related by the Dold-Kan correspondence
The extra structure of fibrations and cofibrations in a model category is, while convenient if it exists, not carried by many categories with weak equivalences which still admit many constructions in homotopy theory. These are notably categories of presheaves with values in a model category.
A category of fibrant objects is essentially like a model category but with all axioms concerning the cofibrations dropped, while at the same time assuming that all objects are fibrant (hence the name). It turns out that this is sufficient for many useful constructions. In particular, it is sufficient for giving a convenient construction of the homotopy category in terms of spans of length one. This makes categories of fibrant objects useful in homotopical cohomology theory.
A category of fibrant objects is
a category with weak equivalences, i.e equipped with a subcategory
where is called a weak equivalence;
equipped with a further subcategory
where is called a fibration
Those morphisms which are both weak equivalences and fibrations are called acyclic fibrations .
This data has to satisfy the following properties:
has finite products;
has a terminal object ;
fibrations are preserved under pullback;
acyclic fibrations are preserved under pullback;
weak equivalences satisfy 2-out-of-3
for every object there exists a path object
all objects are fibrant, i.e. all morphisms to the terminal object are fibrations.
This includes notably all models for categories of infinity-groupoids:
The morphism is given by the degeneracy map as
This map, one checks, has the right lifting property with respect to all boundary of a simplex-inclusions . By a lemma discussed at Kan fibration this means that is an acyclic fibration. Hence , being its right inverse, is a weak equivalence.
The remaining morphism of the path space object is
One checks that this is indeed a Kan fibration.
See for instance section 1 of
Concerning the example of Kan complexes, notice that SSet is also a category of co-fibrant objects (i.e. is a category of fibrant objects) so that Kan complexes are in fact cofibrant and fibrant. That makes much of the technology discussed below superfluous, since it means that the right notion of -morphism between Kan complexes is already the ordinary notion.
But then, often it is useful to model Kan complexes using the Dold-Kan correspondence, and then the second example becomes relevant, where no longer ever object is cofibrant.
The point of the axioms of a category of fibrant objects is that when passing from infinity-groupoids to infinity-stacks, i.e. to sheaves with values in infinity-groupoids, the obvious naïve way to lift the model structure from -groupoids to sheaves of -groupoids fails, as the required lifting axioms will be satisfied only locally (e.g. stalkwise).
One can get around this by employing a more sophisticated model category structure as described at model structure on simplicial presheaves, but often it is useful to use a more lightweight solution and consider sheaves with values in -groupoids just as a category of fibrant objects, thereby effectively dispensing with the troublesome lifting property (as all mention of cofibrations is dropped):
be the full subcategory of
Define a morphism to be a fibration or a weak equivalence, if on each stalk is a fibration or weak equivalence, respectively, of Kan complexes (in terms of the standard model structure on simplicial sets).
Equipped with its structure as a category of fibrant objects, simplicial sheaves on are a model for infinity-stacks living over (the way an object is a sheaf “over ”).
Or let Diff be a (small model of) the site of smooth manifolds. The corresponding sheaf topos, that of smooth spaces has, up to isomorphism, one point per natural number, corresponding to the -dimensional ball .
Equipped with its structure as a category of fibrant objects, simplicial sheaves on are a model for smooth infinity-stacks.
with this structure is a category of fibrant objects.
The terminal object is the sheaf constant on the 0-simplex , which represents the space itself as a sheaf.
For every simplicial sheaf and every point the stalk of the unique morphism is , which is the unique morphism from the Kan complex to . Since Kan complexes are fibrant, this is a Kan fibration for every . So every is a fibrant object by the above definition.
The fact that fibrations and acyclic fibrations are preserved under pullback follows from the fact that the stalk operation
is a pullback diagram in , then for any point of also
is a pullback diagram, now of Kan complexes. Since Kan complexes form a category of fibrant objects, by the above, it follows that is a fibration or acyclic fibration of Kan complexes, respectively. Since this holds for every , it follows that is a fibration or acyclic fibration, respectively, in .
where and denote the degeneracy and face maps, respectively.
For let denote the sheaf
We want to claim that is a path object for .
To check that is fibrant, let be any point and consider the stalk . We compute laboriously
first step is the general formula for the stalk;
third step is the fact that colimits of presheaves are computed objectwise (see examples at colimit);
the fifth step uses that
the sixth step uses that the set is finite, hence a compact object so that the colimit can be taken into the hom;
the seventh step uses again that colimits of presheaves are computed objectwise
the remaining steps then just rewind the first ones, only that now has been replaced by .
That the morphism is a weak equivalence and that is a fibration follows similarly by taking the stalk colimit inside to reduce to the statement that is a weak equivalence and is a fibration, using that is a path object for the Kan complex .
The category of fibrant objects is in fact the motivating example in BrownAHT. Notice that the homotopy category in question coincides with that using the model structure on simplicial presheaves, so that the category of fibrant objects of stalk-wise Kan sheaves is a model for the homotopy category of infinity-stacks.
For a topological space and an open subset, let be the set of continuous maps from into . This set naturally is itself a group, so that to each we may associuate the one-object groupoid
In degree 0 this is the constant sheaf
while in degree 1 this is the sheaf of -valued functions
When the context is understood, we will just write again for this -groupoid valued sheaf
Let be a category of fibrant objects, with fibrations and weak equivalences .
For any object in , let be the category of fibrations over (a full subcategory of the slice category ):
objects are fibrations in ,
morphisms are commuting triangles
There is an obvious forgetful functor , which induces notions of weak equivalence and fibration in .
With this structure, becomes a category of fibrant objects.
Below is proven the factorization lemma that holds in any category of fibrant objects. This implies in particular that every morphism
may be factored as
This provides the path space objects in .
Before looking at more sophisticated constructions, we record the following direct consequences of the definition of a category of fibrant objects.
For every two objects , the two projection maps
out of their product are fibrations.
Because by assumption both morphisms are fibrations and fibrations are preserved under pullback
For every object and everey path object of , the two morphisms
(whose product , recall, is required to be a fibration) are each separately acyclic fibrations.
By the above lemma is the composite of two fibrations and hence itself a fibration.
Moreover, from the diagram
one reads off that the 2-out-of-3 property for weak equivalences implies that is also a weak equivalence.
A central lemma in the theory of categories of fibrant objects is the following factorization lemma.
For every morphism in a category of fibrant objects, there is an object such that factors as
This is the analog of one of the factorization axioms in a model category which says that every map factors as an acyclic cofibration followed by a fibration.
Notice that by 2-out-of-3 this in particular implies that every weak equivalence is given by a span of acyclic fibrations.
The way the proof of this lemma works, one sees that this really arises in the wider context of computing homotopy pullbacks in . Therefore we split the proof in two steps that are useful in their own right and will be taken up in the next section on homotopy limits.
For a morphism in , we say that the morphism defined as the composite vertical morphism in the pullback diagram
The universal bundle terminology is best understood from the following example
For an ordinary group write for the corresponding groupoid. When regarding as a constant simplicial group the corresponding Kan complex is often denoted (see simplicial group) but we shall just write also for this Kan complex, for simplicity.
where the right denotes the action groupoid of acting on by left and right multiplication.
Let be the unique morphism from the point into . The corresponding generalized universal bundle is
The morphism is a fibration.
The defining pullback diagram for can be refined to a double pullback diagram as follows
Both squares are pullback squares. Since pullbacks of fibrations are fibrations, the morphism is a fibration.
By one of the lemmas above, also the projection map is a fibration.
The above diagram exibits as the the composite
of two fibrations. Therefore it is itself a fibration.
Using the factorization lemma, one obtaines the following further useful statements about categories of fibrant objects:
Recall that plain weak equivalences, if they are not at the same time fibrations, are not required by the axioms to be preserved by pullback. But it follows from the axioms that weak equivalences are preserved under pullback along fibrations.
This we establish in two lemmas.
be a morphism of fibrations over some object in and let be any morphism in . Let
be the corresponding morphism pulled back along .
if then also ;
if then also .
For the statement follows from the fact that in the diagram
all squares (the two inner ones as well as the outer one) are pullback squares, since pullback squares compose under pasting.
The same reasoning applies for .
To apply this reasoning to the case where , we first make use of the factorization lemma to decompose as a right inverse to an acyclic fibration followed by an acyclic fibration.
(Compare the definition of the category of fibrant objects of fibrations over , discussed in the example section above.)
Using the above this reduces the proof to showing that the pullback of the top horizontal morphism of
(here the fibration on the right is the composite of the fibration with )
along is a weak equivalence. For that consider the diagram
where again all squares are pullback squares. The top two vertical composite morphisms are identities. Hence by 2-out-of-3 the morphism is a weak equivalence.
The pullback of a weak equivalence along a fibration is again a weak equivalence.
Let be a weak equivalence and be a fibration. We want to show that the left vertical morphism in the pullback
is a fibration.
First of all, using the factorization lemma we may always factor as
with the first morphism a weak equivalence that is a right inverse to an acyclic fibration and the right one an acyclic fibration.
Then the pullback diagram in question may be decomposed into two consecutive pullback diagrams
where the morphisms are indicated as fibrations and acyclic fibrations using the stability of these under arbitrary pullback.
This means that the proof reduces to proving that weak equivalences that are right inverse to some acyclic fibration map to a weak equivalence under pullback along a fibration.
Given such with right inverse , consider the pullback diagram
Notice that the indicated universal morphism into the pullback is a weak equivalence by 2-out-of-3.
The above lemma says that weak equivalences between fibrations over are themselves preserved by base extension along . In total this yields the following diagram
so that with a weak equivalence also is a weak equivalence, as indicated.
Notice that is the morphism that we want to show is a weak equivalence. By 2-out-of-3 for that it is now sufficient to show that is a weak equivalence.
That finally follows now since by assumption the total bottom horizontal morphism is the identity. Hence so is the top horizontal morphism. Hence is right inverse to a weak equivalence, hence is a weak equivalence.
Right properness is a crucial assumption in the closely related work
A homotopy fiber product or homotopy pullback of two morphisms
in a category of fibrant objects is the object defined as the (ordinary) limit
This essentially says that is the universal object that makes the diagram
commute up to homotopy (see the section on homotopies for more on that).
out of a homotopy fiber product is a fibration. If is a weak equivalence, then this is an acyclic fibration.
The same is of course true for the map to and the morphism , by symmetry of the diagram.
On the one hand we have
where both squares are pullback squares.
By the above lemma on generalized universal bundles, the map is a fibration. The first claim follows then since fibrations are stable under pullback.
On the other hand we can rewrite the limit diagram also as
where again both inner squares are pullback squares.
Again by the above statement on generalized universal bundles, we have that the morphism is a fibration. By one of the above propositions, weak equivalences are stable under pullback along fibrations, hence the pullback of is a weak equivalence. Since also is a weak equivalence (being the pullback of an acyclic fibration) the entire morphism is.
Two morphism in are
right homotopic, denoted , precisely if they fit into a diagram
for some path space object ;
homotopic, denoted , if they become right homotopic after pulled back to a weakly equivalent domain, i.e. precisely if they fit into a diagram
for some object and for some path space object of
So this says that there is a right homotopy between the two morphisms after both are pulled back to a sufficiently good resolution of their domain.
For , right homotopy is an equivalence relation on the hom-set .
This follows by “piecing path spaces together”:
Let and be two path space objects of . Then the pullback
defines a new path object, with structure maps
So given two right homotopies with respect to and we can paste them next to each other and deduce a homotopy through
We next similarly want to deduce that not only right homotopy but also true homtopy defines an equivalence relation on hom-sets . For that we need the following to lemmas.
may be refined to a diagram
Consider the pullback square
and apply the factorization lemma to factor the universal morphism into the pullback as
to obtain the diagram
where the middle vertical morphism is still a fibration, being the composite of two fibrations. By 2-out-of-3 it follows that it is also a weak equivalence.
For a morphism and , choices of path objects, there is always another path object with an acyclic fibration and a span of morphisms of path space objects
Apply the lemma above to the square
Right homotopy between morphisms is preserved under pre- and postcomposition with a given morphism.
More precisely, let be two homotopic morphisms. Then
for all morphisms and the composites and are still right homotopic.
moreover, the right homotopy may be realized with every given choice of
path space object for .
We decompose this into two statements:
for any the morphisms are right homotopic.
for any and choice of path object there is an acyclic fibration such that is right homotopic to by a right homotopy .
The first of these follows trivially.
The second one follows by using the weak functoriality property of path objects from above: let be the pullback in the following diagram
We need one more intermediate result for seeing that homotopy is an equivalence relation
in extends to a (right) homtopy-commutative diagram
For every pair of morphisms
and weak equivalence such that there is a right homotopy , there exists a weak equivalence such that .
The first point we accomplish this by letting be the homotopy fiber product in of a representative of the pullback diagram. The lemma about morphisms out of the homotopy fiber product says that is a weak equivalence.
The second point is more work. Let the right homotopy in question. We start by considering the homotopy fiber product
where the long morphisms are weak equivalences by the lemma on morphisms out of homotopy fiber products.
Then consider the two universal morphisms
into that. It follows by 2-out-of-3 that the latter is a weak equivalence. Factoring this using the factorization lemma produces hence
We know moreover that the product map is a fibration, as we can rewrite the homotopy limit as the pullback
It follows that the composite is a fibration and hence a path space object for .
Finally, by setting we obtaine the desired right homotopy .
The relation ” are homotopic”, , is an equivalence relation on .
The nontrivial part is to show transitivity. This now follows with the above lemma about homtopy commutative composition of pullback diagrams and then using the “piecing together of path objects” used above to show that right homotopy is an equivalence relation.
For a category of fibrant objects the category is defined to be the category
with the same objects as ;
with hom-sets the set of equivalence classes
under the above equivalence relation.
Composition in is given by composition of representatives in .
The obvious functor
Let be the image of the weak equivalences of in under this functor, and the image of the fibrations.
The weak equivalences in form a left multiplicative system.
This is now a direct consequence of the above lemma on homotopy-commutative completions of diagrams.
We discuss now that the structure of a category of fibrant objects on a homotopical category induces
a related category
with a morphism
that is the identity on objects,
and induces on a notion of weak equivalences
This implies the following convenient construction of the homotopy category of :
the same objects as ;
the hom-set for all given naturally by
Here the colimit is, as described at multiplicative system, over the opposite category of the category or whose objects are weak equivalences or acyclic fibrations in , and whose morphisms are commuting triangles
in (i.e. for arbitrary ).
So more in detail the above colimit is over the functor
where the first functor is the obvious forgetful functor.
It is again the factorization lemma above (and using 2-out-of-3 that implies that inverting just the acyclic fibrations in is already equivalent to inverting all weak equivalences. This means that the above theorem remains valid if the weak equivalences are replaced by acyclic fibrations:
out of a weak equivalence is refines by a cocycle out of an acyclic fibrantion, namely
Using acyclic fibrations has the advantage that these are preserved under pullback. This allows to consistently compose spans whose left leg is an acyclic fibration by pullback. See also the discussion at anafunctor.
A discussion of this point of using weak equivalences versus acyclic fibrations in the construction of the homotopy category is also in Jardine: Cocycle categories.
We now provide the missing definitions and then the proof of this theorem.
The homotopy categories of and coincide:
By one of the lemmas above, the morphisms are weak equivalences and become isomorphisms in . The section then becomes an inverse for both of them, hence the images of and in coincide. Therefore the above diagram says that homotopic morphisms in become equal in .
But this means that the localization morphism
factors through as
where sends weak equivalences in to isomorphisms in .
The universal property of then implies the universal property for
If the category of fibrant objects has an initial object which coincides with the terminal object , i.e. a zero object, then is a pointed category. In this case we have the following additional concepts and structures.
For a fibration, the pullback in
is the fibre of and is the fibre inclusion. (This is the kernel of the morphism of pointed objects)
(See also fibration sequence)
For any object and any of its path objects, the fiber of is the loop object of with respect to the chosen path object. This construction becomes independent up to canonical isomorphism of the chosen path space after mapping to the homotopy category and hence there is a functor
which sends any object of to its canonical loop object .
Any loop object becomes a group object in , i.e. a group internal to in a natural way.
There is an explicit simplicial construction of the derived hom spaces for a homotopical category that is equipped with the structure of a category of fibrant objects. This is described in (Cisinksi 10) and (Nikolaus-Schreiber-Stevenson 12, section 3.6.2).
At abelian sheaf cohomology is a detailed discussion of how the ordinary notion of sheaf cohomology arises as a special case of that.
The notion of category of fibrant objects was introduced and the above results obtained in
for application to homotopical cohomology theory.
A review is in section I.9 of
There is a description and discussion of this theory and its dual (using cofibrant objects) in
Discussion of embeddings of categories of fibrant objects into model categories is in
Also discussion of the derived hom spaces in categories of fibrant objects is in that article, as well as in section 6.3.2 of
Usage of categories of fibrant objects for the homotopical structure on C*-algebras is in :