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A category of fibrant objects is a category with weak equivalences equipped with extra structure somewhat weaker than that of a model category.
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 nevertheless admit many constructions in homotopy theory. Even if they do, sometimes the cofibrations are intractable in practice.
A category of fibrant objects is essentially like a model category but with all axioms concerning the cofibrations dropped, the concept of fibrations retained (“fibration category”) and assuming that all objects are fibrant (whence 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, and in particular a terminal object ;
the pullback of a fibration along an arbitrary morphism exists, and is again a fibration;
acyclic fibrations are preserved under pullback;
weak equivalences satisfy 2-out-of-3
for every object there exists a path object
where is a weak equivalence and is a fibration;
all objects are fibrant, i.e. all morphisms to the terminal object are fibrations.
The tautological example is the full subcategory of any model category on all objects which are fibrant.
Let be a right proper model category, let be the class of weak equivalences, and let be the class of morphisms in such that any pullback of in is also a homotopy pullback. Then together with and satisfy all the conditions to be a category of fibrant objects except possibly the condition that every morphism in is in ; so if we restrict to the full subcategory of those objects in such that is in , then we do get a category of fibrant objects.
For example, sSet via its standard model structure is a category of fibrant objects in this way. The fibrations in this case are not the Kan fibrations (these also yields a category-of-fibrant-objects structure, via the above, but a different one) but are the sharp maps.
This includes notably all models for categories of infinity-groupoids:
the category of Kan complexes (a full subcategory of SSet)
the category of strict omega-groupoids using the model structure on strict omega-groupoids
The path object of any can be chosen to be the internal hom
in with respect to the closed monoidal structure on SSet with the simplicial 1-simplex .
The morphism is given by the degeneracy map as
This is indeed a weak equivalence, since by the simplicial identities it is a section (a right inverse) for the morphism
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.
The stability of fibrations and acyclic fibrations follows from the above fact that both are characterized by a right lifting property (as described a model structure on simplicial sets).
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):
(-groupoid valued sheaves)
For be a site such that the sheaf topos has enough points, i.e. so that a morphism in is an isomorphism precisely if its image
is a bijection of sets for all points (geometric morphisms from )
Then let
be the full subcategory of
sheaves on with values in the category SSet of simplicial sets
equivalently: simplicial objects in the category of sheaves on
on those sheaves for which each stalk is a Kan complex.
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).
Remarks
The points of this topos precisely correspond to the ordinary points of .
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 the inverse image of a geometric morphism and hence preserves finite limits and in particular pullbacks. So if is a fibration or acyclic fibration in and
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 .
Recall that a functorial choice of path object for a Kan complexe is the internal hom with respect to the closed monoidal structure on simplicial sets:
where and denote the degeneracy and face maps, respectively.
For let denote the sheaf
where on the left we have new notation and on the right we have the internal hom in SSet.
(The notation on the left defines the way in which is copowerered over SSet).
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
Where the
first step is the general formula for the stalk;
second step is the formula for the internal hom in the closed monoidal structure on simplicial sets;
third step is the fact that colimits of presheaves are computed objectwise (see examples at colimit);
the fourth step is the definition of the SSet-enriched functor category by an end
the fifth step uses that
the end truncates to a finite limit with since is -skeletal
and that the colimit is over a filtered category
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.
Let be a topogical group and recall that denotes the corresponding one-object groupoid.
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
By postcomposition this with the nerve operation we obtain an assignment of Kan complexes to open subsets:
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
in .
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.
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
with
a fibration
a weak equivalence that is a section ( a right inverse) of an acyclic fibration:
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.
In the context of Lie groupoid theory these are known as the Morita equivalences between groupoids. There here arise as a special case. Compare also the notion of anafunctor.
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
for some path space object is the generalized universal bundle over relative to .
The universal bundle terminology is best understood from the following example
Consider the category of fibrant objects given by Kan complexes or just strict omega-groupoids.
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.
The corresponding path object is given by the groupoid (or its corresponding Kan complex)
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 action groupoid of acting on itself from just the right. (The corresponding Kan complex is traditionally denoted when thought of as a simplicial group).
That is indeed the universal -principal bundle (under the Quillen equivalence of Kan complexes and topological spaces) is an old result of Segal (as described at generalized universal bundle).
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.
The morphism has a section (a right inverse) and its composite with is :
The section
is the morphism induced via the universal property of the pullback by the section of :
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.
Let
be a morphism of fibrations over some object in and let be any morphism in . Let
be the corresponding morphism pulled back along .
Then
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 fibration and let be a weak equivalence. Without loss of generality we can assume that is a section of an acyclic fibration – otherwise we first decompose with the factorization lemma and then pull back along the factors individually.
Consider the following diagram where the four tiled squares are pullbacks, , and the bent back rectangle is also a pullback. is a weak equivalence since is a acyclic fibration, and by 2-out-of-3 we can conclude that is a weak equivalence as well. By the preceding lemma , so is . The goal is to show that is a weak equivalence, and by 2-out-of-3 it’s enough to check that for . This follows again from 2-out-of-3 since is a section of (because the bottom row of the diagram is an identity).
that satisfy this property are called right proper model categories.
Right properness is a crucial assumption in the closely related work
Using the existence of path space objects one can construct specific homotopy pullbacks called homotopy fiber products .
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).
These homotopy pullbacks present indeed the correct (infinity,1)-limits, this is the content of prop. below.
The projection
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.
One may compute this limit in terms of two consecutive pullbacks in two different ways.
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.
Every diagram
may be refined to a diagram
Consider the pullback square
and apply the factorization lemma, 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
Every diagram
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
and
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
is the identity on objects and the projection to equivalence classes on hom-set.
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
and fibrations
such that
This implies the following convenient construction of the homotopy category of :
For a category of fibrant objects, its homotopy category is (equivalent to) the category with
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:
every cocycle
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
The above theorem on the description of now follows from the general formula for localization at a left multiplicative system of weak equivalences.
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).
For a category of fibrant objects, write for any
for the categories (“categories of cocycles on with coefficients in ”) whose objects are correspondences
with the left leg an acyclic fibration (for ) or just a weak equivalence (for ); and whose morphisms are morphisms of spans
Write for the simplicial localization of the category of fibrant objects at its weak equivalences (hence essentially the (infinity,1)-category that it presents). Then for all objects the canonical maps
of simplicial sets (on the left the nerves of the cocycle categories of def. , on the right the derived hom space given by the simplicial localization) are weak homotopy equivalences.
In other words, is a model for the correct derived hom space.
From this it follows for instance that
The homotopy fiber products in as defined in def. present indeed the correct (infinity,1)-limits.
Observe that for each object the 2-functor of def. sends fibrations to Kan fibrations of simplicial sets (the horn-filling condition comes down to factoring maps through the given fibration, which is possible by pullback along the fibration). Moreover, it is evident that preserves ordinary pullbacks. This means that takes pullbacks along a fibration in to pullbacks in sSet one of whose maps is a Kan fibration. Since the standard model structure on simplicial sets is a right proper model category, this means that these are homotopy pullbacks (as discussed there) in . Finally by prop. this means that the derived hom-space functor sends pullbacks along fibrations to homotopy pullbacks of the correct derived hom-spaces. This means (as discussed for instance at homotopy Kan extension) that the original pullbacks in are the correct homotopy pullbacks.
When the catgegory of fibrant objects is that of locally Kan simplicial sheaves, the hom-sets of its homotopy category compute generalized notions of cohomology.
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
par équivalences dérivées_ (pdf)
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
and also in
Usage of categories of fibrant objects for the homotopical structure on C*-algebras is in :
Categories of fibrant objects form a convenient setting for the study of homotopy type theory:
It is shown in the following paper that categories of fibrant objects are themselves fibrant in the model structure on categories with weak equivalences:
Last revised on February 23, 2024 at 23:16:08. See the history of this page for a list of all contributions to it.