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 of algebraic fibrant objects in a model category forms itself naturally a model category which is Quillen equivalent to , and in which all objects are fibrant. Notably, is always a category of fibrant objects.
Let be a cofibrantly generated model category such that
Choose a set of acyclic cofibrations such that
an object is fibrant precisely if it has the right lifting property against these morphisms.
An algebraic fibrant object in is a fibrant object of together with a choice of lifts (“fillers”) in
for each morphism , .
Write for the category whose objects are algebraic fibrant objects in and whose morphisms are morphisms in that respect the chosen lifts.
The set 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.
induces the transferred model structure on : the fibrations and weak equivalences in are those of the underlying morphisms in .
In particular, every object in is fibrant.
This is (Nikolaus, theorem 2.18)
Since fibrations in are created in , and any algebraically fibrant object is, in particular, fibrant, every object in the model category is fibrant. Thus almost any model category is equivalent to one in which all objects are fibrant. However, in general not all objects in will be cofibrant, even if this was true in 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 .
For define a new object inductively as follows.
is the pushout (in )
here the disjoint unition is taken over all and all morphisms .
This pushout adds all possible “horn” fillers to , given by the . However, after adding the new fillers there may also appear new horns. So we continue this procedure iteratively by filling these new horns.
Since is an acyclic cofibration in , so is its its pushout, hence by assumption the pushout is a monomorphism. This implies that if an factors through then it does so uniquely.
Let be the pushout
where now the coproduct is over those morphisms that do not factor throuth .
Proceeding this way yields a sequence of acyclic cofibrations
be the colimit over this sequence.
The object inherits canonical fillers: since by all are small objects we have that every morphism factors through one of the intermediate . Then for any of the morphisms in the corresponding morphism provides a filler.
The unit of this adjunction is the inclusion given by the above construction and is hence a weak equivalence.
Since by assumption has all fillers we have commuting diagrams
By the universality of the pushout these induce unique morphisms
By induction this induces morphisms that form a cocone under . Therefore there is a unique morphusm as indicated.
The morphisms have retracts.
being the unit of the adjunction . Take to be the (unique) filler for this morphism.
We now give a formal justification for calling the objects of algebraic objects by showing that is the category of algebras over an algebraic theory in , or more precisely that it is the category of algebra over the monad on
The functor 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 is a morphism in such that has an inverse , then it follows that must preserve fillers, hence itself constitutes the underlying morphism of an inverse if in .
For the third item, let be two morphisms in such that the coequalizer
exists in , together with sections of and of such that . We need to equip with fillers such that it becomes the coequalizer of in .
For a horn, declare that its filler is the the chosen filler of
We check now that this choice indeed makes the coequalizer in .
First of all we need to check that preserves the chosen fillers.
So given a filler
it is sent by to the filler
of . This is supposed to be the same as
Now use the relation to rewrite the first diagram equivalently as
and the relation to rewrite the second diagram equivalently as
This show that the fillers and are images under and , respectively, of a single filler in . Since coequalizes, it follows that .
Finally we need to show that the coequalizing morphism in thus established is indeed universal with this property.
For a coequalizing morphism in , the fact that is a coequalizer in already gives a unique morphism in
So it is sufficient to observe that preserves chosen fillers. But this is true be the above construction, which says that the chosen fillers in are the images of the chosen fillers in .
We discuss the proof for a slightly simpler statement, from which the full statement follows easily.
Let be an algebraic fibrant object, an object in and
a morphism in (really: ). Then there is an algebraic fibrant object and a morphism such that the composite
is a morphism of algebraic fibrant objects and it is initial with this property:
for every morphism in with an algebraic fibrant object such that the composite preserves distinguished fillers, there exists a unique morphism such that we have a commutative diagram
Moreover, if is a monomorphism in , then is an acyclic cofibration.
This is (Nikolaus, prop. 2.6).
The naive idea would be to equip with distinguished fillers
for each distinguished filler . But since the composite may factor through in many ways, this will not give a unique notion of filler. Therefore we shall iteratetively form colimits that equate these potentially different fillers.
Let be the set of morphisms of the form that factor through . For each let be the set of images of distinuished fillers .
where on the right we take the colimit over the diagram with one object per morphism that factors through , and one morphism per induced distinguished filler obtained from a choice of factorization through .
This comes with a canonica cocone-morphism . Continue this way to build by coequalizing the different ways morphisms into may factor through , etc. to obtain a sequence
The object 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 . But we start with letting be the pushout
where now the coproduct is over all morphisms that do not factor through . After that we proceed exactly as we did for the construction for and obtain a sequence
Finally we set
One checks that this has the claimed properties (…).
As before, the inclusion is an acyclic cofibration. Hence by the above remarks in the case that is a monomorphism also is an acyclic cofibration.
By replacing in this proof factorization through a single by factorization through a family of s one finds that is a solid functor.
Consider first limits: for a diagram and its limit in , let be a horn. Since all the composite maps have distinguished fillers and since the morphisms between these respect these fillers, we get a cone of fillers with tip over . The universal cone morphism provides then a distinguished filler for the limit. By the same argument, any cone in whose underlying cone in is a limiting cone is also limiting in .
Now consider filtered colimits. By assumption the domains are small objects. Therefore for a filtered diagram and its filtered colimit, any morphism factors through one of the . This provides a filler . Observe that while neither nor the filler are uniquely determined, its image in is. This makes an object in . By the same argument one finds that it is the universal cocone under in .
Now for a general diagram with colimit in , we use the above fact that is a solid functor to deduce that is the colimit of in .
While this gives a general prescription for computing colimits, the following lemma asserts a slightly simpler way for computing certain pushouts in . This will be needed for establishing the model category structure.
Let be a morphism in and
a diagram in . Then
its pushout in is , where is the canonical morphism;
if is an acyclic cofibration then so is .
This is (Nikolaus, prop 2.14).
First check that has the universal property of the pushout: for any we have by the above proposition that morphisms are in bijection to morphisms in such that the composition
preserves distinguished fillers.
To show that is an acyclic cofibration of is, recall that this morphism is the composite
Here the first morphism is an acyclic cofibration because it is the pushout of the acyclic cofibration . Therefore by assumption is a monomorphism and by the remarks in the solidity lemma the morphism is an acyclic cofibration. Hence so is the composite of the two.
This appears as (Nikolaus, def. 2.15).
This follows from the above result that 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 above, using the assumption that all acyclic cofibrations in are monomorphisms.
We spell out the argument, following (Nikolaus).
By the conditions listed at transferred model structure it is sufficient to check that
The first item is true by the above discussion of filtered colimits in .
That the second item holds follows from the above pushout lemma which asserts that -images of pushouts of -images of generating acyclic cofibrations are acylic cofibrations, and again by the fact that preserves filtered colimits, which implies that it preserves the transfinite composition of these acyclic cofibrations.
The functor preserves acyclic cofibrations.
This appears as (Nikolaus, theorem 2.18)
We observe that the unit and counit of are both weak equivalences:
the unit of the adjunction is 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 is a weak equivalence.
This implies that induces an equivalence of categories on the homotopy categories. Since equals its total right derived functor (since every object in is fibrant) this means that is a Quillen equivalence.
If is a locally presentable category then so is .
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 is an accessible functor, hence that it preserves filtered colimits. Now is a left adjoint and hence preserves all colimits, while we checked above that preserves filtered colimits.
If is a combinatorial model category then so is .
The standard model structure on simplicial sets models the (∞,1)-category ∞Grpd of ∞-groupoids: its fibrant objects are precisely the Kan complexes. But a Kan complex is a model for an -groupoid in which composites and inverses of k-morphisms are only guaranteed to exist, but are not specifically chosen .
An algebraic fibrant object in 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
Similarly, the model structure for quasi-categories models (∞,1)-categories: its fibrant objects are precisely the quasi-categories: an algebraic quasi-category. Again, these form a model for -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 -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 is along the lines of the discussion in