on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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 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 $\mathbf{C}$ is
a category with weak equivalences, i.e equipped with a subcategory
where $f \in Mor(W)$ is called a weak equivalence;
equipped with a further subcategory
where $f \in Mor(F)$ 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:
$C$ 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 $\sigma$ is a weak equivalence and $d_0 \times d_1$ is a fibration;
all objects are fibrant, i.e. all morphisms $B \to {*}$ to the terminal object are fibrations.
The tautological example is the full subcategory of any model category on all objects which are fibrant.
Let $M$ be a right proper model category, let $W$ be the class of weak equivalences, and let $F_+$ be the class of morphisms $f$ in $M$ such that any pullback of $f$ in $M$ is also a homotopy pullback. Then $M$ together with $W$ and $F_+$ satisfy all the conditions to be a category of fibrant objects except possibly the condition that every morphism $X \to {*}$ in $M$ is in $F_+$; so if we restrict to the full subcategory of those objects $X$ in $M$ such that $X \to {*}$ is in $F_+$, 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 $X$ can be chosen to be the internal hom
in with respect to the closed monoidal structure on SSet with the simplicial 1-simplex $\Delta^1$.
The morphism $X \to X^I$ is given by the degeneracy map $\sigma_0 : \Delta^0 \to \Delta^1$ 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 $\partial \Delta^n \to \Delta^n$. By a lemma discussed at Kan fibration this means that $[\delta_0,X]$ is an acyclic fibration. Hence $[\sigma_0, X]$, being its right inverse, is a weak equivalence.
The remaining morphism of the path space object $X^I \to X \times X$ 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. $SSet^{op}$ 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 $\infty$-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 $\infty$-groupoids to sheaves of $\infty$-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 $\infty$-groupoids just as a category of fibrant objects, thereby effectively dispensing with the troublesome lifting property (as all mention of cofibrations is dropped):
For $C$ be a site such that the sheaf topos $Sh(C)$ has enough points, i.e. so that a morphism $f : A \to B$ in $Sh(X)$ is an isomorphism precisely if its image
is a bijection of sets for all points (geometric morphisms from $Sh({*}) \simeq Set$)
Then let
be the full subcategory of
sheaves on $C$ with values in the category SSet of simplicial sets
equivalently: simplicial objects in the category of sheaves on $C$
on those sheaves $A$ for which each stalk $x^* A \in SSet$ is a Kan complex.
Define a morphism $f : A \to B$ to be a fibration or a weak equivalence, if on each stalk $x^* f : x^* A \to x^* B$ is a fibration or weak equivalence, respectively, of Kan complexes (in terms of the standard model structure on simplicial sets).
Remarks
For instance for $X$ any topological space we may take $C = Op(X)$ to be the category of open subsets of $X$. The points of this topos precisely correspond to the ordinary points of $X$.
Equipped with its structure as a category of fibrant objects, simplicial sheaves on $X$ are a model for infinity-stacks living over $X$ (the way an object $A \in Sh(X)$ is a sheaf “over $X$”).
Or let $C =$ 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 $n$-dimensional ball $D^n$.
Equipped with its structure as a category of fibrant objects, simplicial sheaves on $Diff$ are a model for smooth infinity-stacks.
$SSh(X)$ with this structure is a category of fibrant objects.
The terminal object ${*} = X$ is the sheaf constant on the 0-simplex $\Delta^0$, which represents the space $X$ itself as a sheaf.
For every simplicial sheaf $A$ and every point $x \in X$ the stalk of the unique morphism $A \to {*}$ is $x^* A \to x^* {X}$, which is the unique morphism from the Kan complex $x^* A$ to $\Delta^0$. Since Kan complexes are fibrant, this is a Kan fibration for every $x \in X$. So every $A$ 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 $f : A \to B$ is a fibration or acyclic fibration in $SSh(X)$ and
is a pullback diagram in $SSh(X)$, then for $x \in X$ any point of $X$ also
is a pullback diagram, now of Kan complexes. Since Kan complexes form a category of fibrant objects, by the above, it follows that $x^* (h^* f)$ is a fibration or acyclic fibration of Kan complexes, respectively. Since this holds for every $x$, it follows that $h^* f$ is a fibration or acyclic fibration, respectively, in $SSh(X)$.
Recall that a functorial choice of path object for a Kan complexe $K$ is the internal hom $[\Delta^1, K]$ with respect to the closed monoidal structure on simplicial sets:
where $s_i$ and $d_i$ denote the degeneracy and face maps, respectively.
For $A \in SSh(X)$ let $[\Delta^1,A]$ 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 $SSh(X)$ is copowerered over SSet).
We want to claim that $[\Delta^1,A]$ is a path object for $A$.
To check that $[\Delta^1,A]$ is fibrant, let $x \in X$ be any point and consider the stalk $x^* [\Delta^1,A] \in SSet$. 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 $k \leq n+1$ since $\Delta^1 \times \Delta^n$ is $(n+1)$-skeletal
and that the colimit is over a filtered category
the sixth step uses that the set $\Delta([k],[1])\times \Delta([k],[n])$ 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 $A(U)$ has been replaced by $colim_{U \ni x} A(U)$.
That the morphism $A \to [\Delta^1,A]$ is a weak equivalence and that $[\Delta^1,A] \stackrel{d_0 \times d_1}{\to} A \times A$ is a fibration follows similarly by taking the stalk colimit inside to reduce to the statement that $x^* A \to [\Delta^1,x^* A]$ is a weak equivalence and $[\Delta^1,x^* A] \stackrel{d_0 \times d_1}{\to} x^*A \times x^* A$ is a fibration, using that $[\Delta^1,x^*A]$ is a path object for the Kan complex $x^* A$.
The category of fibrant objects $SSh(X)$ 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 $G$ be a topogical group and recall that $\mathbf{B} G$ denotes the corresponding one-object groupoid.
For $X$ a topological space and $U$ an open subset, let $C(U, G) \in Set$ be the set of continuous maps from $U$ into $G$. This set naturally is itself a group, so that to each $U \subset X$ 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 $G$-valued functions
When the context is understood, we will just write $\mathbf{B}G$ again for this $\infty$-groupoid valued sheaf
Let $\mathbf{C}$ be a category of fibrant objects, with fibrations $F \subset Mor(\mathbf{C})$ and weak equivalences $W \subset Mor(\mathbf{C})$.
For any object $B$ in $\mathbf{C}$, let $\mathbf{C}_B^F$ be the category of fibrations over $B$ (a full subcategory of the slice category $\mathbf{C}/B$):
objects are fibrations $A \to B$ in $\mathbf{C}$,
morphisms are commuting triangles
in $\mathbf{C}$.
There is an obvious forgetful functor $\mathbf{C}_B^F \to \mathbf{C}$, which induces notions of weak equivalence and fibration in $\mathbf{C}_B^F$.
With this structure, $\mathbf{C}_B^F$ 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 $\mathbf{C}^F_B$.
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 $A_1, A_2 \in \mathbf{C}$, the two projection maps
out of their product are fibrations.
Because by assumption both morphisms $A_i \to {*}$ are fibrations and fibrations are preserved under pullback
For every object $B \in \mathbf{C}$ and everey path object $B^I$ of $B$, the two morphisms
(whose product $d_0 \times d_1$, recall, is required to be a fibration) are each separately acyclic fibrations.
By the above lemma $d_i : B^I \stackrel{d_0 \times d_1}{\to} B \times B \stackrel{p_i}{\to} B$ 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 $d_i$ is also a weak equivalence.
A central lemma in the theory of categories of fibrant objects is the following factorization lemma.
For every morphism $f : C \to D$ in a category $\mathbf{C}$ of fibrant objects, there is an object $\mathbf{E}_f B$ such that $f$ factors as
with
$p_f$ a fibration
$\sigma_f$ a weak equivalence that is a section ( a right inverse):
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 $f : C \stackrel{\in W}{\to} B$ 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. Compar 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 $C$. 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 $f : C \to B$ a morphism in $\mathbf{C}$, we say that the morphism $p_f : \mathbf{E}_f B \to B$ defined as the composite vertical morphism in the pullback diagram
for some path space object $B^I$ is the generalized universal bundle over $B$ relative to $f$.
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 $G$ an ordinary group write $\mathbf{B} G$ for the corresponding groupoid. When regarding $G$ as a constant simplicial group the corresponding Kan complex is often denoted $\bar W G$ (see simplicial group) but we shall just write $\mathbf{B} G$ 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 $G \times G$ acting on $G$ by left and right multiplication.
Let ${*} : {*} \to \mathbf{B} G$ be the unique morphism from the point into $\mathbf{B} G$. The corresponding generalized universal bundle is
the action groupoid of $G$ acting on itself from just the right. (The corresponding Kan complex is traditionally denoted $W G$ when thought of as a simplicial group).
That $G//G \to \mathbf{B}G$ is indeed the universal $G$-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 $p_f : \mathbf{E}_f B \to B$ is a fibration.
The defining pullback diagram for $\mathbf{E}_f B$ can be refined to a double pullback diagram as follows
Both squares are pullback squares. Since pullbacks of fibrations are fibrations, the morphism $\mathbf{E}_f B \to C \times B$ is a fibration.
By one of the lemmas above, also the projection map $p_i : B \times B \to B$ is a fibration.
The above diagram exibits $p_f$ as the the composite
of two fibrations. Therefore it is itself a fibration.
The morphism $\mathbf{E}_f B \stackrel{\simeq}{\to} C$ has a section (a right inverse) $\sigma_f : C \stackrel{\simeq}{\to} \mathbf{E}_f B$ and its composite with $p_f$ is $f$:
The section
is the morphism induced via the universal property of the pullback by the section $\sigma : B \to B^I$ of $d_0 : B^I \to B$:
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 $B$ in $\mathbf{C}$ and let $u : B' \to B$ be any morphism in $\mathbf{C}$. Let
be the corresponding morphism pulled back along $u$.
Then
if $f \in F$ then also $u^* f \in F$;
if $f \in W$ then also $u^* f \in W$.
For $f \in F$ 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 $f \in W \cap F$.
To apply this reasoning to the case where $f \in W$, we first make use of the factorization lemma to decompose $f$ as a right inverse to an acyclic fibration followed by an acyclic fibration.
(Compare the definition of the category of fibrant objects $\mathbf{C}_B^F$ of fibrations over $B$, 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 $\mathbf{E}_f A_2 \to A_2$ with $A_2 \to B$)
along $u$ 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 $B' \times_B E_1 \to B' \times_B \mathbf{E}$ is a weak equivalence.
The pullback of a weak equivalence along a fibration is again a weak equivalence.
Let $u : B' \to B$ be a weak equivalence and $p : E \to B$ 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 $B' \to B$ as
$B ' \stackrel{\in W}{\to} \mathbf{E}_u B \stackrel{\in W \cap F}{\to} B$
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 $u : B' \stackrel{\in W}{\to} B$ that are right inverse to some acyclic fibration $v : B \stackrel{\in W \cap F}{\to} B'$ map to a weak equivalence under pullback along a fibration.
Given such $u$ with right inverse $v$, consider the pullback diagram
Notice that the indicated universal morphism $p \times Id : E \stackrel{\in W}{\to} E_1$ into the pullback is a weak equivalence by 2-out-of-3.
The above lemma says that weak equivalences between fibrations over $B$ are themselves preserved by base extension along $u : B' \to B$. In total this yields the following diagram
so that with $p \times Id : E \to E_1$ a weak equivalence also $u^* (p \times Id)$ is a weak equivalence, as indicated.
Notice that $u^* E = B' \times_B E \to E$ 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 $u^* E_1 \to E_1$ 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 $u^* E_1 \to E_1$ is right inverse to a weak equivalence, hence is a weak equivalence.
Model categories 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 $A \times_C C^I \times_C B$ defined as the (ordinary) limit
This essentially says that $A \times_C C^I \times_C B$ 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. 4 below.
The projection
out of a homotopy fiber product is a fibration. If $v : B \to C$ is a weak equivalence, then this is an acyclic fibration.
The same is of course true for the map to $B$ and the morphism $u : A \to C$, 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 $\mathbf{E}_v C \to C$ 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 $\mathbf{E}_u C \to C$ is a fibration. By one of the above propositions, weak equivalences are stable under pullback along fibrations, hence the pullback $A \times_C C^I \times_C B \to \mathbf{E}_u C$ of $v$ is a weak equivalence. Since also $\mathbf{E}_u C \to A$ is a weak equivalence (being the pullback of an acyclic fibration) the entire morphism $A \times_C C^I \times_C B \to A$ is.
Two morphism $f,g : A \to B$ in $C(A,B)$ are
right homotopic, denoted $f \simeq g$, precisely if they fit into a diagram
for some path space object $B^I$;
homotopic, denoted $f \sim g$, 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 $\hat A$ and for some path space object $B^I$ of $I$
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 $A,B \in \mathbf{C}$, right homotopy is an equivalence relation on the hom-set $\mathbf{C}(A,B)$.
This follows by “piecing path spaces together”:
Let $B^{I_1}$ and $B^{I_2}$ be two path space objects of $B$. Then the pullback
defines a new path object, with structure maps
So given two right homotopies with respect to $B^{I_1}$ and $B^{i_2}$ we can paste them next to each other and deduce a homotopy through $B^{I_1 \vee I_2}$
We next similarly want to deduce that not only right homotopy $f \simeq g$ but also true homtopy $f \sim g$ defines an equivalence relation on hom-sets $\mathbf{C}(A,B)$. 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 5, to factor the universal morphism $A \to X \times_B E \to E$ 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 $u : B \to C$ a morphism and $B^I$, $C^I$ choices of path objects, there is always another path object $B^{I'}$ with an acyclic fibration $B^I \stackrel{\in W \cap F}{\leftarrow} B^{I'}$ and a span of morphisms of path space objects
Apply the lemma above to the square
Right homotopy $f \simeq g$ between morphisms is preserved under pre- and postcomposition with a given morphism.
More precisely, let $f, g : B \to C$ be two homotopic morphisms. Then
for all morphisms $A \to B$ and $C \to D$ the composites $A \to B \stackrel{f}{\to} C \to D$ and $A \to B \stackrel{g}{\to} C \to D$ are still right homotopic.
moreover, the right homotopy may be realized with every given choice of
path space object $D^I$ for $D$.
We decompose this into two statements:
for any $A \to B$ the morphisms $A \to B \stackrel{f,g}{\to} B$ are right homotopic.
for any $u : C \to D$ and choice $D^I$ of path object there is an acyclic fibration $B' \to B$ such that $B' \to B \stackrel{f}{\to} C \to D$ is right homotopic to $B' \to B \stackrel{g}{\to} C \to D$ by a right homotopy $\eta : B' \to D^I$.
The first of these follows trivially.
The second one follows by using the weak functoriality property of path objects from above: let $B' := B \times_{C^I} C^{I'}$ be the pullback in the following diagram
We need one more intermediate result for seeing that homotopy is an equivalence relation
Every diagram
in $\mathbf{C}$ extends to a (right) homtopy-commutative diagram
For every pair of morphisms
and weak equivalence $t : B \stackrel{\in W}{\to} C$ such that there is a right homotopy $t \circ f \simeq t \circ g$, there exists a weak equivalence $t' : A' \to A$ such that $f \circ t' \simeq g \circ t'$.
The first point we accomplish this by letting $A' := A \times_C C^I \times_C B$ be the homotopy fiber product in $C$ of a representative of the pullback diagram. The lemma about morphisms out of the homotopy fiber product says that $A' \to A$ is a weak equivalence.
The second point is more work. Let $\eta : A \to C^I$ 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 $D \stackrel{\in F}{\to} B \times B$ is a fibration, as we can rewrite the homotopy limit as the pullback
It follows that the composite $D' \to D \to B \times B$ is a fibration and hence $D'$ a path space object for $B$.
Finally, by setting $A' = A \times_D D'$ we obtaine the desired right homotopy $f \circ t' \simeq g \circ t'$.
The relation “$f, g \in C(A,B)$ are homotopic”, $f \sim g$, is an equivalence relation on $C(A,B)$.
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 $C$ a category of fibrant objects the category $\pi C$ is defined to be the category
with the same objects as $C$;
with hom-sets the set of equivalence classes
under the above equivalence relation.
Composition in $\pi C$ is given by composition of representatives in $C$.
The obvious functor
is the identity on objects and the projection to equivalence classes on hom-set.
Let $\pi W \subset Mor(\pi C)$ be the image of the weak equivalences of $C$ in $\pi C$ under this functor, and $\pi F$ the image of the fibrations.
The weak equivalences in $\pi C$ 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 $C$ induces
a related category $\pi C$
with a morphism $C \to \pi C$
that is the identity on objects,
and induces on $\pi C$ a notion of weak equivalences
and fibrations
such that
This implies the following convenient construction of the homotopy category of $C$:
For $C$ a category of fibrant objects, its homotopy category is (equivalent to) the category $Ho_C$ with
the same objects as $C$;
the hom-set $Ho_C(A,B)$ for all $A, B \in Obj(C)$ given naturally by
Here the colimit is, as described at multiplicative system, over the opposite category of the category $\pi W_A$ or $(\pi F\cap \pi W)_A$ whose objects are weak equivalences $\hat A \stackrel{w \in \pi W}{\to} A$ or acyclic fibrations $\hat A \stackrel{f \in \pi W\cap F}{\to} A$ in $\pi C$, and whose morphisms are commuting triangles
in $\pi C$ (i.e. for arbitrary $h$).
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 $C$ is already equivalent to inverting all weak equivalences. This means that the above theorem remains valid if the weak equivalences $t : A' \to A$ are replaced by acyclic fibrations:
every cocycle $\array{ Y &\stackrel{g}{\to}& A \\ {}^\simeq \downarrow^{\mathrlap{f}} \\ X }$
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 $C$ and $\pi C$ coincide:
By one of the lemmas above, the morphisms $d_i : B^I \to B$ are weak equivalences and become isomorphisms in $Ho_C$. The section $\sigma : B \to B^I$ then becomes an inverse for both of them, hence the images of $d_0$ and $d_1$ in $Ho_C$ coincide. Therefore the above diagram says that homotopic morphisms in $C$ become equal in $Ho_C$.
But this means that the localization morphism
factors through $\pi C$ as
where $Q_{\pi C}$ sends weak equivalences in $\pi C$ to isomorphisms in $Ho_C$.
The universal property of $Q$ then implies the universal property for $Q_{\pi C}$
The above theorem on the description of $Ho_C$ now follows from the general formula for localization at a left multiplicative system of weak equivalences.
If the category $C$ of fibrant objects has an initial object which coincides with the terminal object $e$, i.e. a zero object, then $C$ is a pointed category. In this case we have the following additional concepts and structures.
For $p : Y \to X$ a fibration, the pullback $F$ in
is the fibre of $p$ and $i$ is the fibre inclusion. (This is the kernel of the morphism $f$ of pointed objects)
(See also fibration sequence)
For $B$ any object and $B^I$ any of its path objects, the fiber of $B^I \stackrel{d_0 \times d_1}{\to} B \times B$ is the loop object $\Omega^{(I)} B$ of $B$ 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 $B$ of $C$ to its canonical loop object $\Omega B$.
Any loop object $\Omega B$ becomes a group object in $Ho_C$, i.e. a group internal to $Ho_C$ 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 $\mathcal{C}$ a category of fibrant objects, write for any $X, A \in Obj(\mathcal{C})$
for the categories (“categories of cocycles on $X$ with coefficients in $A$”) whose objects are correspondences
with the left leg an acyclic fibration (for $Cocycle(X,A)$) or just a weak equivalence (for $wCocycle(X,A)$); and whose morphisms are morphisms of spans
Write $L^H_{we} \mathcal{C}$ for the simplicial localization of the category of fibrant objects $\mathcal{C}$ at its weak equivalences (hence essentially the (infinity,1)-category that it presents). Then for all objects $X,A \in Obj(\mathcal{C})$ the canonical maps
of simplicial sets (on the left the nerves of the cocycle categories of def. 8, on the right the derived hom space given by the simplicial localization) are weak homotopy equivalences.
In other words, $N Cocycle(X,A) \simeq_{whe} N wCocycle(X,A)$ is a model for the correct derived hom space.
From this it follows for instance that
The homotopy fiber products in $\mathcal{C}$ as defined in def. 4 present indeed the correct (infinity,1)-limits.
Observe that for each object $X$ the 2-functor $N Cocycle(X,-) \colon \mathcal{C} \to sSet$ of def. 8 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 $N Cocycle(X,-)$ preserves ordinary pullbacks. This means that $N Cocycle(X,-)$ takes pullbacks along a fibration in $\mathcal{C}$ to pullbacks in sSet one of whose maps is a Kan fibration. Since the standard model structure on simplicial sets $sSet_{Quillen}$ is a right proper model category, this means that these are homotopy pullbacks (as discussed there) in $sSet_{Quillen}$. Finally by prop. 3 this means that the derived hom-space functor $\mathbb{R}Hom(X,-)$ 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 $\mathcal{C}$ 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
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 :