terminal object is the zero ring, and the two must not be conflated when cofibrancy and fibrancy are read off from the structure maps.