model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
An object in a model category is fibrant if all morphisms into it have extensions along acyclic cofibrations. An algebraic fibrant object is a fibrant object equipped with a choice of such extensions.
Under mild conditions, the category $Alg C$ of algebraic fibrant objects in a model category $C$ forms itself naturally a model category which is Quillen equivalent to $C$, and in which all objects are fibrant. Notably, $Alg C$ is always a category of fibrant objects.
Let $C$ be a cofibrantly generated model category such that
Choose a set $\{A_j \to B_j\}_{j \in J}$ of acyclic cofibrations such that
the domains $A_j$ are small objects;
an object $X \in C$ is fibrant precisely if it has the right lifting property against these morphisms.
An algebraic fibrant object in $C$ is a fibrant object of $C$ together with a choice of lifts (“fillers”) $\sigma_j : B_j \to X$ in
for each morphism $A_j \to X$, $j \in J$.
Write $Alg C$ for the category whose objects are algebraic fibrant objects in $C$ and whose morphisms are morphisms in $C$ that respect the chosen lifts.
The set $J$ can be taken to be that of all generating acyclic cofibrations, if their domains are small. But often there are smaller subsets that still characterize all fibrant objects.
The forgetful functor adjunction
induces the transferred model structure on $Alg C$: the fibrations and weak equivalences in $Alg C$ are those of the underlying morphisms in $C$.
In particular, every object in $Alg C$ is fibrant.
The Quillen adjunction $(F \dashv U)$ is a Quillen equivalence.
This is (Nikolaus, theorem 2.20)
Since fibrations in $Alg C$ are created in $C$, and any algebraically fibrant object is, in particular, fibrant, every object in the model category $Alg C$ is fibrant. Thus almost any model category is equivalent to one in which all objects are fibrant. However, in general not all objects in $Alg C$ will be cofibrant, even if this was true in $C$ itself.
We spell out the constructions and lemmas that yield the theorem on the model structure. In the course of this we also discuss a few results of interest in their own right, such as the monadicity, local presentability and combinatoriality of $Alg C$.
We describe explicitly the left adjoint $F : C \to Alg C$ to the forgetful functor $U : Alg C \to C$. This follows Richard Garner‘s improved version of the small object argument (Garner).
For $X \in C$ define a new object $X_\infty$ inductively as follows.
$X_1$ is the pushout (in $C$)
here the disjoint unition is taken over all $j$ and all morphisms $A_j \to X$.
This pushout adds all possible “horn” fillers to $X$, given by the $B_j \to X_1$. However, after adding the new fillers there may also appear new horns. So we continue this procedure iteratively by filling these new horns.
Since $\coprod A_j \to \coprod B_j$ is an acyclic cofibration in $C$, so is its its pushout, hence by assumption the pushout $X \to X_1$ is a monomorphism. This implies that if an $A_j \to X_1$ factors through $X \to X_1$ then it does so uniquely.
Let $X_2$ be the pushout
where now the coproduct is over those morphisms $A_j \to X$ that do not factor throuth $X \to X_1$.
Proceeding this way yields a sequence of acyclic cofibrations
We set
be the colimit over this sequence.
The object $X_\infty$ inherits canonical fillers: since by all $A_j$ are small objects we have that every morphism $A_j \to X_\infty$ factors through one of the intermediate $X_i$. Then for any of the morphisms $A_j \to B_j$ in $J$ the corresponding morphism $B_j \to X_{i+1} \to X_\infty$ provides a filler.
The assignment $F : X \to X_\infty$ with $X_\infty$ regarded as an algebraic fibrant objects as above constitutes a left adjoint functor $F : C \to Alg C$ to the forgetful functor $U : Alg C \to C$.
The unit of this adjunction is the inclusion $X \to X_\infty$ given by the above construction and is hence a weak equivalence.
We establish the hom-isomorphism of the adjunction and its construction by the unit of the adjunction: For any $Z \in Alg C$ and $\phi : X \to U(Z)$ any morphism in $C$, we need to show that there is a unique commuting diagram
Since by assumption $Z$ has all fillers we have commuting diagrams
By the universality of the pushout these induce unique morphisms
By induction this induces morphisms $\phi_n : X_n \to Z$ that form a cocone under $(X \to X_1 \to X_2 \to \cdots )$. Therefore there is a unique morphusm $\phi_\infty$ as indicated.
The morphisms $F(j) : F A_j \to F B_j$ have retracts.
Since $F$ is left adjoint to $U$ the condition that $F A_j \to F B_j \stackrel{r}{\to} F A_j$ is the identity is equivalent to its adjunct, the diagonal in
being the unit of the adjunction $A_j \to U F A_j$. Take $\tilde r$ to be the (unique) filler for this morphism.
We now give a formal justification for calling the objects of $Alg C$ algebraic objects by showing that $Alg C$ is the category of algebras over an algebraic theory in $C$, or more precisely that it is the category of algebra over the monad $F \circ U$ on $C$
The functor $U : Alg C \to C$ is monadic.
This is (Nikolaus, prop. 24)
By one of the equivalent statements of Beck's monadicity theorem we have to check that
The first item we already checked above.
For the second item it is sufficient to observe that if $f : X \to Y$ is a morphism in $Alg C$ such that $U f$ has an inverse $g : U Y \to U X$, then it follows that $g$ must preserve fillers, hence itself constitutes the underlying morphism of an inverse if $f$ in $Alg C$.
For the third item, let $f,g : X \to Y$ be two morphisms in $Alg C$ such that the coequalizer
exists in $C$, together with sections $s$ of $\pi$ and $t$ of $U(f)$ such that $s \circ \pi = U(t) \circ t$. We need to equip $Q$ with fillers such that it becomes the coequalizer of $f,g$ in $Alg C$.
For $h : A_j \to Q$ a horn, declare that its filler is the the chosen filler of $s \circ h$
We check now that this choice indeed makes $Q$ the coequalizer in $Alg C$.
First of all we need to check that $\pi$ preserves the chosen fillers.
So given a filler
it is sent by $\pi$ to the filler
of $\pi k$. This is supposed to be the same as
Now use the relation $U(f) t = Id$ to rewrite the first diagram equivalently as
and the relation $U(g) t = s \pi$ to rewrite the second diagram equivalently as
This show that the fillers $\hat k$ and $\hat r$ are images under $U(f)$ and $U(g)$, respectively, of a single filler in $X$. Since $\pi$ coequalizes, it follows that $\pi \hat k = \pi \hat r$.
Finally we need to show that the coequalizing morphism $\pi$ in $Alg C$ thus established is indeed universal with this property.
For $\phi : Y \to Z$ a coequalizing morphism in $Alc C$, the fact that $Q$ is a coequalizer in $C$ already gives a unique morphism $\phi_Q$ in
So it is sufficient to observe that $\phi_Q$ preserves chosen fillers. But this is true be the above construction, which says that the chosen fillers in $Q$ are the images of the chosen fillers in $Y$.
The forgetful functor $U : Alg C \to C$ is a solid functor.
We discuss the proof for a slightly simpler statement, from which the full statement follows easily.
Let $Y$ be an algebraic fibrant object, $X$ an object in $C$ and
a morphism in $C$ (really: $U(Y) \to X$). Then there is an algebraic fibrant object $X^f_\infty$ and a morphism $X \to X^f_\infty$ such that the composite
is a morphism of algebraic fibrant objects and it is initial with this property:
for every morphism $\phi : X \to Z$ in $C$ with $Z$ an algebraic fibrant object such that the composite $Y \to X \to Z$ preserves distinguished fillers, there exists a unique morphism $\phi_\infty : X^f_\infty \to Z$ such that we have a commutative diagram
Moreover, if $f$ is a monomorphism in $C$, then $X \to X^f_\infty$ is an acyclic cofibration.
This is (Nikolaus, prop. 2.6).
The naive idea would be to equip $X$ with distinguished fillers $f \hat k$
for each distinguished filler $\hat k$. But since the composite may factor through $Y$ in many ways, this will not give a unique notion of filler. Therefore we shall iteratively form colimits that equate these potentially different fillers.
Let $H$ be the set of morphisms of the form $h : A_j \to X$ that factor through $f : Y \to X$. For each $h \in H$ let $F_h$ be the set of images of distinuished fillers $B_j \to X$.
Set
where on the right we take the colimit over the diagram with one object $(B_j)_h$ per morphism $h : A_j \to X$ that factors through $Y$, and one morphism $(B_j)_h \to X$ per induced distinguished filler obtained from a choice of factorization through $Y$.
This comes with a canonica cocone-morphism $X \to X_H$. Continue this way to build $X_{H'}$ by coequalizing the different ways morphisms into $X_H$ may factor through $X$, etc. to obtain a sequence
and set
Notice that if $f$ is a monomorphism then all factorizations are unique and hence $X \to X_0^f$ is an isomorphism.
The object $X_0^f$ can be seen to have the desired universal factorization property. But it may not yet be itself algebraically fibrant. So we conclude by essentially applying the construction of the left adjoint $F$. But we start with letting $X_1^f$ be the pushout
where now the coproduct is over all morphisms that do not factor through $Y$. After that we proceed exactly as we did for the construction for $F$ and obtain a sequence
Finally we set
One checks that this has the claimed properties (…).
As before, the inclusion $X_0^f \to X_\infty^f$ is an acyclic cofibration. Hence by the above remarks in the case that $f$ is a monomorphism also $X \to X_\infty^f$ is an acyclic cofibration.
By replacing in this proof factorization through a single $Y$ by factorization through a family of $Y$s one finds that $U$ is a solid functor.
The category $Alg C$ has all limits and colimits in $C$.
The limits and the filtered colimits (but not the general colimit)s are created by $U$.
Consider first limits: for $K : J \to Alg C$ a diagram and $\lim_{\leftarrow} U K$ its limit in $C$, let $A_j \to \lim_{\leftarrow} U K$ be a horn. Since all the composite maps $A_J \to \lim_{\leftarrow} U K \to U K(j)$ have distinguished fillers and since the morphisms between these respect these fillers, we get a cone of fillers with tip $B_j$ over $U K$. The universal cone morphism $B_j \to \lim_{\leftarrow} U K$ provides then a distinguished filler for the limit. By the same argument, any cone $const_T \to \lim_{\leftarrow} U K$ in $Alg_C$ whose underlying cone in $C$ is a limiting cone is also limiting in $Alg C$.
Now consider filtered colimits. By assumption the domains $A_j$ are small objects. Therefore for $K : I \to C$ a filtered diagram and $L := \lim_{\to} K$ its filtered colimit, any morphism $A_j \to L$ factors through one of the $K_i$. This provides a filler $B_j \to K_i \to L$. Observe that while neither $i$ nor the filler $B_j \to K_i$ are uniquely determined, its image in $L$ is. This makes $L$ an object in $Alg C$. By the same argument one finds that it is the universal cocone under $K$ in $Alg C$.
Now for $K : J \to Alg C$ a general diagram with colimit $X := \lim_\to U K$ in $C$, we use the above fact that $U$ is a solid functor to deduce that $X_\infty^f$ is the colimit of $K$ in $Alg C$.
While this gives a general prescription for computing colimits, the following lemma asserts a slightly simpler way for computing certain pushouts in $Alg C$. This will be needed for establishing the model category structure.
Let $i : A \to B$ be a morphism in $C$ and
a diagram in $Alg C$. Then
its pushout in $Alg C$ is $(B \coprod_A U Y)_\infty^f$, where $f : U Y \to B \coprod_A U Y$ is the canonical morphism;
if $i : A \to B$ is an acyclic cofibration then so is $Y\to (B \coprod_A U Y)_\infty^f$.
This is (Nikolaus, prop 2.14).
First check that $(B \coprod_A U Y)_\infty^f$ has the universal property of the pushout: for any $Z \in Alg C$ we have by the above proposition that morphisms $(B \coprod_A U Y)_\infty^f \to Z$ are in bijection to morphisms $B \coprod_A U Y \to U Z$ in $C$ such that the composition
preserves distinguished fillers.
This in turn is the same as a morphism $g_1 : Y \to Z$ in $Alg C$ and a morphism $g_2 : B \to U Z$ in $C$ that agree on $A$. By adjunction $(F \dashv U)$ this corresponds to an adjunct $\tilde g_2: F B \to Z$. This establishes the pushout property of $(B \coprod_A U Y)_\infty^f$.
To show that $Y \to (B \coprod_A U Y)_\infty^f$ is an acyclic cofibration of $i : A \to B$ is, recall that this morphism is the composite
Here the first morphism $f$ is an acyclic cofibration because it is the pushout of the acyclic cofibration $i$. Therefore by assumption $f$ is a monomorphism and by the remarks in the solidity lemma the morphism $B \coprod_A U Y \to (B \coprod_A U Y)_\infty^f$ is an acyclic cofibration. Hence so is the composite of the two.
Under the above conditions, $Alg C$ becomes a model category with the $U$-transferred model structure: weak equivalences and fibrations in $Alg C$ are those morphisms that are so in $C$.
This appears as (Nikolaus, def. 2.15).
This follows from the above result that $U$ is a solid functor as described in detail at solid functor by observing that the acyclicity condition required there follows as in the discussion of the solidity of $U$ above, using the assumption that all acyclic cofibrations in $C$ are monomorphisms.
We spell out the argument, following (Nikolaus).
By the conditions listed at transferred model structure it is sufficient to check that
$U$ preserves small object, which is the case in particular if it preserves filtered colimits;
$U$ sends sequential colimits of pushouts of images under $F$ of generating acyclic cofibrations to weak equivalences.
The first item is true by the above discussion of filtered colimits in $Alg C$.
That the second item holds follows from the above pushout lemma which asserts that $U$-images of pushouts of $F$-images of generating acyclic cofibrations are acylic cofibrations, and again by the fact that $U$ preserves filtered colimits, which implies that it preserves the transfinite composition of these acyclic cofibrations.
The functor $U$ preserves acyclic cofibrations.
is a Quillen equivalence.
This appears as (Nikolaus, theorem 2.18)
We observe that the unit and counit of $(F \dashv U)$ are both weak equivalences:
the unit of the adjunction is $X \to X_\infty$ is by the above construction a transfinite composition of acyclic cofibrations (in fact a fibrant replacement) hence in particular a weak equivalence. Moreover, the unit is a section of the counit in that
Hence by 2-out-of-3 also $X \to F X$ is a weak equivalence.
This implies that $(F \dashv U)$ induces an equivalence of categories on the homotopy categories. Since $R$ equals its total right derived functor (since every object in $Alg C$ is fibrant) this means that $(F \dashv)$ is a Quillen equivalence.
If $C$ is a locally presentable category then so is $Alg C$.
By the above we know that $Alg C$ is the category of algebras over a monad $U\circ F : C \to C$.
We invoke the theorem on local presentability of categories of algebras discussed at locally presentable category. This says that it is sufficient to check that $U \circ F$ is an accessible functor, hence that it preserves filtered colimits. Now $F$ is a left adjoint and hence preserves all colimits, while we checked above that $U$ preserves filtered colimits.
If $C$ is a combinatorial model category then so is $Alg C$.
The standard model structure on simplicial sets $sSet_{Quillen}$ models the (∞,1)-category ∞Grpd of ∞-groupoids: its fibrant objects are precisely the Kan complexes. But a Kan complex is a model for an $\infty$-groupoid in which composites and inverses of k-morphisms are only guaranteed to exist, but are not specifically chosen .
An algebraic fibrant object in $sSet_{Quillen}$ is a Kan complex with a chosen filler for each horn: an algebraic Kan complex (see simplicial T-complex for a related but much stricter notion). This means precisely that all possible composites and all possible inverses are chosen. So the Quillen equivalence
induces an equivalence from an algebraic definition of ∞-groupoids to a geometric definition.
Similarly, the model structure for quasi-categories $sSet_{Joyal}$ models (∞,1)-categories: its fibrant objects are precisely the quasi-categories: an algebraic quasi-category. Again, these form a model for $(\infty,1)$-categories in which composition is only a relation, not an operation.
But equipping a quasi-category with the structure of an algebraic fibrant object precisely means choosing such composites. Accordingly, the Quillen equivalence
establishes an equivalence of an algebraic with the standard geometric model for $(\infty,1)$-categories.
The model structure on algebraic fibrant objects was introduced and discussed in
A survey is in
The refined small object argument that is being used in the construction of the left adjoint $F : C \to Alg C$ is along the lines of the discussion in