model structure on algebraic fibrant objects


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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 AlgCAlg C of algebraic fibrant objects in a model category CC forms itself naturally a model category which is Quillen equivalent to CC, and in which all objects are fibrant. Notably, AlgCAlg C is always a category of fibrant objects.



Let CC be a cofibrantly generated model category such that

Choose a set {A jB j} jJ\{A_j \to B_j\}_{j \in J} of acyclic cofibrations such that

An algebraic fibrant object in CC is a fibrant object of CC together with a choice of lifts (“fillers”) σ j:B jX\sigma_j : B_j \to X in

A j X σ j B j \array{ A_j &\to& X \\ \downarrow & \nearrow_{\mathrlap{\sigma_j}} \\ B_j }

for each morphism A jXA_j \to X, jJj \in J.

Write AlgCAlg C for the category whose objects are algebraic fibrant objects in CC and whose morphisms are morphisms in CC that respect the chosen lifts.


The set JJ 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

(FU):AlgCUFC (F \dashv U) : Alg C \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C

induces the transferred model structure on AlgCAlg C: the fibrations and weak equivalences in AlgCAlg C are those of the underlying morphisms in CC.

In particular, every object in AlgCAlg C is fibrant.

The Quillen adjunction (FU)(F \dashv U) is a Quillen equivalence.

This is (Nikolaus, theorem 2.18)


Since fibrations in AlgCAlg C are created in CC, and any algebraically fibrant object is, in particular, fibrant, every object in the model category AlgCAlg 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 AlgCAlg C will be cofibrant, even if this was true in CC 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 AlgCAlg C.

The left adjoint

We describe explicitly the left adjoint F:CAlgCF : C \to Alg C to the forgetful functor U:AlgCCU : Alg C \to C. This follows Richard Garner’s improved version of the small object argument (Garner).

For XCX \in C define a new object X X_\infty inductively as follows.


X 1X_1 is the pushout (in CC)

A j X B j X 1, \array{ \coprod A_j &\to& X \\ \downarrow && \downarrow \\ \coprod B_j &\to& X_1 } \,,

here the disjoint unition is taken over all jj and all morphisms A jXA_j \to X.


This pushout adds all possible “horn” fillers to XX, given by the B jX 1B_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 A jB j\coprod A_j \to \coprod B_j is an acyclic cofibration in CC, so is its its pushout, hence by assumption the pushout XX 1X \to X_1 is a monomorphism. This implies that if an A jX 1A_j \to X_1 factors through XX 1X \to X_1 then it does so uniquely.


Let X 2X_2 be the pushout

A j X 1 B j X 2, \array{ \coprod A_j &\to& X_1 \\ \downarrow && \downarrow \\ \coprod B_j &\to& X_2 } \,,

where now the coproduct is over those morphisms A jXA_j \to X that do not factor throuth XX 1X \to X_1.

Proceeding this way yields a sequence of acyclic cofibrations

XX 1X 2X 3. X \to X_1 \to X_2 \to X_3 \to \cdots \,.

We set

X :=lim (XX 1X 2X 3) X_\infty := \lim_\to (X \to X_1 \to X_2 \to X_3 \to \cdots )

be the colimit over this sequence.


The object X X_\infty inherits canonical fillers: since by all A jA_j are small objects we have that every morphism A jX A_j \to X_\infty factors through one of the intermediate X iX_i. Then for any of the morphisms A jB jA_j \to B_j in JJ the corresponding morphism B jX i+1X B_j \to X_{i+1} \to X_\infty provides a filler.


The assignment F:XX F : X \to X_\infty with X X_\infty regarded as an algebraic fibrant objects as above constitutes a left adjoint functor F:CAlgCF : C \to Alg C to the forgetful functor U:AlgCCU : Alg C \to C.

The unit of this adjunction is the inclusion XX 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 ZAlgCZ \in Alg C and ϕ:XU(Z)\phi : X \to U(Z) any morphism in CC, we need to show that there is a unique commuting diagram

X ϕ Z !ϕ X . \array{ X &\stackrel{\phi}{\to}& Z \\ \downarrow & \nearrow_{\exists ! \phi_\infty} \\ X_\infty } \,.

Since by assumption ZZ has all fillers we have commuting diagrams

A j X ϕ B j Z. \array{ A_j &\to& X \\ \downarrow && \downarrow^{\mathrlap{\phi}} \\ B_j &\to& Z } \,.

By the universality of the pushout these induce unique morphisms

X ϕ Z ϕ 1 X 1. \array{ X &\stackrel{\phi}{\to}& Z \\ \downarrow & \nearrow_{\phi_1 } \\ X_1 } \,.

By induction this induces morphisms ϕ n:X nZ\phi_n : X_n \to Z that form a cocone under (XX 1X 2)(X \to X_1 \to X_2 \to \cdots ). Therefore there is a unique morphusm ϕ \phi_\infty as indicated.


The morphisms F(j):FA jFB jF(j) : F A_j \to F B_j have retracts.


Since FF is left adjoint to UU the condition that FA jFB jrFA jF A_j \to F B_j \stackrel{r}{\to} F A_j is the identity is equivalent to its adjunct, the diagonal in

A j j B j r A j r˜ UFA j UF(j) UFB j UFr UFA j \array{ A_j &\stackrel{j}{\to}& B_j &\stackrel{r}{\to}& A_j \\ \downarrow && \downarrow &\searrow^{\mathrlap{\tilde r}}& \downarrow \\ U F A_j &\stackrel{U F (j)}{\to}& U F B_j &\stackrel{U F r}{\to} & U F A_j }

being the unit of the adjunction A jUFA jA_j \to U F A_j. Take r˜\tilde r to be the (unique) filler for this morphism.

Monadicity of the forgetful functor

We now give a formal justification for calling the objects of AlgCAlg C algebraic objects by showing that AlgCAlg C is the category of algebras over an algebraic theory in CC, or more precisely that it is the category of algebra over the monad FUF \circ U on CC


The functor U:AlgCCU : 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

  1. UU has a left adjoint,
  2. UU reflects isomorphisms (i.e. it is conservative), and
  3. DD has, and UU preserves, coequalizers of UU-split pairs.

The first item we already checked above.

For the second item it is sufficient to observe that if f:XYf : X \to Y is a morphism in AlgCAlg C such that UfU f has an inverse g:UYUXg : U Y \to U X, then it follows that gg must preserve fillers, hence itself constitutes the underlying morphism of an inverse if ff in AlgCAlg C.

For the third item, let f,g:XYf,g : X \to Y be two morphisms in AlgCAlg C such that the coequalizer

UXUgUfUYπQ U X \stackrel{\overset{U f}{\to}}{\underset{U g}{\to}} U Y \stackrel{\pi}{\to} Q

exists in CC, together with sections ss of π\pi and tt of U(f)U(f) such that sπ=U(t)ts \circ \pi = U(t) \circ t. We need to equip QQ with fillers such that it becomes the coequalizer of f,gf,g in AlgCAlg C.

For h:A jQh : A_j \to Q a horn, declare that its filler is the the chosen filler of shs \circ h

A j h Q s Id B j Y π Q. \array{ A_j &\stackrel{h}{\to}& Q \\ \downarrow && \downarrow^{\mathrlap{s}} & \searrow^{\mathrlap{Id}} \\ B_j &\to& Y &\stackrel{\pi}{\to} & Q } \,.

We check now that this choice indeed makes QQ the coequalizer in AlgCAlg C.

First of all we need to check that π\pi preserves the chosen fillers.

So given a filler

A j k Y k^ B j \array{ A_j &\stackrel{k}{\to}& Y \\ \downarrow & \nearrow_{\mathrlap{\hat k}} \\ B_j }

it is sent by π\pi to the filler

A j k Y π Q πk^ B j \array{ A_j &\stackrel{k}{\to}& Y &\stackrel{\pi}{\to}& Q \\ \downarrow & \nearrow && \nearrow_{\pi \hat k} \\ B_j }

of πk\pi k . This is supposed to be the same as

A j k Y π Q s Y π Q r^ πr^ B j. \array{ A_j &\stackrel{k}{\to}& Y &\stackrel{\pi}{\to}& Q &\stackrel{s}{\to}& Y &\stackrel{\pi}{\to}& Q \\ \downarrow && && & \nearrow_{\hat r} && \nearrow_{\pi \hat r} \\ B_j } \,.

Now use the relation U(f)t=IdU(f) t = Id to rewrite the first diagram equivalently as

A j k Y t X U(f) Y π Q g^ πk^ B j \array{ A_j &\stackrel{k}{\to}& Y &\stackrel{t}{\to}& X &\stackrel{U(f)}{\to}& Y &\stackrel{\pi}{\to}& Q \\ \downarrow &&&&& \nearrow_{\mathrlap{\hat g}} && \nearrow_{\pi \hat k} \\ B_j }

and the relation U(g)t=sπU(g) t = s \pi to rewrite the second diagram equivalently as

A j k Y t X U(g) Y π Q r^ πr^ B j. \array{ A_j &\stackrel{k}{\to}& Y &\stackrel{t}{\to}& X &\stackrel{U(g)}{\to}& Y &\stackrel{\pi}{\to}& Q \\ \downarrow && && & \nearrow_{\hat r} && \nearrow_{\pi \hat r} \\ B_j } \,.

This show that the fillers k^\hat k and r^\hat r are images under U(f)U(f) and U(g)U(g), respectively, of a single filler in XX. Since π\pi coequalizes, it follows that πk^=πr^\pi \hat k = \pi \hat r.

Finally we need to show that the coequalizing morphism π\pi in AlgCAlg C thus established is indeed universal with this property.

For ϕ:YZ\phi : Y \to Z a coequalizing morphism in AlcCAlc C, the fact that QQ is a coequalizer in CC already gives a unique morphism ϕ Q\phi_Q in

X gf Y π Q ϕ !ϕ Q Z. \array{ X & \stackrel{\overset{f}{\to}}{\underset{g}{\to}} & Y &\stackrel{\pi}{\to} & Q \\ && {}^{\mathllap{\phi}}\downarrow & \swarrow_{\mathrlap{\exists ! \phi_Q}} \\ && Z } \,.

So it is sufficient to observe that ϕ Q\phi_Q preserves chosen fillers. But this is true be the above construction, which says that the chosen fillers in QQ are the images of the chosen fillers in YY.

Solidity of the forgetful functor


The forgetful functor U:AlgCCU : 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 YY be an algebraic fibrant object, XX an object in CC and

f:YX f : Y \to X

a morphism in CC (really: U(X)YU(X) \to Y). Then there is an algebraic fibrant object X fX^f_\infty and a morphism XX fX \to X^f_\infty such that the composite

YXX f Y\to X \to X^f_{\infty}

is a morphism of algebraic fibrant objects and it is initial with this property:

for every morphism ϕ:XZ\phi : X \to Z in CC with ZZ an algebraic fibrant object such that the composite YXZY \to X \to Z preserves distinguished fillers, there exists a unique morphism ϕ :X fZ\phi_\infty : X^f_\infty \to Z such that we have a commutative diagram

Y f alg X ϕ Z !ϕ X f. \array{ Y \\ {}^{\mathllap{f}}\downarrow & \searrow^{\mathrlap{alg}} \\ X &\stackrel{\phi}{\to}& Z \\ \downarrow & \nearrow_{\mathrlap{\exists ! \phi_\infty}} \\ X^f_\infty } \,.

Moreover, if ff is a monomorphism in CC, then XX fX \to X^f_\infty is an acyclic cofibration.

This is (Nikolaus, prop. 2.6).

The naive idea would be to equip XX with distinguished fillers fk^f \hat k

A j k Y f X k^ fk^ B j \array{ A_j &\stackrel{k}{\to}& Y &\stackrel{f}{\to}& X \\ \downarrow & \nearrow_{\mathrlap{\hat k}} && \nearrow_{f \mathrlap{\hat k}} \\ B_j }

for each distinguished filler k^\hat k. But since the composite may factor through YY 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 HH be the set of morphisms of the form h:A jXh : A_j \to X that factor through f:YXf : Y \to X. For each hHh \in H let F hF_h be the set of images of distinuished fillers B jXB_j \to X.


X H:=lim ((B j) h (B j) h ϕF h ϕF h X) X_H := \lim_\to \left( \array{ (B_j)_h && (B_{j'})_{h'} && \cdots \\ & {}_{\mathllap{\phi \in F_h}}\searrow & \downarrow^{\mathrlap{\phi' \in F_{h'}}} & \cdots \\ && X } \right)

where on the right we take the colimit over the diagram with one object (B j) h(B_j)_h per morphism h:A jXh : A_j \to X that factors through YY, and one morphism (B j) hX(B_j)_h \to X per induced distinguished filler obtained from a choice of factorization through YY.

This comes with a canonica cocone-morphism XX HX \to X_H. Continue this way to build X HX_{H'} by coequalizing the different ways morphisms into X HX_H may factor through XX, etc. to obtain a sequence

XX HX HX H X \to X_H \to X_{H'} \to X_{H''} \to \cdots

and set

X 0 f:=lim (XX HX HX H). X_0^f := \lim_\to (X \to X_H \to X_{H'} \to X_{H''} \to \cdots) \,.

Notice that if ff is a monomorphism then all factorizations are unique and hence XX 0 fX \to X_0^f is an isomorphism.

The object X 0 fX_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 FF. But we start with letting X 1 fX_1^f be the pushout

jA j X 0 f jB j X 1 f \array{ \coprod_j A_j &\to& X_0^f \\ \downarrow && \downarrow \\ \coprod_j B_j &\to& X_1^f }

where now the coproduct is over all morphisms that do not factor through YY. After that we proceed exactly as we did for the construction for FF and obtain a sequence

X 0 fX 1 fX 2 f. X_0^f \to X_1^f \to X_2^f \to \cdots \,.

Finally we set

X f:=lim (X 0 fX 1 fX 2 f). X_\infty^f := \lim_\to(X_0^f \to X_1^f \to X_2^f \to \cdots) \,.

One checks that this has the claimed properties (…).

As before, the inclusion X 0 fX fX_0^f \to X_\infty^f is an acyclic cofibration. Hence by the above remarks in the case that ff is a monomorphism also XX fX \to X_\infty^f is an acyclic cofibration.

By replacing in this proof factorization through a single YY by factorization through a family of YYs one finds that UU is a solid functor.

Limits and colimits of algebraic objects


The category AlgCAlg C has all limits and colimits in CC.

The limits and the filtered colimits (but not the general colimit)s are created by UU.


Consider first limits: for K:JAlgCK : J \to Alg C a diagram and lim UK\lim_{\leftarrow} U K its limit in CC, let A jlim UKA_j \to \lim_{\leftarrow} U K be a horn. Since all the composite maps A Jlim UKUK(j)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 jB_j over UKU K. The universal cone morphism B jlim UKB_j \to \lim_{\leftarrow} U K provides then a distinguished filler for the limit. By the same argument, any cone const Tlim UKconst_T \to \lim_{\leftarrow} U K in Alg CAlg_C whose underlying cone in CC is a limiting cone is also limiting in AlgCAlg C.

Now consider filtered colimits. By assumption the domains A jA_j are small objects. Therefore for K:ICK : I \to C a filtered diagram and L:=lim KL := \lim_{\to} K its filtered colimit, any morphism A jLA_j \to L factors through one of the K iK_i. This provides a filler B jK iLB_j \to K_i \to L. Observe that while neither ii nor the filler B jK iB_j \to K_i are uniquely determined, its image in LL is. This makes LL an object in AlgCAlg C. By the same argument one finds that it is the universal cocone under KK in AlgCAlg C.

Now for K:JAlgCK : J \to Alg C a general diagram with colimit X:=lim UKX := \lim_\to U K in CC, we use the above fact that UU is a solid functor to deduce that X fX_\infty^f is the colimit of KK in AlgCAlg C.

While this gives a general prescription for computing colimits, the following lemma asserts a slightly simpler way for computing certain pushouts in AlgCAlg C. This will be needed for establishing the model category structure.


Let i:ABi : A \to B be a morphism in CC and

FA Y Fi FB \array{ F A &\to& Y \\ {}^{\mathllap{F i}}\downarrow \\ F B }

a diagram in AlgCAlg C. Then

  • its pushout in AlgCAlg C is (B AUY) f(B \coprod_A U Y)_\infty^f, where f:UYB AUYf : U Y \to B \coprod_A U Y is the canonical morphism;

  • if i:ABi : A \to B is an acyclic cofibration then so is Y(B AUY) fY\to (B \coprod_A U Y)_\infty^f.

This is (Nikolaus, prop 2.14).


First check that (B AUY) f(B \coprod_A U Y)_\infty^f has the universal property of the pushout: for any ZAlgCZ \in Alg C we have by the above proposition that morphisms (B AUY) fZ(B \coprod_A U Y)_\infty^f \to Z are in bijection to morphisms B AUYUZB \coprod_A U Y \to U Z in CC such that the composition

UYfB AUYUZ U Y \stackrel{f}{\to} B \coprod_A U Y \to U Z

preserves distinguished fillers.

This in turn is the same as a morphism g 1:YZg_1 : Y \to Z in AlgCAlg C and a morphism g 2:BUZg_2 : B \to U Z in CC that agree on AA. By adjunction (FU)(F \dashv U) this corresponds to an adjunct g˜ 2:FBZ\tilde g_2: F B \to Z. This establishes the pushout property of (B AUY) f(B \coprod_A U Y)_\infty^f.

To show that Y(B AUY) fY \to (B \coprod_A U Y)_\infty^f is an acyclic cofibration of i:ABi : A \to B is, recall that this morphism is the composite

YfB AUY(B AUY) f. Y \stackrel{f}{\to} B \coprod_A U Y \to (B \coprod_A U Y)_\infty^f \,.

Here the first morphism ff is an acyclic cofibration because it is the pushout of the acyclic cofibration ii. Therefore by assumption ff is a monomorphism and by the remarks in the solidity lemma the morphism B AUY(B AUY) fB \coprod_A U Y \to (B \coprod_A U Y)_\infty^f is an acyclic cofibration. Hence so is the composite of the two.

Transferred model structure


Under the above conditions, AlgCAlg C becomes a model category with the UU-transferred model structure: weak equivalences and fibrations in AlgCAlg C are those morphisms that are so in CC.

This appears as (Nikolaus, def. 2.15).


This follows from the above result that UU 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 UU above, using the assumption that all acyclic cofibrations in CC are monomorphisms.

We spell out the argument, following (Nikolaus).

By the conditions listed at transferred model structure it is sufficient to check that

  1. UU preserves small object, which is the case in particular if it preserves filtered colimits;

  2. UU sends sequential colimits of pushouts of images under FF of generating acyclic cofibrations to weak equivalences.

The first item is true by the above discussion of filtered colimits in AlgCAlg C.

That the second item holds follows from the above pushout lemma which asserts that UU-images of pushouts of FF-images of generating acyclic cofibrations are acylic cofibrations, and again by the fact that UU preserves filtered colimits, which implies that it preserves the transfinite composition of these acyclic cofibrations.


The functor UU preserves acyclic cofibrations.


The Quillen adjunction

(FU):AlgCUFC (F \dashv U) : Alg C \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C

is a Quillen equivalence.

This appears as (Nikolaus, theorem 2.18)


We observe that the unit and counit of (FU)(F \dashv U) are both weak equivalences:

the unit of the adjunction is XX 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

X FX Id X. \array{ X &\to& F X \\ & {}_{\mathllap{Id}}\searrow & \downarrow \\ && X } \,.

Hence by 2-out-of-3 also XFXX \to F X is a weak equivalence.

This implies that (FU)(F \dashv U) induces an equivalence of categories on the homotopy categories. Since RR equals its total right derived functor (since every object in AlgCAlg C is fibrant) this means that (F)(F \dashv) is a Quillen equivalence.

Local presentability and combinatoriality


If CC is a locally presentable category then so is AlgCAlg C.


By the above we know that AlgCAlg C is the category of algebras over a monad UF:CCU\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 UFU \circ F is an accessible functor, hence that it preserves filtered colimits. Now FF is a left adjoint and hence preserves all colimits, while we checked above that UU preserves filtered colimits.


If CC is a combinatorial model category then so is AlgCAlg C.


Algebraic \infty-groupoids

The standard model structure on simplicial sets sSet QuillensSet_{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 QuillensSet_{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

sSet QuillenAlgsSet Quillen sSet_{Quillen} \stackrel{\leftarrow}{\to} Alg sSet_{Quillen}

induces an equivalence from an algebraic definition of ∞-groupoids to a geometric definition.

Algebraic (,1)(\infty,1)-categories

Similarly, the model structure for quasi-categories sSet JoyalsSet_{Joyal} models (∞,1)-categories: its fibrant objects are precisely the quasi-categories: an algebraic quasi-category. Again, these form a model for (,1)(\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

sSet JoyalAlgsSet Joyal sSet_{Joyal} \stackrel{\leftarrow}{\to} Alg sSet_{Joyal}

establishes an equivalence of an algebraic with the standard geometric model for (,1)(\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:CAlgCF : C \to Alg C is along the lines of the discussion in

  • Richard Garner, Understanding the small object argument Applied Categorical Structures, 17(3):247–285, 2009 (pdf)

Revised on May 10, 2012 04:22:45 by Mike Shulman (