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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{category of fibrant objects} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory} [[!include model category theory - contents]] \hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory} [[!include homotopy - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{FullSubcategoriesOfModelCategories}{Full subcategories of model categories}\dotfill \pageref*{FullSubcategoriesOfModelCategories} \linebreak \noindent\hyperlink{RightProperModelCategories}{Right proper model categories}\dotfill \pageref*{RightProperModelCategories} \linebreak \noindent\hyperlink{groupoids}{$\infty$-Groupoids}\dotfill \pageref*{groupoids} \linebreak \noindent\hyperlink{simplicial_sheaves}{Simplicial sheaves}\dotfill \pageref*{simplicial_sheaves} \linebreak \noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak \noindent\hyperlink{slice_categories}{Slice categories}\dotfill \pageref*{slice_categories} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{SimpleConsequences}{Simple consequences of the definition}\dotfill \pageref*{SimpleConsequences} \linebreak \noindent\hyperlink{generalized_universal_bundles_and_the_factorization_lemma}{Generalized universal bundles and the factorization lemma}\dotfill \pageref*{generalized_universal_bundles_and_the_factorization_lemma} \linebreak \noindent\hyperlink{more_sophisticated_consequences_of_the_definition}{More sophisticated consequences of the definition}\dotfill \pageref*{more_sophisticated_consequences_of_the_definition} \linebreak \noindent\hyperlink{homotopy_fiber_product}{Homotopy fiber product}\dotfill \pageref*{homotopy_fiber_product} \linebreak \noindent\hyperlink{Homotopies}{Homotopies}\dotfill \pageref*{Homotopies} \linebreak \noindent\hyperlink{HomotopyCategory}{The homotopy category}\dotfill \pageref*{HomotopyCategory} \linebreak \noindent\hyperlink{pointed_category_of_fibrant_objects}{Pointed category of fibrant objects}\dotfill \pageref*{pointed_category_of_fibrant_objects} \linebreak \noindent\hyperlink{fibers}{Fibers}\dotfill \pageref*{fibers} \linebreak \noindent\hyperlink{fibration_sequences}{Fibration Sequences}\dotfill \pageref*{fibration_sequences} \linebreak \noindent\hyperlink{DerivedHomSpaces}{Derived hom-spaces}\dotfill \pageref*{DerivedHomSpaces} \linebreak \noindent\hyperlink{application_in_cohomology_theory}{Application in cohomology theory}\dotfill \pageref*{application_in_cohomology_theory} \linebreak \noindent\hyperlink{related_concept}{Related concept}\dotfill \pageref*{related_concept} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{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 \emph{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 \emph{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 [[span]]s of length one. This makes categories of fibrant objects useful in [[homotopical cohomology theory]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{}\hypertarget{}{} A \textbf{category of fibrant objects} $\mathbf{C}$ is \begin{itemize}% \item a [[category with weak equivalences]], i.e equipped with a subcategory \begin{displaymath} Core(\mathbf{C}) \hookrightarrow W \hookrightarrow C \end{displaymath} where $f \in Mor(W)$ is called a \textbf{weak equivalence}; \item equipped with a further subcategory \begin{displaymath} Core(\mathbf{C}) \hookrightarrow F \hookrightarrow C \,, \end{displaymath} where $f \in Mor(F)$ is called a \textbf{fibration} Those morphisms which are both weak equivalences and fibrations are called \textbf{acyclic fibrations} . \end{itemize} This data has to satisfy the following properties: \begin{itemize}% \item $C$ has [[finite products]], and in particular a [[terminal object]] ${*}$; \item the [[pullback]] of a fibration along an arbitrary morphism exists, and is again a fibration; \item acyclic fibrations are preserved under [[pullback]]; \item weak equivalences satisfy [[category with weak equivalences|2-out-of-3]] \item for every object there exists a [[path object]] \begin{itemize}% \item this means: for every object $B$ there exists at least one object denoted $B^I$ (not necessarily but possibly the [[internal hom]] with an [[interval object]]) that fits into a diagram\begin{displaymath} (B \stackrel{Id \times Id}{\to} B \times B) = (B \stackrel{\sigma}{\to} B^I \stackrel{d_0 \times d_1}{\to} B \times B) \end{displaymath} where $\sigma$ is a weak equivalence and $d_0 \times d_1$ is a fibration; \end{itemize} \item all objects are \emph{fibrant}, i.e. all morphisms $B \to {*}$ to the terminal object are fibrations. \end{itemize} \end{defn} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{FullSubcategoriesOfModelCategories}{}\subsubsection*{{Full subcategories of model categories}}\label{FullSubcategoriesOfModelCategories} The tautological example is the [[full subcategory]] of any [[model category]] on all objects which are fibrant. \hypertarget{RightProperModelCategories}{}\subsubsection*{{Right proper model categories}}\label{RightProperModelCategories} 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 \emph{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 [[model structure on simplicial sets|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 \hyperlink{FullSubcategoriesOfModelCategories}{the above}, but a different one) but are the [[sharp maps]]. \hypertarget{groupoids}{}\subsubsection*{{$\infty$-Groupoids}}\label{groupoids} This includes notably all models for categories of [[infinity-groupoid]]s: \begin{itemize}% \item the category of [[Kan complex]]es (a full [[subcategory]] of [[SSet]]) \item the category of [[strict omega-groupoid]]s using the [[model structure on strict omega-groupoids]] \end{itemize} \begin{prop} \label{}\hypertarget{}{} [[nLab:Kan complex|Kan complexes]] form a [[nLab:category of fibrant objects|Brown category of fibrant objects]]. \end{prop} \begin{proof} The [[path object]] of any $X$ can be chosen to be the [[nLab:internal hom|internal hom]] \begin{displaymath} X^I = [\Delta^1, X] \end{displaymath} in with respect to the [[closed monoidal structure on presheaves|closed monoidal structure on]] [[SSet]] with the simplicial 1-[[simplex]] $\Delta^1$. The morphism $X \to X^I$ is given by the [[simplex category|degeneracy map]] $\sigma_0 : \Delta^0 \to \Delta^1$ as \begin{displaymath} X \stackrel{\simeq}{\to} [\Delta^0, X] \stackrel{[\sigma_0, X]}{\to} [\Delta^1, X] \,. \end{displaymath} This is indeed a weak equivalence, since by the [[simplicial identities]] it is a [[section]] (a [[inverse|right inverse]]) for the morphism \begin{displaymath} [\Delta^1, X] \stackrel{[\delta_0,X]}{\to} [\Delta^0, X] \,. \end{displaymath} This map, one checks, has the [[weak factorization system|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 object|path space object]] $X^I \to X \times X$ is \begin{displaymath} [\Delta^1, X] \stackrel{[\delta_0 \sqcup \delta_1, X]}{\to} [\Delta^0 \sqcup \Delta^0, X] \stackrel{\simeq}{\to} X \times X \,. \end{displaymath} 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 [[nLab:weak factorization system|right lifting property]] (as described a [[model structure on simplicial sets]]). See for instance \href{http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-1.dvi}{section 1} of \begin{itemize}% \item Goerss, Jardine, \emph{Simplicial homotopy theory} . \end{itemize} \end{proof} Concerning the example of [[Kan complex]]es, notice that [[SSet]] is also a \emph{category of co-fibrant objects} (i.e. $SSet^{op}$ is a category of fibrant objects) so that [[Kan complex]]es 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. \hypertarget{simplicial_sheaves}{}\subsubsection*{{Simplicial sheaves}}\label{simplicial_sheaves} The point of the axioms of a category of fibrant objects is that when passing from [[infinity-groupoid]]s to [[infinity-stack]]s, i.e. to [[sheaf|sheaves]] with values in [[infinity-groupoid]]s, the obvious na\"i{}ve 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. [[stalk]]wise). 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): \begin{defn} \label{}\hypertarget{}{} For $C$ be a [[site]] such that the [[sheaf]] [[topos]] $Sh(C)$ has [[point of a topos|enough points]], i.e. so that a morphism $f : A \to B$ in $Sh(X)$ is an isomorphism precisely if its image \begin{displaymath} x^* f : x^* A \to x^* B \end{displaymath} is a bijection of sets for all [[point of a topos|points]] ([[geometric morphism]]s from $Sh({*}) \simeq Set$) \begin{displaymath} x : Set \simeq Sh({*}) \stackrel{\stackrel{x^*}{\leftarrow}} {\stackrel{x_*}{\to}} Sh(C) \,. \end{displaymath} Then let \begin{displaymath} \mathbf{C} = SSh(C) \end{displaymath} be the full [[subcategory]] of \begin{itemize}% \item [[sheaf|sheaves]] on $C$ with values in the category [[SSet]] of [[simplicial set]]s \item equivalently: [[simplicial object]]s in the [[category of sheaves]] on $C$ \end{itemize} 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]]). \end{defn} \textbf{Remarks} \begin{itemize}% \item 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-stack]]s living \textbf{over $X$} (the way an object $A \in Sh(X)$ is a sheaf ``over $X$''). \item Or let $C =$ [[Diff]] be a (small model of) the [[site]] of smooth [[manifold]]s. The corresponding [[sheaf]] [[topos]], that of [[smooth space]]s 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-stack]]s. \end{itemize} \begin{lemma} \label{}\hypertarget{}{} $SSh(X)$ with this structure is a category of fibrant objects. \end{lemma} \begin{proof} 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 complex]]es 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 \begin{displaymath} x^* : SSh(X) \to SSh(pt) \simeq SSet \end{displaymath} is the [[inverse image]] of a [[geometric morphism]] and hence preserves finite [[limit]]s and in particular [[pullback]]s. So if $f : A \to B$ is a fibration or acyclic fibration in $SSh(X)$ and \begin{displaymath} \itexarray{ A \times_B C &\to& B \\ \downarrow^{\mathrlap{h^* f}} && \downarrow^\mathrlap{f} \\ C &\stackrel{h}{\to}& B } \end{displaymath} is a [[pullback]] diagram in $SSh(X)$, then for $x \in X$ any point of $X$ also \begin{displaymath} \itexarray{ x^*(A \times_{B} C) &\to& x^*B \\ \downarrow^{\mathrlap{x^* (h^* f)}} && \downarrow^{\mathrlap{x^* f}} \\ x^*C &\stackrel{x^*}{\to}& B } \end{displaymath} is a [[pullback]] diagram, now of [[Kan complex]]es. 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 complex]]e $K$ is the [[internal hom]] $[\Delta^1, K]$ with respect to the [[closed monoidal structure on presheaves|closed monoidal structure on]] [[simplicial set]]s: \begin{displaymath} K \stackrel{=}{\to} [\Delta^0, K] \stackrel{s_0}{\to} [\Delta^1, K] \stackrel{d_0 \times d_1}{\to} [\Delta^0, K] \times [\Delta^0, K] \stackrel{=}{\to} K \times K \,, \end{displaymath} where $s_i$ and $d_i$ denote the [[simplicial set|degeneracy and face maps]], respectively. For $A \in SSh(X)$ let $[\Delta^1,A]$ denote the sheaf \begin{displaymath} [\Delta^1,A] : U \mapsto [\Delta^1,A(U)] \,, \end{displaymath} 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 [[copower]]ered 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 \begin{displaymath} \begin{aligned} x^* [\Delta^1,A] &\simeq colim_{U \ni x} [\Delta^1,A(U)] \\ &\simeq colim_{U \ni x} SSet(\Delta^1 \times \Delta^\bullet, A(U)) \\ &\simeq ([n] \mapsto colim_{U \ni x} SSet(\Delta^1 \times \Delta^\n, A(U)) \\ & \simeq ([n] \mapsto colim_{U \ni x} \int_{[k] \in \Delta} Set(\Delta([k],[1])\times\Delta([k],[n]), A(U)_k) \\ & \simeq ([n] \mapsto \int_{[k \leq n+1] \in \Delta}( colim_{U \ni x} Set(\Delta([k],[1])\times\Delta([k],[n]), A(U)_k ) \\ &\simeq ([n] \mapsto \int_{[k] \in \Delta|_{n+1}}( Set(\Delta([k],[1])\times\Delta([k],[n]), colim_{U \ni x} A(U)_k ) \\ &\simeq ([n] \mapsto \int_{[k] \in \Delta|_{n+1}}( Set(\Delta([k],[1])\times\Delta([k],[n]), (colim_{U \ni x} A(U))_k ) \\ &\simeq [\Delta^1, colim_{U \ni x} A(U)] \\ & \simeq [\Delta^1, x^* A] ) \end{aligned} \end{displaymath} Where the \begin{itemize}% \item first step is the general formula for the [[stalk]]; \item second step is the formula for the [[internal hom]] in the [[closed monoidal structure on presheaves|closed monoidal structure on simplicial sets]]; \item third step is the fact that colimits of presheaves are computed objectwise (see examples at [[colimit]]); \item the fourth step is the definition of the [[SSet]]-[[enriched functor category]] by an [[end]] \item the fifth step uses that \begin{itemize}% \item the end truncates to a finite limit with $k \leq n+1$ since $\Delta^1 \times \Delta^n$ is $(n+1)$-[[simplicial skeleton|skeletal]] \item and that the colimit is over a [[filtered category]] \item and that [[limits and colimits by example|filtered colimits commute with finite limits]]; \end{itemize} \item 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; \item the seventh step uses again that colimits of presheaves are computed objectwise \item the remaining steps then just rewind the first ones, only that now $A(U)$ has been replaced by $colim_{U \ni x} A(U)$. \end{itemize} 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]] $x^* A$. \end{proof} 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-stack]]s. \hypertarget{example}{}\paragraph*{{Example}}\label{example} 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 \begin{displaymath} U \mapsto \mathbf{B} C(U,G) \,. \end{displaymath} By postcomposition this with the [[nerve]] operation we obtain an assignment of [[Kan complex]]es to open subsets: \begin{displaymath} U \mapsto N \mathbf{B} C(U,G) \,. \end{displaymath} In degree 0 this is the constant [[sheaf]] \begin{displaymath} (N \mathbf{B}(-,G))_0 : U \mapsto {*} \end{displaymath} while in degree 1 this is the [[sheaf]] of $G$-valued functions \begin{displaymath} (N \mathbf{B}(-,G))_1 : U \mapsto C(U,G) \,. \end{displaymath} When the context is understood, we will just write $\mathbf{B}G$ again for this $\infty$-groupoid valued sheaf \begin{displaymath} \mathbf{B}G := N \mathbf{B} C(-,G) \,. \end{displaymath} \hypertarget{slice_categories}{}\subsubsection*{{Slice categories}}\label{slice_categories} 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]] $\mathbf{C}/B$): \begin{itemize}% \item objects are fibrations $A \to B$ in $\mathbf{C}$, \item morphisms are commuting triangles \begin{displaymath} \itexarray{ A &&\to&& A' \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B } \end{displaymath} in $\mathbf{C}$. \end{itemize} There is an obvious [[forgetful functor]] $\mathbf{C}_B^F \to \mathbf{C}$, which induces notions of weak equivalence and fibration in $\mathbf{C}_B^F$. \begin{lemma} \label{}\hypertarget{}{} With this structure, $\mathbf{C}_B^F$ becomes a category of fibrant objects. \end{lemma} \begin{proof} Below is proven the \emph{[[factorization lemma]]} that holds in any category of fibrant objects. This implies in particular that every morphism \begin{displaymath} \itexarray{ A &&\stackrel{Id \times Id}{\to}&& A \times_B A \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B } \end{displaymath} may be factored as \begin{displaymath} \itexarray{ A &\stackrel{\in W}{\to}& A^I &\stackrel{\in F}{\to}& A \times_B A \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B } \,. \end{displaymath} This provides the [[path object|path space objects]] in $\mathbf{C}^F_B$. \end{proof} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{SimpleConsequences}{}\subsubsection*{{Simple consequences of the definition}}\label{SimpleConsequences} Before looking at more sophisticated constructions, we record the following direct consequences of the definition of a category of fibrant objects. \begin{lemma} \label{}\hypertarget{}{} For every two objects $A_1, A_2 \in \mathbf{C}$, the two projection maps \begin{displaymath} p_i : A_1 \times A_2 \stackrel{\in F}{\to} A_i \end{displaymath} out of their [[product]] are fibrations. \end{lemma} \begin{proof} Because by assumption both morphisms $A_i \to {*}$ are fibrations and fibrations are preserved under pullback \begin{displaymath} \itexarray{ A_1 \times A_2 &\to& A_2 \\ \;{}^{p_1}\downarrow^{\mathrlap{\Rightarrow \in F}} && \downarrow^{\mathrlap{\in F}} \\ A_1 &\to& {*} } \,. \end{displaymath} \end{proof} \begin{lemma} \label{}\hypertarget{}{} For every object $B \in \mathbf{C}$ and everey [[path object]] $B^I$ of $B$, the two morphisms \begin{displaymath} d_i : B^I \stackrel{\in W \cap F}{\to} B \end{displaymath} (whose product $d_0 \times d_1$, recall, is required to be a fibration) are each separately acyclic fibrations. \end{lemma} \begin{proof} 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 \begin{displaymath} \itexarray{ B &\stackrel{\simeq}{\to}& B^I &\stackrel{d_0 \times d_1}{\to}& B \times B \\ &&&\searrow^{d_i}& \downarrow^{\mathrlap{p_i}} \\ & \searrow^{Id}&&& B } \end{displaymath} one reads off that the 2-out-of-3 property for weak equivalences implies that $d_i$ is also a weak equivalence. \end{proof} \hypertarget{generalized_universal_bundles_and_the_factorization_lemma}{}\subsubsection*{{Generalized universal bundles and the factorization lemma}}\label{generalized_universal_bundles_and_the_factorization_lemma} A central lemma in the theory of categories of fibrant objects is the following [[factorization lemma]]. \begin{lemma} \label{FactorizationLemma}\hypertarget{FactorizationLemma}{} 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 \begin{displaymath} \itexarray{ && \mathbf{E}_f B \\ & {}^{\sigma_f \in W}\nearrow && \searrow^{p_f \in F} \\ C &&\stackrel{f}{\to}&& B } \end{displaymath} with \begin{itemize}% \item $p_f$ a fibration \item $\sigma_f$ a weak equivalence that is a [[section]] ( a [[inverse|right inverse]]): \begin{displaymath} Id_{\mathbf{E}_f B} = ( C \stackrel{\sigma_f}{\to} \mathbf{E}_f B \stackrel{\simeq}{\to} C ) \,. \end{displaymath} \end{itemize} \end{lemma} \begin{remark} \label{}\hypertarget{}{} This is the analog of one of the \emph{factorization axioms} in a [[model category]] which says that every map factors as an acyclic cofibration followed by a fibration. Notice that by [[category with weak equivalences|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. \begin{displaymath} \itexarray{ && \mathbf{E}_f B \\ & {}^{\sigma_f \in W}\nearrow && \searrow^{p_f \in F\cap W} \\ C &&\stackrel{f \in W}{\to}&& B } \,. \end{displaymath} In the context of [[Lie groupoid]] theory these are known as the [[Morita equivalence]]s between groupoids. There here arise as a special case. Compar also the notion of [[anafunctor]]. \end{remark} The way the proof of this lemma works, one sees that this really arises in the wider context of computing [[homotopy pullback]]s 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. \begin{defn} \label{}\hypertarget{}{} 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 \begin{displaymath} \itexarray{ \mathbf{E}_f B &\stackrel{\simeq}{\to}\gt& C \\ \downarrow && \downarrow^{\mathrlap{f}} \\ B^I &\stackrel{d_0}{\to}& B \\ \downarrow^{\mathrlap{d_1}} \\ B } \end{displaymath} for some [[path object|path space object]] $B^I$ is the [[generalized universal bundle]] over $B$ relative to $f$. \end{defn} The universal bundle terminology is best understood from the following example \begin{example} \label{}\hypertarget{}{} Consider the category of fibrant objects given by [[Kan complex]]es or just [[strict omega-groupoid]]s. 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) \begin{displaymath} (\mathbf{B} G)^I = [\Delta^1, \mathbf{B} G ] = G \backslash \backslash G//G \end{displaymath} 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 \begin{displaymath} \mathbf{E}_{*} G = G//G \end{displaymath} 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 [[homotopy hypothesis|Quillen equivalence of]] [[Kan complex]]es and [[topological space]]s) is an old result of Segal (as described at [[generalized universal bundle]]). \end{example} \begin{lemma} \label{}\hypertarget{}{} The morphism $p_f : \mathbf{E}_f B \to B$ is a fibration. \end{lemma} \begin{proof} The defining [[pullback]] diagram for $\mathbf{E}_f B$ can be refined to a double pullback diagram as follows \begin{displaymath} \itexarray{ \mathbf{E}_f B &\stackrel{\in F}{\to}& C \times B &\stackrel{p_1}{\to}& C \\ \downarrow && \downarrow^{\mathrlap{f \times Id}} && \downarrow^{\mathrlap{f}} \\ B^I &\stackrel{d_0 \times d_1 \in F}{\to}& B \times B &\stackrel{p_1}{\to}& B \\ \downarrow^{\mathrlap{d_1}} & \swarrow_{p_2 \in F} \\ B } \,. \end{displaymath} 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 \begin{displaymath} \begin{aligned} p_f &: \mathbf{E}_f B \to C \times B \stackrel{f \times Id}{\to} B \times B \stackrel{p_2}{\to} B \\ & = \mathbf{E}_f B \to C \times B \stackrel{p_2}{\to} B \end{aligned} \end{displaymath} of two fibrations. Therefore it is itself a fibration. \end{proof} \begin{lemma} \label{}\hypertarget{}{} The morphism $\mathbf{E}_f B \stackrel{\simeq}{\to} C$ has a [[section]] (a [[inverse|right inverse]]) $\sigma_f : C \stackrel{\simeq}{\to} \mathbf{E}_f B$ and its composite with $p_f$ is $f$: \begin{displaymath} \itexarray{ \mathbf{E}_f B &\stackrel{\sigma_f}{\leftarrow}&& C \\ \downarrow^{\mathrlap{p_f}} && \swarrow_{f} \\ B } \end{displaymath} \end{lemma} \begin{proof} The [[section]] \begin{displaymath} \sigma_f = Id \times \sigma \circ f \end{displaymath} is the morphism induced via the universal property of the [[pullback]] by the [[section]] $\sigma : B \to B^I$ of $d_0 : B^I \to B$: \begin{displaymath} \itexarray{ C &\stackrel{\sigma_f \in W}{\to}& \mathbf{E}_f B &\stackrel{\in W \cap F}{\to}& C \\ \downarrow^{\mathrlap{f}} && \downarrow && \downarrow^{\mathrlap{f}} \\ B &\stackrel{\sigma}{\to}& B^I &\stackrel{d_1 \in W \cap F}{\to}& B \\ & {}_{Id}\searrow & \downarrow^{\mathrlap{d_0}} \\ && B } \;\;\;\; = \;\;\;\; \itexarray{ C &\stackrel{Id}{\to}& C \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f}} \\ B &\stackrel{Id}{\to}& B } \,. \end{displaymath} \end{proof} \hypertarget{more_sophisticated_consequences_of_the_definition}{}\subsubsection*{{More sophisticated consequences of the definition}}\label{more_sophisticated_consequences_of_the_definition} 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. \begin{lemma} \label{BaseChangePreservesFibrationsAndWeakEquivalences}\hypertarget{BaseChangePreservesFibrationsAndWeakEquivalences}{} Let \begin{displaymath} \itexarray{ A_1 &&\stackrel{f}{\to}&& A_2 \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B } \end{displaymath} 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 \begin{displaymath} \itexarray{ u^*A_1 &&\stackrel{u^* f}{\to}&& u^* A_2 \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B' } \end{displaymath} be the corresponding morphism pulled back along $u$. Then \begin{itemize}% \item if $f \in F$ then also $u^* f \in F$; \item if $f \in W$ then also $u^* f \in W$. \end{itemize} \end{lemma} \begin{proof} \label{}\hypertarget{}{} For $f \in F$ the statement follows from the fact that in the diagram \begin{displaymath} \itexarray{ B' \times_B A_1 &\to& A_1 \\ \;\;\downarrow^{\mathrlap{u^* f \in F}} && \;\;\downarrow^{\mathrlap{f \in F}} \\ B' \times_B A_2 &\to& A_2 \\ \;\downarrow^{\mathrlap{\in F}} && \;\downarrow^{\mathrlap{\in F}} \\ B' &\stackrel{u}{\to}& B } \end{displaymath} 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. \begin{displaymath} f : A_1 \stackrel{\in W}{\to} \mathbf{E}_f A_2 \stackrel{\in W \cap F}{\to} A_2 \,. \end{displaymath} (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 \begin{displaymath} \itexarray{ A_1 &&\stackrel{}{\to}&& \mathbf{E}_f A_2 \\ & {}_{\in F}\searrow && \swarrow_{\in F} \\ && B } \end{displaymath} (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 \begin{displaymath} \itexarray{ B' \times_B A_1 &\to& A_1 \\ \downarrow && \downarrow \\ B' \times_B \mathbf{E}_f A_2 &\to& \mathbf{E}_f A_2 \\ \;\;\downarrow^{\mathrlap{\in W \cap F}} && \;\;\downarrow^{\mathrlap{\in W \cap F}} \\ B' \times_B A_1 &\to& A_1 \\ \;\downarrow^{\mathrlap{\in F}} && \;\downarrow^{\mathrlap{\in F}} \\ B' &\to& B } \end{displaymath} 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. \end{proof} \begin{lemma} \label{}\hypertarget{}{} The pullback of a weak equivalence along a fibration is again a weak equivalence. \end{lemma} \begin{proof} 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]] \begin{displaymath} \itexarray{ E \times_B B' &\to& B' \\ \;\;\;\;\downarrow^{\mathrlap{\Rightarrow \in W} } && \;\downarrow^{\mathrlap{\in W}} \\ E &\stackrel{\in F}{\to}& B } \end{displaymath} 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 \begin{displaymath} \itexarray{ E \times_B B' &\to& B' \\ \downarrow && \downarrow \\ Q &\stackrel{\in F}{\to}& \mathbf{E}_u B \\ \;\;\downarrow^{\mathrlap{\in W \cap F}} && \;\;\downarrow^{\mathrlap{\in W \cap F}} \\ E &\stackrel{\in F}{\to}& B } \,, \end{displaymath} 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 \begin{displaymath} \itexarray{ E \\ & {}_{\in W}\searrow^{p \times Id} && \searrow^{Id} \\ && E_1 := B \times_{B'} E & \stackrel{\in W \cap F}{\to} & E \\ &&\downarrow^{\mathrlap{\in F}} && \downarrow^{\mathrlap{p \in F}} \\ &&&& B \\ &&\downarrow && \downarrow^{\mathrlap{v \in F \cap W}} \\ &&B &\stackrel{v \in W \cap F}{\to}& B' } \,. \end{displaymath} Notice that the indicated universal morphism $p \times Id : E \stackrel{\in W}{\to} E_1$ into the pullback is a weak equivalence by [[category with weak equivalences|2-out-of-3]]. The previous lemma \ref{BaseChangePreservesFibrationsAndWeakEquivalences} 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 \begin{displaymath} \itexarray{ u^* E = B' \times_B E &\to &E \\ &{}_{\in W}\searrow^{u^*(p \times Id)} && {}_{\in W}\searrow^{p \times Id} && \searrow^{Id} \\ && u^* E_1 &\to& E_1 & \stackrel{\in W \cap F}{\to} & E \\ &&\downarrow^{\mathrlap{\in F}}&&\downarrow^{\mathrlap{\in F}} && \downarrow^{\mathrlap{p \in F}} \\ &&&&&& B \\ &&\downarrow&&\downarrow && \downarrow^{\mathrlap{v \in F \cap W}} \\ && B' &\stackrel{u}{\to}& B &\stackrel{v \in W \cap F}{\to}& B' } \end{displaymath} 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. \end{proof} \begin{remark} \label{}\hypertarget{}{} [[model category|Model categories]] that satisfy this property are called [[right proper model category|right proper model categories]]. Right properness is a crucial assumption in the closely related work \begin{itemize}% \item Jardine, \emph{Cocycle categories} (\href{http://arxiv.org/abs/math.AT/0605198}{arXiv}) \end{itemize} \end{remark} \hypertarget{homotopy_fiber_product}{}\subsubsection*{{Homotopy fiber product}}\label{homotopy_fiber_product} Using the existence of [[path object|path space objects]] one can construct specific [[homotopy pullback]]s called \emph{homotopy fiber products} . \begin{defn} \label{HomotopyFiberProducts}\hypertarget{HomotopyFiberProducts}{} A \textbf{homotopy fiber product} or \textbf{homotopy pullback} of two morphisms \begin{displaymath} A \stackrel{u}{\to} C \stackrel{v}{\leftarrow} B \end{displaymath} in a category of fibrant objects is the object $A \times_C C^I \times_C B$ defined as the (ordinary) [[nLab:limit|limit]] \begin{displaymath} \itexarray{ A \times_C C^I \times_C B &&&\to & B \\ &&&& \downarrow^v \\ & &C^I & \stackrel{d_0}{\to}& C \\ \downarrow && \downarrow^{\mathrlap{d_1}} \\ A &\stackrel{u}{\to} & C } \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} This essentially says that $A \times_C C^I \times_C B$ is the universal object that makes the diagram \begin{displaymath} \itexarray{ A \times_C C^I \times_C B &\to& B \\ \downarrow && \downarrow^{\mathrlap{v}} \\ A &\stackrel{u}{\to}& C } \end{displaymath} commute up to [[homotopy]] (see the section on homotopies for more on that). \end{remark} \begin{remark} \label{}\hypertarget{}{} These homotopy pullbacks present indeed the correct [[(infinity,1)-limits]], this is the content of prop. \ref{PullbacksAlongFibrationsAreCorrectHomotopyPullbacks} below. \end{remark} \begin{lemma} \label{}\hypertarget{}{} The projection \begin{displaymath} A \times_C C^I \times_C B \to A \end{displaymath} out of a homotopy fiber product is a fibration. If $v : B \to C$ is a weak equivalence, then this is an acyclic fibration. \end{lemma} The same is of course true for the map to $B$ and the morphism $u : A \to C$, by symmetry of the diagram. \begin{proof} One may compute this [[nLab:limit|limit]] in terms of two consecutive [[nLab:pullback|pullback]]s in two different ways. On the one hand we have \begin{displaymath} \itexarray{ A \times_C C^I \times_C B &\to& \mathbf{E}_v C &\to & B \\ && \downarrow && \downarrow^{\mathrlap{v}} \\ & &C^I & \stackrel{d_0}{\to}& C \\ \downarrow && \downarrow^{\mathrlap{d_1}} \\ A &\stackrel{u}{\to}& C } \end{displaymath} 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 \begin{displaymath} \itexarray{ A \times_C C^I \times_C B &\to& && B \\ \downarrow && && \downarrow^{\mathrlap{v}} \\ \mathbf{E}_u C & \stackrel{\in W \cap F}{\to} &C^I & \stackrel{d_0 \in W \cap F}{\to}& C \\ \downarrow^{\mathrlap{\in W \cap F}} && \;\;\downarrow^{\mathrlap{d_1\in W \cap F}} \\ A &\stackrel{u}{\to} & C } \end{displaymath} 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. \end{proof} \hypertarget{Homotopies}{}\subsubsection*{{Homotopies}}\label{Homotopies} \begin{defn} \label{}\hypertarget{}{} Two morphism $f,g : A \to B$ in $C(A,B)$ are \begin{itemize}% \item \textbf{right [[homotopy|homotopic]]}, denoted $f \simeq g$, precisely if they fit into a diagram \begin{displaymath} \itexarray{ && B \\ & {}^f\nearrow & \uparrow^{d_0} \\ A &\stackrel{\eta}{\to}& B^I \\ & {}_g\searrow & \downarrow^{\mathrlap{d_1}} \\ && B } \end{displaymath} for some [[path object|path space object]] $B^I$; \item \textbf{[[homotopy|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 \end{itemize} \begin{displaymath} \itexarray{ && A &\stackrel{f}{\to}& B \\ &{}^{w \in W}\nearrow&&& \uparrow^{d_0} \\ \hat A && \stackrel{\eta}{\to} && B^I \\ &{}_{w\in W}\searrow & && \downarrow^{d_1} \\ && A &\stackrel{g}{\to}& B } \end{displaymath} for some object $\hat A$ and for some [[path object|path space object]] $B^I$ of $I$ \end{defn} \begin{remark} \label{}\hypertarget{}{} 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. \end{remark} \begin{lemma} \label{}\hypertarget{}{} For $A,B \in \mathbf{C}$, right homotopy is an equivalence relation on the [[hom-set]] $\mathbf{C}(A,B)$. \end{lemma} \begin{proof} 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 \begin{displaymath} \itexarray{ B^{I_1 \vee I_2} &\to& B^{I_2} \\ \downarrow && \downarrow^{d_0} \\ B^{I_1} &\stackrel{d_1}{\to}& B } \end{displaymath} defines a new path object, with structure maps \begin{displaymath} B \stackrel{\sigma_1 \times \sigma_2}{\to} B^{I_1 \vee I_2} \stackrel{(d_0 \circ p_1) \times (d_1\circ p_2)}{\to} B \times B \,. \end{displaymath} 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 $B^{I_1 \vee I_2}$ \begin{displaymath} \itexarray{ && B \\ & {}^f\nearrow & \uparrow^{d_0^1} \\ A &\stackrel{\eta_1}{\to}& B^{I_1} \\ & {}_{g}\searrow& \downarrow^{\mathrlap{d_1^1}} & \nwarrow \\ && B && B^{I_1 \vee I_2} \\ & {}^{g}\nearrow & \downarrow^{\mathrlap{d_0^2}} & \swarrow \\ A &\stackrel{\eta_2}{\to}& B^{I_2} \\ & {}_h\searrow & \downarrow^{\mathrlap{d_1^2}} \\ &&B } \end{displaymath} \end{proof} 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-set]]s $\mathbf{C}(A,B)$. For that we need the following to lemmas. \begin{lemma} \label{}\hypertarget{}{} Every diagram \begin{displaymath} \itexarray{ A &\to& E \\ \;\;\downarrow^{\mathrlap{i \in W}} && \;\;\downarrow^{\mathrlap{p \in F}} \\ X &\to& B } \end{displaymath} may be refined to a diagram \begin{displaymath} \itexarray{ A &\to & X' &\to& E \\ & {}_{i}\searrow & \;\;\downarrow^{\mathrlap{t \in W \cap F}} && \;\;\downarrow^{\mathrlap{p \in F}} \\ && X &\to& B } \end{displaymath} \end{lemma} \begin{proof} Consider the pullback square \begin{displaymath} \itexarray{ A &\to& X \times_B E &\to& E \\ &{}_{i \in W}\searrow& \;\; \downarrow^{\mathrlap{\in F}} && \;\; \downarrow^{\mathrlap{\in F}} \\ && X &\to& B } \end{displaymath} and apply the [[factorization lemma]], lemma \ref{FactorizationLemma}, to factor the universal morphism $A \to X \times_B E \to E$ into the pullback as \begin{displaymath} A \stackrel{\in W}{\to} \mathbf{E} E \stackrel{\in F}{\to} E \end{displaymath} to obtain the diagram \begin{displaymath} \itexarray{ A &\stackrel{\simeq}{\to}& \mathbf{E} E &\to& E \\ &{}_{i \in W}\searrow& \;\; \downarrow^{\mathrlap{\in F}} && \;\; \downarrow^{\mathrlap{\in F}} \\ && X &\to& B } \,, \end{displaymath} where the middle vertical morphism is still a fibration, being the composite of two fibrations. By [[category with weak equivalences|2-out-of-3]] it follows that it is also a weak equivalence. \end{proof} \begin{lemma} \label{}\hypertarget{}{} 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 \begin{displaymath} \itexarray{ B &\stackrel{=}{\leftarrow}& B &\stackrel{u}{\to}& C \\ \downarrow^{\mathrlap{\sigma}} && \downarrow^{\sigma'} && \downarrow^{\mathrlap{\sigma_C}} \\ B^I &\stackrel{\in W \cap F}{\leftarrow}& B^{I'} &\stackrel{\bar u}{\to}& C^I \\ \;\;\downarrow^{d_0 \times d_1} && \;\;\downarrow^{\mathrlap{d'_0 \times d'_1}} && \;\;\downarrow^{\mathrlap{d_0^C \times d_1^C}} \\ B \times B &\stackrel{=}{\leftarrow}& B \times B &\stackrel{u \times u}{\to}& C \times C } \end{displaymath} \end{lemma} \begin{proof} Apply the lemma above to the square \begin{displaymath} \itexarray{ B &\stackrel{u}{\to}& C &\stackrel{\sigma_C}{\to}& C^I \\ \downarrow^{\mathrlap{\sigma}} &&&& \downarrow^{\mathrlap{d_0 \times d_1}} \\ B^I &\stackrel{d_0 \times d_1}{\to}& B \times B &\stackrel{u \times u}{\to}& C \times C } \,. \end{displaymath} \end{proof} \begin{prop} \label{}\hypertarget{}{} 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 \begin{itemize}% \item 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. \item moreover, the right homotopy may be realized with every given choice of \newline [[nLab:path object|path space object]] $D^I$ for $D$. \end{itemize} \end{prop} \begin{proof} We decompose this into two statements: \begin{enumerate}% \item for any $A \to B$ the morphisms $A \to B \stackrel{f,g}{\to} B$ are right homotopic. \item 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 $\eta : B' \to D^I$. \end{enumerate} 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 \begin{displaymath} \itexarray{ B' &\to& C^{I'} &\stackrel{\bar u}{\to}& D^I \\ \;\;\;\downarrow^{\mathrlap{\in W \cap F}} && \;\;\;\downarrow^{\mathrlap{\in W \cap F}} && \downarrow \\ B &\stackrel{\eta}{\to}& C^I \\ &{}_{f \times g}\searrow & \downarrow && \downarrow \\ && C \times C &\stackrel{u \times u}{\to}& D \times D } \end{displaymath} \end{proof} We need one more intermediate result for seeing that homotopy is an equivalence relation \begin{lemma} \label{}\hypertarget{}{} \begin{itemize}% \item Every diagram \begin{displaymath} \itexarray{ && B \\ && \downarrow^{\mathrlap{w \in W}} \\ A &\to& C } \end{displaymath} in $\mathbf{C}$ extends to a (right) homtopy-commutative diagram \begin{displaymath} \itexarray{ A' &\to & B \\ \downarrow^{\mathrlap{w' \in W}} && \downarrow^{\mathrlap{w \in W}} \\ A &\to& C } \,. \end{displaymath} \item For every pair of morphisms \begin{displaymath} f, g, A \stackrel{\to}{\to} B \end{displaymath} 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 $f \circ t' \simeq g \circ t'$. \end{itemize} \end{lemma} \begin{proof} \begin{itemize}% \item 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. \item 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 \begin{displaymath} \itexarray{ D := B \times_C C^I \times_C B &\to&&\stackrel{\in W}{\to}& B \\ \downarrow^{\mathrlap{\in W}} &&&& \downarrow^{\mathrlap{t \in W}} \\ && C^I &\stackrel{d_0}{\to}& C \\ \downarrow && \downarrow^{d_1} \\ B &\stackrel{t \in W}{\to}& C } \,, \end{displaymath} where the long morphisms are weak equivalences by the lemma on morphisms out of homotopy fiber products. \end{itemize} Then consider the two universal morphisms \begin{displaymath} (f,\eta,g) : A \to B \times_C C^I \times_C B \end{displaymath} and \begin{displaymath} (Id, \sigma \circ t, Id) : B \stackrel{\in W}{\to} B \times_C C^I \times_C B \end{displaymath} into that. It follows by 2-out-of-3 that the latter is a weak equivalence. Factoring this using the factorization lemma produces hence \begin{displaymath} B \stackrel{\in W}{\to} D' \stackrel{\in W \cap F}{\to} D \,. \end{displaymath} 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 \begin{displaymath} \itexarray{ D &\to& C^I \\ \downarrow && \downarrow^{\mathrlap{\in F}} \\ B \times B &\stackrel{f \times g}{\to}& C \times C } \,. \end{displaymath} 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 $f \circ t' \simeq g \circ t'$. \begin{displaymath} \itexarray{ A' &\to& D' \\ \downarrow^{\mathrlap{t'}} && \downarrow \\ A &\to & D &\to & C^I \\ & {}_{f \times g}\searrow & \downarrow && \downarrow \\ && B \times B &\stackrel{t \times t}{\to}& C \times C } \,. \end{displaymath} \end{proof} \begin{lemma} \label{}\hypertarget{}{} The relation ``$f, g \in C(A,B)$ are homotopic'', $f \sim g$, is an [[equivalence relation]] on $C(A,B)$. \end{lemma} \begin{proof} 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. \end{proof} \begin{defn} \label{}\hypertarget{}{} For $C$ a category of fibrant objects the category $\pi C$ is defined to be the category \begin{itemize}% \item with the same objects as $C$; \item with [[hom-set]]s the set of equivalence classes \begin{displaymath} \pi C(A,B) := C(A,B)/_\sim \end{displaymath} under the above equivalence relation. \item Composition in $\pi C$ is given by composition of representatives in $C$. \end{itemize} \end{defn} \begin{defn} \label{}\hypertarget{}{} The obvious functor \begin{displaymath} C \to \pi C \end{displaymath} is the identity on objects and the projection to [[equivalence relation|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. \end{defn} \begin{theorem} \label{}\hypertarget{}{} The weak equivalences in $\pi C$ form a left multiplicative system. \end{theorem} \begin{proof} This is now a direct consequence of the above lemma on homotopy-commutative completions of diagrams. \end{proof} \hypertarget{HomotopyCategory}{}\subsubsection*{{The homotopy category}}\label{HomotopyCategory} We discuss now that the structure of a category of fibrant objects on a [[homotopical category]] $C$ induces \begin{itemize}% \item a related category $\pi C$ \item with a morphism $C \to \pi C$ \begin{itemize}% \item that is the identity on objects, \item and induces on $\pi C$ a notion of weak equivalences \begin{displaymath} \pi W \subset Mor(\pi C) \end{displaymath} and fibrations \begin{displaymath} \pi F \subset Mor(\pi C) \end{displaymath} \end{itemize} \item such that \begin{itemize}% \item the weak equivalences in $\pi C$ form a [[calculus of fractions|left multiplicative system]] . \end{itemize} \end{itemize} This implies the following convenient construction of the [[homotopy category]] of $C$: \begin{theorem} \label{}\hypertarget{}{} For $C$ a category of fibrant objects, its [[homotopy category]] is ([[equivalence of categories|equivalent]] to) the category $Ho_C$ with \begin{itemize}% \item the same objects as $C$; \item the [[hom-set]] $Ho_C(A,B)$ for all $A, B \in Obj(C)$ given naturally by \begin{displaymath} \begin{aligned} Ho_C(A,B) & \simeq colim_{\hat A \stackrel{w\in \pi W}{\to} A} \pi C (\hat A,B) \\ & = colim_{\hat A \stackrel{f\in \pi W\cap F}{\to} A} \pi C (\hat A,B) \end{aligned} \,. \end{displaymath} \end{itemize} \end{theorem} Here the colimit is, as described at [[calculus of fractions|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 \begin{displaymath} \itexarray{ \hat A &&\stackrel{h}{\to}&& \hat A' \\ & \searrow && \swarrow \\ && A } \end{displaymath} in $\pi C$ (i.e. for arbitrary $h$). So more in detail the above colimit is over the functor \begin{displaymath} \pi C(-, B)_A : (\pi W_A)^{op} \to (\pi C)^{op} \stackrel{\pi C(-, B)}{\to} Set \,, \end{displaymath} where the first functor is the obvious forgetful functor. \begin{remark} \label{}\hypertarget{}{} 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 \emph{acyclic fibrations}: every cocycle $\itexarray{ Y &\stackrel{g}{\to}& A \\ {}^\simeq \downarrow^{\mathrlap{f}} \\ X }$ out of a weak equivalence is refines by a cocycle out of an acyclic fibrantion, namely \begin{displaymath} \itexarray{ \mathbf{E}_f X &\stackrel{\simeq}{\to}& Y &\stackrel{g}{\to}& A \\ &{}_{\in F \cap W}\searrow& {}^\simeq \downarrow^{\mathrlap{f}} \\ && X } \,. \end{displaymath} 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: \href{http://www.math.uiuc.edu/K-theory/0782/}{Cocycle categories}. \end{remark} We now provide the missing definitions and then the proof of this theorem. \begin{lemma} \label{}\hypertarget{}{} The homotopy categories of $C$ and $\pi C$ coincide: \begin{displaymath} Ho_C \simeq Ho_{\pi C} \,. \end{displaymath} \end{lemma} \begin{proof} By one of the lemmas above, the morphisms $d_i : B^I \to B$ are weak equivalences and become [[isomorphism]]s 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 \begin{displaymath} Q_C : C \to Ho_C \end{displaymath} factors through $\pi C$ as \begin{displaymath} Q_C : C \to \pi C \stackrel{Q_{\pi C}}{\to} Ho_C \end{displaymath} 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 $Q_{\pi C}$ \begin{displaymath} \itexarray{ C &\to& \pi C &\to & A \\ \downarrow^{\mathrlap{Q_C}} & \swarrow^{Q_{\pi C}} && \swarrow \\ Ho_C } \,. \end{displaymath} \end{proof} The above theorem on the description of $Ho_C$ now follows from the general formula for [[localization]] at a [[calculus of fractions|left multiplicative system]] of weak equivalences. \hypertarget{pointed_category_of_fibrant_objects}{}\subsubsection*{{Pointed category of fibrant objects}}\label{pointed_category_of_fibrant_objects} If the category $C$ of fibrant objects has an initial object which \emph{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. \hypertarget{fibers}{}\subsubsection*{{Fibers}}\label{fibers} For $p : Y \to X$ a fibration, the pullback $F$ in \begin{displaymath} \itexarray{ F &\stackrel{i}{\to}& Y \\ \downarrow && \downarrow \\ e &\to& X } \end{displaymath} is the \textbf{fibre} of $p$ and $i$ is the \emph{fibre inclusion}. (This is the \emph{kernel} of the morphism $f$ of [[pointed object]]s) \hypertarget{fibration_sequences}{}\subsubsection*{{Fibration Sequences}}\label{fibration_sequences} (See also [[fibration sequence]]) For $B$ any object and $B^I$ any of its [[path object]]s, the fiber of $B^I \stackrel{d_0 \times d_1}{\to} B \times B$ is the \textbf{[[loop space object|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 \begin{displaymath} \Omega : Ho_C \to Ho_C \end{displaymath} 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 [[internalization|internal to]] $Ho_C$ in a natural way. \hypertarget{DerivedHomSpaces}{}\subsubsection*{{Derived hom-spaces}}\label{DerivedHomSpaces} 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 (\hyperlink{Cisinski10}{Cisinksi 10}) and (\hyperlink{NSS12}{Nikolaus-Schreiber-Stevenson 12, section 3.6.2}). \begin{defn} \label{CocycleCategories}\hypertarget{CocycleCategories}{} For $\mathcal{C}$ a category of fibrant objects, write for any $X, A \in Obj(\mathcal{C})$ \begin{displaymath} Cocycle(X,A) , wCocycle(X,A) \in Cat \end{displaymath} for the [[categories]] (``categories of [[cocycles]] on $X$ with [[coefficients]] in $A$'') whose [[objects]] are [[correspondences]] \begin{displaymath} X \stackrel{\simeq}{\longleftarrow} \hat X \stackrel{}{\longrightarrow} A \end{displaymath} 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 \begin{displaymath} \itexarray{ && \hat X \\ & \swarrow && \searrow \\ X && \downarrow && A \\ & \nwarrow && \nearrow \\ && \hat X' } \end{displaymath} \end{defn} \begin{prop} \label{CocycleCategoriesPresentDerivedHomSpaces}\hypertarget{CocycleCategoriesPresentDerivedHomSpaces}{} 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 \begin{displaymath} N Cocycle(X,A) \to N wCocycle(X,A) \to L^H_{we} \mathcal{C}(X,A) \end{displaymath} of [[simplicial sets]] (on the left the [[nerves]] of the cocycle categories of def. \ref{CocycleCategories}, on the right the [[derived hom space]] given by the [[simplicial localization]]) are [[weak homotopy equivalences]]. \end{prop} 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 \begin{prop} \label{PullbacksAlongFibrationsAreCorrectHomotopyPullbacks}\hypertarget{PullbacksAlongFibrationsAreCorrectHomotopyPullbacks}{} The [[homotopy fiber products]] in $\mathcal{C}$ as defined in def. \ref{HomotopyFiberProducts} present indeed the correct [[(infinity,1)-limits]]. \end{prop} \begin{proof} Observe that for each object $X$ the 2-functor $N Cocycle(X,-) \colon \mathcal{C} \to sSet$ of def. \ref{CocycleCategories} 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. \ref{CocycleCategoriesPresentDerivedHomSpaces} 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 \emph{[[homotopy Kan extension]]}) that the original pullbacks in $\mathcal{C}$ are the correct homotopy pullbacks. \end{proof} \hypertarget{application_in_cohomology_theory}{}\subsection*{{Application in cohomology theory}}\label{application_in_cohomology_theory} When the catgegory of fibrant objects is that of locally Kan simplicial sheaves, the [[hom-set]]s 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. \hypertarget{related_concept}{}\subsection*{{Related concept}}\label{related_concept} \begin{itemize}% \item [[category of cofibrant objects]] \item [[Waldhausen category]] \item [[model category]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The notion of \emph{category of fibrant objects} was introduced and the above results obtained in \begin{itemize}% \item [[Kenneth Brown]], \emph{[[BrownAbstractHomotopyTheory.pdf:file]]}, Transactions of the American Mathematical Society, Vol. 186 (1973), 419-458 ([[BrownAHT]]). \end{itemize} for application to [[homotopical cohomology theory]]. A review is in \href{http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-1.dvi}{section I.9} of \begin{itemize}% \item [[Paul Goerss]] and [[Rick Jardine]], 1999, \emph{[[Simplicial homotopy theory]]}, number 174 in Progress in Mathematics, Birkhauser. (\href{http://www.maths.abdn.ac.uk/~bensondj/html/archive/goerss-jardine.html}{ps}) \end{itemize} There is a description and discussion of this theory and its dual (using cofibrant objects) in \begin{itemize}% \item [[K. H. Kamps]] and [[Tim Porter]], \emph{Abstract Homotopy and Simple Homotopy Theory} (\href{http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1}{GoogleBooks}) \end{itemize} Discussion of embeddings of categories of fibrant objects into model categories is in \begin{itemize}% \item [[Denis-Charles Cisinski]], \emph{Invariance de la K-Th\'e{}orie par \'e{}quivalences d\'e{}riv\'e{}es} (\href{http://www.math.univ-toulouse.fr/~dcisinsk/eqderkth.pdf}{pdf}) \end{itemize} 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 \begin{itemize}% \item [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], \emph{[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- Presentations]]} (\href{http://arxiv.org/abs/1207.0249}{arXiv:1207.0249}) \end{itemize} and also in \begin{itemize}% \item [[Geoffroy Horel]], \emph{Brown categories and bicategories}, \href{https://arxiv.org/abs/1506.02851}{arxiv} \end{itemize} Usage of categories of fibrant objects for the [[homotopical structure on C\emph{-algebras]] is in :} \begin{itemize}% \item [[Otgonbayar Uuye]], \emph{Homotopy theory for $C^\ast$-algebras}, Journal of Noncommutative Geometry, (\href{http://arxiv.org/abs/1011.2926}{arxiv:1011.2926}) \end{itemize} Categories of fibrant objects form a convenient setting for the study of [[homotopy type theory]]: \begin{itemize}% \item Jeremy Avigad, Chris Kapulkin, Peter LeFanu Lumsdaine, \emph{Homotopy limits in type theory} \href{http://arxiv.org/abs/1304.0680}{1304.0680} \end{itemize} It is shown in the following paper that categories of fibrant objects are themselves fibrant in the [[model structure on categories with weak equivalences]]: \begin{itemize}% \item Lennart Meier, \emph{Fibration Categories are Fibrant Relative Categories}, \href{https://arxiv.org/abs/1503.02036}{arxiv} \end{itemize} [[!redirects categories of fibrant objects]] \end{document}