### Algebraic Model Structures

#### Posted by Tom Leinster

*Guest post by Emily Riehl*

Here’s a quick definition of a model structure on a complete and cocomplete category $M$: a model structure consists of three classes of morphisms $(C,F,W)$ - the *cofibrations*, *fibrations*, and *weak equivalences* - such that $W$ satisfies the 2-of-3 property and $(C \cap W, F)$ and $(C, F\cap W)$ are weak factorization systems.

A weak factorization system (henceforth *wfs*) $(\mathcal{L},\mathcal{R})$ consists of two classes of maps, closed under retracts, such that elements of the left class lift against elements of the right, as depicted below,

$\array{ \qquad\cdot & \stackrel{u}{\to} & \cdot\qquad \\ {}^{\mathcal{L} \ni f}\downarrow & {}^{\exists}\nearrow & \downarrow^{g \in \mathcal{R}} \\ \qquad\cdot & \stackrel{v}{\to} & \cdot\qquad}$

and such that every morphism can be factored as an arrow in $\mathcal{L}$ followed by an arrow in $\mathcal{R}$.

In examples, we typically think of the right class as a collection of morphisms satisfying some *property*, but it is also possible to conceive of them *algebraically*, that is, to incorporate the defining lifting property into a piece of *structure* attached to each given morphism. In one familiar example, the unalgebraicized perspective defines a Hurewicz fibration to be a map of spaces $p \colon E \rightarrow B$ with the homotopy lifting property, while the algebraicized perspective equips each such $p$ with the structure of a path lifting function. Using this idea, we define a new notion of algebraic model structure, introduced below, whose wfs are replaced with natural weak factorization systems. This algebraization is inobtrusive - it can be given in many familiar situations simply by making choices that are known to exist and does not affect the underlying model structure, except possibly by producing better factorizations - but has many interesting features because the algebraicized wfs satisfy better categorical properties.

### An algebraization

Often, the factorization in a wfs $(\mathcal{L},\mathcal{R})$ is assumed to be functorial, meaning that there exists a functor $\vec{E}\colon M^{2} \rightarrow M^{3}$ from the category $M^{2}$ of arrows in $M$ and commutative squares to the category $M^3$ of composable pairs of arrows in $M$ that is a section of the canonical composition functor $d_1 \colon M^{3} \rightarrow M^{ 2}$. The functorial factorization is most conveniently described by a pair of functors $L,R \colon M^{2} \rightarrow M^{2}$ and a functor $E\colon M^{2} \rightarrow M$, needed later, as depicted below

$\array{ \cdot & \stackrel{u}{\to} & \cdot \\ {}^{f}\downarrow & & \downarrow^{g} \\ \cdot & \stackrel{v}{\to} & \cdot} \quad \mapsto \quad \array{\cdot & \stackrel{u}{\to} & \cdot \\ {}^{L f}\downarrow & & \downarrow^{L g} \\ E f & \stackrel{E(u,v)}{\to} & E g \\ {}_{R f}\downarrow & & \downarrow_{R g} \\ \cdot & \stackrel{v}{\to} & \cdot}$

which then must satisfy various conditions which encode the fact that they fit together to give a functor $\vec{E}$ as above.

Note that the functors $L$ and $R$ are *pointed*, with the canonical natural transformations $\vec{\epsilon} \colon L \Rightarrow 1$ and $\vec{\eta} \colon 1 \Rightarrow R$ given by the functorial factorizations. Furthermore, the elements of the classes $\mathcal{L}$ and $\mathcal{R}$ are precisely those objects of $M^{ 2}$ which can be given the structure of algebras for the pointed endofunctors $L$ and $R$, constructed by solving the lifting problems

$f \in \mathcal{L} \quad \text{iff} \quad \array{ \cdot & \stackrel{L f}{\to} & \cdot\qquad \\ {}^{f}\downarrow & {}^{s}\nearrow & \downarrow^{R f} \\ \cdot & \stackrel{=}{\to} & \cdot\qquad} \quad \quad g \in \mathcal{R} \quad \text{iff} \quad \array{\cdot & \stackrel{=}{\to} & \cdot \\ {}^{L g}\downarrow & {}^{t}\nearrow & \downarrow^{g} \\ \cdot & \stackrel{R g}{\to} & \cdot}$

The algebra structures $s$ and $t$ can be used to solve any lifting problem of $(L,\vec{\epsilon})$-coalgebra against a $(R,\vec{\eta})$-algebra as depicted below

$\array{\cdot & \stackrel{u}{\to} & \cdot \\ {}^{L f}\downarrow\qquad & & {}^{t}\uparrow\downarrow^{L g} \\ E f & \stackrel{E(u,v)}{\to} & E g \\ {}_{R f}\downarrow \uparrow{}_s & & \downarrow_{R g} \\ \cdot & \stackrel{v}{\to} & \cdot}$

Categorically, wfs have two principle defects, both of which are rectified by their algebraization below. For one, a wfs on a category $M$ does not induce a wfs on the diagram category $M^{A}$ because a natural transformation whose components lie in $\mathcal{L}$ won’t lift naturally against a natural transformation whose components lie in $\mathcal{R}$. For another, the classes $\mathcal{L}$ and $\mathcal{R}$ are not closed under all colimits and limits, respectively, in the arrow category $M^{2}$, something that is true of the stronger, but homotopically useless, notion of *orthogonal factorization system*. Both of these problems disappear if we ask that the pointed endofunctors arising from the functorial factorization underlie a comonad and monad, respectively. The comultiplication and multiplication natural transformations are exactly what’s needed to insure that lifting problems of left factors against right factors can be solved naturally. Also, coalgebras for the comonad and algebras for the monad are closed under colimits and limits respectively.

**Definition.** A natural weak factorization system (henceforth, *nwfs*) **(L,R)** consists of a comonad **L**$= (L,\vec{\epsilon},\vec{\delta})$ and a monad **R**$=(R, \vec{\eta}, \vec{\mu})$ such that $(L,\vec{\epsilon})$ and $(R,\vec{\eta})$ arise as the pointed endofunctors of a functorial factorization $\vec{E}$ and such that the canonical natural transformation $L R \Rightarrow R L$ is a distributive law.

Note the distributive law condition was not in the original definition; we won’t worry about it here. The underlying wfs $(\overline{\mathcal{L}}, \overline{\mathcal{R}})$ is given by retract closures of the classes of maps admitting a **L**-coalgebra or **R**-algebra structure, respectively; alternatively, these are the arrows admitting pointed endofunctor (co)algebra structures.

Importantly, due to recent work of Richard Garner, cofibrantly generated nwfs can be produced by a modified version of Quillen’s small object argument that works for locally presentable categories $M$ and also for categories, like **Top** or **TopGp** whose objects are bounded with respect to some proper, well-copowered orthogonal factorization system. Interestingly, Garner’s construction works for small categories $J$ over the arrow category $M^{2}$, not just for generating sets. The nwfs **(L,R)** that is produced satisfies two universal properties.

There is a canonical functor $\lambda \colon J \rightarrow$

**L-coalg**that is universal among*morphisms of nwfs*(described below).There is a canonical isomorphism

**R-alg**$\cong J^{\square}$, where the latter category has as objects arrows of $M$ with chosen (coherent) solutions to any lifting problem against an object of $J$.

### Features of algebraic model structures

We can now introduce algebraic model structures

**Definition.** An *algebraic model structure* on a complete and cocomplete category $M$ with a class of weak equivalences ${W}$ satisfying the 2-of-3 property consists of a pair of nwfs **(C _{t},F)** and

**(C,F**together with a morphism of nwfs $\xi \colon$

_{t})**(C**$\rightarrow$

_{t},F)**(C,F**which we call the

_{t})*comparison map*such that the underlying wfs $(\overline{ {C}_t},\overline{ {F}})$ and $(\overline{ {C}}, \overline{ {F}_t})$ form a model structure with weak equivalences ${W}$

If we let $R, Q \colon M^{2} \rightarrow M$ be the functors accompanying the functorial factorizations of **(C _{t},F)** and

**(C,F**, respectively, then the comparison map provides a natural arrow

_{t})$\array{ & \text{dom} f & \\ {}^{C_{t}f}\swarrow & & \searrow {}^{C f} \\ R f & \stackrel{\xi_f}{\to} & Q f \\ {}_{F f}\searrow & & \swarrow {}_{F_{t}f} \\ & \text{cod} f & }$

A morphism of nwfs must also satisfy two pentagons, which say that $\xi$ induces functors

$\quad \xi_* \colon$