### 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

$\xi_\ast \colon\mathbf{C_t}$**-coalg** $\rightarrow$ **C-coalg**

$\xi^\ast \colon \mathbf{F_t}$**-alg** $\to$ **F-alg**

that provide an algebraic way of regarding a trivial cofibration/fibration as a cofibration/fibration. Naturality says that the two natural ways of solving a lifting problem of a trivial cofibration against a trivial fibration (apply $\xi_\ast$ and use **(C,F _{t})** or apply $\xi^\ast$ and use

**(C**) produce the same lifts.

_{t},F)By the universal property of Garner’s small object argument, cofibrantly generated algebraic model structures exist whenever the underlying ordinary model structures are cofibrantly generated; we’ll see below that these aren’t the only examples.

Any algebraic model structure has an induced fibrant replacement monad **R** obtained from **F** by slicing over the terminal object. A cofibrant replacement comonad **Q** is obtained from **C** dually. As in the classical setting, one can set up lifting problems to obtain an arrow $R Q X \rightarrow Q R X$ comparing the two bifibrant replacements of an object $X$. Here, the comparisons fit together to give a natural transformation $\chi\colon R Q \Rightarrow Q R$.

$\array{ & \emptyset & \\& \swarrow {} \searrow& \\ Q X & \stackrel{Q\eta_X}{\to} & Q R X \\ {}^{\eta_{Q X}}\downarrow & {}^{\exists \chi_X}\nearrow & \downarrow^{\epsilon_{RX}} \\ R Q X & \stackrel{R\epsilon_X}{\to} & R X\\ & \searrow {} \swarrow &\\ & \ast & }$

Furthermore, $\chi$ is a distributive law of the monad over the comonad, which says that **Q** lifts to a comonad on **R-alg**, the category of *algebraically fibrant objects*, and **R** lifts to a monad on **Q-coalg**. The categories of (co)algebras for these lifted (co)monads are isomorphic and give a category of *algebraically bifibrant objects*. By recent work of Thomas Nikolaus and the theorems described in the next section, either Quillen’s or Joyal’s model structure of simplicial sets can be lifted to an algebraic model structure on the appropriate category of algebraically fibrant objects in such a way that the canonical (monadic) adjunction **sSet** $\leftrightarrow$ **R-alg** is an algebraic Quillen adjunction, defined below.

We’ll turn to these mysterious algebraic Quillen adjunctions in a moment, but first let’s highlight a few theorems which illustrate some interesting aspects of this theory.

Unlike ordinary wfs, a nwfs on $M$ induces a levelwise nwfs on any diagram category $M^{A}$, where $A$ is small. A bit surprisingly, the levelwise nwfs is cofibrantly generated whenever the inducing one is. One example where these factorizations appear in Steve Lack’s trivial model structures on certain diagram 2-categories, and although he proves that these are not cofibrantly generated in the classical sense, we have:

**Theorem.** Lack’s trivial model structure on the 2-category **Cat**${}^{A}$ is a cofibrantly generated algebraic model structure.

A second theorem, which is joint work with Richard Garner and Mike Shulman, deserves mention both for the unusual result and also for its proof.

** Theorem.** If $M$ has an algebraic model structure generated by categories $J$ and $I$ which come with a functor $J \rightarrow I$ over $M^{2}$ and if the cofibrations are monomorphisms, then the components of the comparison map $\xi$ are **C**-coalgebras.

In the discrete case, the first condition means simply that we include the generating trivial cofibrations in the set of generating cofibrations, which can of course be done. The point of the proof is to exploit the fact that the forgetful functor **C-coalg**$\rightarrow M^{2}$ creates colimits. The components of the comparison map are constructed via a colimiting process, following the steps in the small object argument. We can inductively show that they inherit a canonical **C**-coalgebra structure by checking that the objects in the colimiting diagram are themselves coalgebras and the morphisms are maps of such. We cannot see how a result such as this, which after all refers to morphisms present in the underlying ordinary model category, could be proven in the non-algebraic theory.

As a corollary, the fibrant replacement monad constructed via the small object argument in this context preserves algebraic trivial cofibrations.

### Algebraic Quillen adjunctions

Many ordinary model structures are constructed by passing a previously known cofibrantly generated model structure across an adjunction. This classical theorem can be extended to algebraic model structures.

**Theorem.** Suppose $T \colon M \leftrightarrow K \colon S$ is an adjunction with $T \dashv S$ and $M$ has a algebraic model structure generated by $J$ and $I$. If $S$ takes the left class of the underlying wfs generated by $TJ$ into the weak equivalences of $M$, then $TJ$ and $TI$ generate an algebraic model structure on $K$ with weak equivalences created by $S$.

The proof relies heavily upon this fact: when a nwfs **(C,F)** is cofibrantly generated, all arrows in the right class of the underlying wfs admit an **F**-algebra structure. This is because **F-alg**$\cong {J}^{\square}$ is retract closed.

As in the classical setting, any such adjunction is automatically a Quillen adjunction, meaning the right adjoint preserves (trivial) fibrations and the left adjoint preserves (trivial) cofibrations. Here, algebraic analogs of these properties hold: the left adjoint lifts to functors **C _{t}-coalg**$\rightarrow$

**L**,

_{t}-coalg**C-coalg**$\rightarrow$

**L-coalg**, and the right adjoint lifts to functors

**R**$\rightarrow$

_{t}-alg**F**,

_{t}-alg**R-alg**$\rightarrow$

**F-alg**, where

**C**and

**F**refer to the algebraic model structure on $M$ and

**L**and

**R**refer likewise to $K$.

We won’t say much about the proofs here, except that they’re rather tricky (at least when it comes to the left adjoints). A direct computational proof is possible, but a better idea, which was suggested by Richard Garner, uses an alternate characterization of a nwfs as a category of algebras for a monad over the functor cod$\colon M^{2} \rightarrow M$ equipped with a composition law that is natural in a suitable double categorical sense. We use this characterization to show that the natural transformations characterizing the lifts of $S$ (such a pair is sometimes called a *lax morphism of monads*) are compatible with the comultiplication for the comonads. Then, their mates will automatically be *colax morphisms of comonads*, which is to say, they specify the desired lifts of $T$.

At this point, we should note that we’ve given a name to describe this situation.

**Definition.** An *adjunction of nwfs* $(T,S,\gamma,\rho) \colon$ **(C,F)** $\rightarrow$ **(L,R)** consists of a colax morphism of comonads $(T,\gamma) \colon$ **C-coalg** $\rightarrow$ **L-coalg** and a lax morphism of monads $(S,\rho) \colon$ **R-alg** $\rightarrow$ **F-alg** such that $T \colon M \leftrightarrow K \colon S$ with $T \dashv S$, with the first nwfs on $M$ and the second on $K$, and such that $\gamma$ and $\rho$ are mates.

**Examples.** An adjunction of nwfs $(1,1,\xi,\xi)$ is exactly a morphism of nwfs $\xi$. Less trivially, if **(C,F)** is generated by ${J}$ and **(L,R)** is generated by $T {J}$, then there is a canonical adjunction of nwfs as above.

Adjunctions of nwfs can canonically be composed. Thus, in the setting of the theorem above, there are (a priori) six algebraic Quillen adjunctions: one for the trivial cofibration-fibration nwfs on each category, one for the cofibration-trivial fibration nwfs, two for the comparison maps, and two adjunctions of nwfs **(C _{t},F)** $\rightarrow$

**(L,R**for the two possible composites. In fact, we might hope that these last two would be the same, which would say precisely that the lifts

_{t})$\array{ C_t-\text{coalg} & \stackrel{\tilde{T}}{\to} & L_t-\text{coalg} \\ {}^{(\xi^M)_\ast}\downarrow & & \downarrow^{(\xi^K)_\ast} \\ C-\text{coalg} & \stackrel{\tilde{T}}{\to} & L-\text{coalg}} \quad \quad \text{and} \quad \quad \array{ R_t-\text{alg} & \stackrel{\tilde{S}}{\to} & F_t-\text{alg} \\ {}^{(\xi^K)^\ast}\downarrow & & \downarrow^{(\xi^M)^\ast} \\ R-\text{alg} & \stackrel{\tilde{S}}{\to} & F-\text{alg}}$

commute. This leads to the following definition.

**Definition.** An *algebraic Quillen adjunction* $T \colon M \leftrightarrow {K} \colon S$ consists of five adjunctions of nwfs

$\array{ (C_t,F) & \stackrel{(T,S,\gamma_t,\rho)}{\to} & (L_t,R) \\ {}^{\mathllap{(1,1,\xi^{M},\xi^{M})}}\downarrow & \searrow {}^{(T,S,\gamma\cdot T\xi^{M} , S\xi^{K}\cdot\rho)} & \downarrow^{\mathrlap{(1,1,\xi^{K},\xi^{K})}} \\ (C,F_t) & \stackrel{(T,S,\gamma,\rho_t)}{\to} & (L,R_t)}$

such that both triangles commute.

Significant examples exist, such as the adjunction between simplicial sets and spaces given by geometric realization and the total singular complex functor.

**Theorem.** The adjunction of the theorem above is canonically an algebraic Quillen adjunction.

To prove this result, it remains only to prove the naturality statement, which again requires a bit of work, the content of which may be of independent categorical interest. The crux of the difficulty is to show that the two canonical ways of assigning **L _{t}**-coalgebra structures to the objects of $T {J}$ are the same: one uses the canonical functor $T {J} \rightarrow$

**L-coalg**arising from small object argument on ${K}$ and the other uses the analogous functor for $M$ and then composes with the lift

**C**$\rightarrow$

_{t}-coalg**L**of $T$.

_{t}-coalgGarner’s small object argument constructs a reflection of small categories ${J}$ along the canonical forgetful functor

$\quad\quad$**(C _{t},F)**$\mapsto$

**C**$\colon$

_{t}-coalg**NWFS**$(M) \rightarrow$

**CAT**/$M^{2}$

The canonical functors $J \rightarrow$ **C _{t}-coalg** described above are the units. Garner proves that these functors are universal with respect to all morphisms of nwfs. We enlarge the categories above to give a forgetful functor

$\quad\quad$**(C _{t},F)**$\mapsto$

**C**$\colon$

_{t}-coalg**NWFS**${}_{ladj} \rightarrow$

**CAT**$/(-)^{2}_{ladj}$

from the category of nwfs and adjuntions of nwfs to the category of categories sliced over arrow categories, whose morphisms are left adjoints of specified adjunctions between the base categories together with lifts to the fibers. We prove that Garner’s construction produces a reflection along this forgetful functor; in other words, that the units are universal with respect to adjunctions of nwfs, which is precisely what is needed to prove the theorem above.

A more leisurely and much more detailed account of this material can be found here. I am still searching applications that will really justify this theory, but this algebraization seems quite natural from the categorical point of view, so I am hopeful that they will be found.

## Re: Algebraic model structures

Nice.

Maybe we now would like to make the following step:

we have from Thomas a notion of algebraic $(\infty,1)$-category (i.e. a fibrant object in $Alg sSet_{Joyal}$) and now from you one of

algebraic model structure. So the obvious next demand seems to be to show that algebraic model structures are models (in some direct sense) for algebraic $(\infty,1)$-categories.Do you know how to say

algebraically enriched algebraic model structure? I imagine it would be useful to now have a notion of $Alg sSet_{Quillen}$-enriched algebraic model structures.And then to redo all the standard theory about how model categories present $(\infty,1)$-categories for algebraic model structures and algebraic $(\infty,1)$-categories.