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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*{codiscrete cofibration} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{2category_theory}{}\paragraph*{{2-category theory}}\label{2category_theory} [[!include 2-category theory - 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{enriched_categories}{Enriched categories}\dotfill \pageref*{enriched_categories} \linebreak \noindent\hyperlink{toposes}{Toposes}\dotfill \pageref*{toposes} \linebreak \noindent\hyperlink{double_categories}{Double categories}\dotfill \pageref*{double_categories} \linebreak \noindent\hyperlink{construction_of_a_proarrow_equipment}{Construction of a proarrow equipment}\dotfill \pageref*{construction_of_a_proarrow_equipment} \linebreak \noindent\hyperlink{equippable_2categories}{Equippable 2-categories}\dotfill \pageref*{equippable_2categories} \linebreak \noindent\hyperlink{the_construction}{The construction}\dotfill \pageref*{the_construction} \linebreak \noindent\hyperlink{canonical_factorization_systems}{Canonical factorization systems}\dotfill \pageref*{canonical_factorization_systems} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A [[2-category]] is a good context for doing a lot of category theory (including [[internal category]] theory, [[enriched category]] theory, and so on), but there are some things that are hard to do there, such as to talk about [[weighted limits]] and colimits. This leads one to introduce the notion of a [[2-category equipped with proarrows]], which is a 2-category along with extra data that plays the role of [[profunctors]], allowing the definition of weighted limits and other aspects of category theory. However, it would also be nice if the extra data in a proarrow equipment were somehow determined by the 2-category we started with. This is especially so when talking about functors between equipments, since functors between 2-categories are often easier to construct. It turns out that in many cases, including the most common ones, this is the case: we can construct the proarrows in terms of the underlying 2-category. This was originally realized by Ross Street. The idea is to identify a profunctor with its [[collage]], aka its [[cograph of a profunctor|cograph]], which is a special sort of [[cospan]] in $Cat$ (or $V Cat$, or whatever other 2-category one wants to start with). One then simply has to characterize, in 2-categorical terms, which cospans are collages, and how to do things like compose them. It turns out that in most cases the characterization is precisely that they are the \textbf{two-sided codiscrete cofibrations} --- i.e. the [[two-sided fibration|two-sided discrete fibrations]] in the [[opposite 2-category]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Suppose that $K$ is a 2-category with finite [[2-colimits]], and $A,C\in K$. A [[cofibration]] from $A$ to $C$ is a [[cospan]] $A\to B \leftarrow C$ which is an [[two-sided fibration|internal two-sided fibration]] in $K^{op}$. As remarked at [[fibration in a 2-category]], there is a [[2-monad]] on $Span_{K^{op}}(A,C)$ whose algebras are such fibrations. In other words, there is a \emph{2-comonad} on $Cospan_K(A,C)$ whose \emph{coalgebras} are such fibrations. This 2-comonad is defined by \begin{displaymath} (A\to B \leftarrow C) \quad \mapsto\quad (A \to (A\times I) +_A B +_C (C\times I) \leftarrow C) \end{displaymath} where $I$ is the [[interval category]] $(0\to 1)$ and $(-\times I)$ denotes the [[copower]] with $I$. In the pushouts, the map $A\to A\times I$ is the inclusion at $0$ and $C\to C\times I$ is the inclusion at $1$. A cospan $A\to B \leftarrow C$ in a 2-category $K$ is \textbf{codiscrete} if it is codiscrete in the 2-category $Cospan_K(A,C)\simeq (A+C)/K$. This means that for any $X\in Cospan(A,C)$, the hom-category $Cospan(A,C)(B,X)$ is equivalent to a [[discrete category]]. Explicitly, it means that given any two morphisms $B \;\rightrightarrows\; X$ of cospans, if there exists a 2-cell from one to the other in $Cospan(A,C)$, then it is unique and invertible. A \textbf{codiscrete cofibration} is a two-sided cofibration which is codiscrete as a cospan. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{enriched_categories}{}\subsubsection*{{Enriched categories}}\label{enriched_categories} We sketch a characterization of cofibrations in $V Cat$, where $V$ is any B\'e{}nabou [[cosmos]]. Let $A\overset{f}{\to} B \overset{g}{\leftarrow} C$ be a cospan and let $D = (A\times I) +_A B +_C (C\times I)$. We claim that $D$ has the following description. \begin{itemize}% \item Its objects are the disjoint union of those of $A$, $B$, and $C$, i.e. $ob(D) = ob(A) \sqcup ob(B) \sqcup ob(C)$. \item $A$ and $B$ and $C$ are (disjoint) full subcategories of $D$. \item There are no morphisms in $D$ from $A$ to $B$, or from $B$ to $C$, or from $A$ to $C$. That is, for $a\in A$, $b\in B$, and $c\in C$ we have $D(a,b) = D(b,c) = D(a,c) = \emptyset$. \item If $a\in A$, $b\in B$, and $c\in C$, we have $D(b,a) = B(b,f a)$, $D(c,b) = B(g c, b)$, and $D(c,a) = B(g c, f a)$. \end{itemize} That $D$ is a $V$-category is immediate, and it is easy to check the universal property. We write $A \overset{i}{\to} D \overset{j}{\leftarrow} C$ for the inclusions. Now suppose that $B$ is a coalgebra for the 2-comonad in question. Therefore, in particular we have a map $h\colon B\to D$ in $Cospan(A,C)$, so that $h f = i$ and $h g = j$ (or perhaps only isomorphic; it really makes no difference here). Moreover, the counit of the comonad is the obvious map $k\colon D\to B$, so we must have $k h = 1_B$. Since $i$ and $j$ are injective on objects and have disjoint images, so must be $f$ and $g$. And since $i$ and $j$ are fully faithful $V$-functors, the action of $f$ and $g$ on homs must be split monic in $V$, and the action of $h$ on homs in $A$ and $B$ must be split epic. But since $h k = 1_B$, the action of $h$ on homs must also be split monic, hence an isomorphism, and hence so must that of $f$ and $g$ be. Therefore, $f$ and $g$ are fully faithful inclusions with disjoint images. Clearly $h$ must take the images of $f$ and $g$ to the images of $i$ and $j$, respectively. Because $k h = 1_B$, it must be that $h$ takes the rest of $B$ to itself, sitting in the canonical copy of $B$ inside $D$. This uniquely defines $h$, as long as $B$ satisfies the condition that \begin{itemize}% \item For $a\in A$, $c\in C$, and $b\in B \setminus (A\cup C)$, we have $B(a,b) = B(b,c) = B(a,c) = \emptyset$. \end{itemize} It is then easy to check that if $f$ and $g$ are fully faithful with disjoint images and this condition holds, then $B$ is in fact a coalgebra for the comonad in question, i.e. a two-sided cofibration from $A$ to $C$. Note that such a cofibration from $A$ to $C$ can be identified with the following data: a category $B' = B\setminus (A\cup C)$, profunctors $m\colon A ⇸ B$, $n\colon B ⇸ C$, and $p\colon A ⇸ C$, and a morphism $n m \to p$ of profunctors. Such a thing is sometimes called a \textbf{gamut} from $A$ to $C$. Now a 2-cell in $Cospan(A,C)$ is simply a natural transformation between functors $B \;\rightrightarrows\; X$ whose components on the images of $A$ and $C$ are isomorphisms. Thus, if $B$ is a cofibration as above with the property that $B \setminus (A\cup C)$ is empty, then it must be codiscrete. The converse is easy to check, taking $X$ to be the ordinal $4 = (0\le 1 \le 2\le 3)$ as a category. But a gamut with $B'=\emptyset$ is nothing but a profunctor $A ⇸C$; hence codiscrete cofibrations in $V Cat$ can be precisely identified with the collages of profunctors. \hypertarget{toposes}{}\subsubsection*{{Toposes}}\label{toposes} A codiscrete cofibration in the 2-category $Topoi$ of [[topoi]] can be identified with a [[left exact functor]]. \hypertarget{double_categories}{}\subsubsection*{{Double categories}}\label{double_categories} Codiscrete cofibrations in the 2-category $Dbl$ of [[double categories]], [[double functors]], and [[horizontal transformations]] can be identified with [[double profunctors]]. \hypertarget{construction_of_a_proarrow_equipment}{}\subsection*{{Construction of a proarrow equipment}}\label{construction_of_a_proarrow_equipment} The examples of profunctors suggest that given any 2-category $K$ with finite 2-colimits, we may try to canonically equip it with proarrows by defining the proarrows $A ⇸C$ to be the codiscrete cofibrations. The sticky point is then how to define units and composition of such proarrows in order to obtain an equipment. The unit is obvious: we should take the unit proarrow of $A$ to be the cospan $A\to A\times I \leftarrow A$, which is always a codiscrete cofibration. Binary composition is subtler. The obvious way to compose codiscrete cofibrations $A \to B \leftarrow C$ and $C\to D \leftarrow E$, of course, is to take a [[pushout]] $B +_C D$. It is not hard to show (see references): \begin{utheorem} In any 2-category with finite 2-colimits, if $B$ and $D$ are cofibrations, then so is $B +_C D$. \end{utheorem} However, $B +_C D$ will not be codiscrete even if $B$ and $D$ are. In $V Cat$, if $B$ and $D$ are collages of profunctors $m$ and $n$, then $B +_C D$ represents the gamut consisting of $m$, $n$, and the composite profunctor $n m$, with the middle category being $C$. Thus, in order to obtain the correct composite, we need to forget about $C$ somehow. The best way to do this seems to be using a [[factorization system in a 2-category]], akin the way in which we construct the [[bicategory of relations]] from any [[regular category]]. \hypertarget{equippable_2categories}{}\subsubsection*{{Equippable 2-categories}}\label{equippable_2categories} We are looking for a [[factorization system in a 2-category|2-categorical factorization system]] $(\mathcal{E},\mathcal{M})$ in $K$ such that if we have a two-sided cofibration $A\to C\leftarrow B$ and we factor $A+B \to C$ into an $\mathcal{E}$-map and an $\mathcal{M}$-map, then the resulting cospan $A\to E \leftarrow B$ is a codiscrete cofibration. Codiscreteness means in particular that the $\mathcal{E}$-map $A+B\to E$ should be codiscrete, i.e. representably cofaithful and co-conservative. Moreover, if $A\to C\leftarrow B$ was already a codiscrete cofibration, then $A+B\to C$ should already be in $\mathcal{E}$. This suggests the following definition. \begin{udefn} A 2-category with finite 2-limits and 2-colimits is \textbf{pre-equippable} if it has a factorization system $(\mathcal{E},\mathcal{M})$ such that \begin{itemize}% \item if $A\to C \leftarrow B$ is a codiscrete cofibration, then $A+B \to C$ is in $\mathcal{E}$, and \item every morphism in $\mathcal{E}$ is representably co-faithful and co-conservative. \end{itemize} It is \textbf{equippable} if in addition it satisfies: \begin{itemize}% \item Morphisms in $\mathcal{M}$ are closed under pushout and tensors with $I$. \end{itemize} \end{udefn} Co-conservative morphisms are also called \textbf{liberal}. Recall that by definition of codiscreteness, if $A\to C \leftarrow B$ is a codiscrete cofibration, then $A+B\to C$ is cofaithful and liberal; thus the first two conditions are compatible. The example to keep in mind is $V Cat$, for any suitable $V$, where $\mathcal{E}$ is the class of essentially surjective $V$-functors and $\mathcal{M}$ is the class of $V$-fully-faithful functors. \begin{uprop} Any morphism which is right orthogonal to codiscrete cofibrations is [[fully faithful morphism|representably fully faithful]]. In particular, if $K$ is pre-equippable, then every morphism in $\mathcal{M}$ is representably fully faithful. \end{uprop} \begin{proof} For any $X$ in $K$, we have a codiscrete cofibration $X\to X \times I \leftarrow X$, and thus $X+X \to X\times I$ is in $\mathcal{E}$. But orthogonality with respect to all such morphisms is precisely representable fully-faithfulness. \end{proof} \begin{uprop} Any representably fully faithful morphism is right orthogonal to any [[cocomma object]]. In particular, $K$ is pre-equippable and every codiscrete cofibration is a cocomma object, then $\mathcal{M}$ is precisely the class of representably fully faithful morphisms. \end{uprop} \begin{proof} Maps out of a cocomma object are in canonical correspondence with 2-cells in $K$. But representable fully-faithfulness means that 2-cells lift uniquely along such a map. Hence so do maps out of a cocomma object, and hence any representably fully faithful map is right orthogonal to all cocomma cospans. \end{proof} \begin{uprop} If $K$ is pre-equippable, then any [[inverter]] or [[equifier]] is in $\mathcal{M}$, and every morphism in $\mathcal{E}$ is cofaithful and liberal. \end{uprop} \begin{proof} Any inverter is always right orthogonal to any liberal morphism, and any equifier is always right orthogonal to any cofaithful morphism. \end{proof} \hypertarget{the_construction}{}\subsubsection*{{The construction}}\label{the_construction} In an equippable 2-category, we can compose cofibrations in the desired way. \begin{uprop} If $K$ is equippable, $A\to E \leftarrow B$ is a two-sided cofibration, and $A+B \to F \to E$ is an $(\mathcal{E},\mathcal{M})$-factorization, then $A\to F \leftarrow B$ is a codiscrete cofibration. In particular, the category $CodCofib(A,B)$ is coreflective in the 2-category $Cofib(A,B)$. \end{uprop} \begin{proof} Since $\mathcal{E}$-morphisms are cofaithful and liberal, $A\to F \leftarrow B$ is certainly codiscrete. That it is a cofibration is proven as in (MB, 4.18). Coreflectivity follows by orthogonality for the factorization system $(\mathcal{E},\mathcal{M})$, since all codiscrete cofibrations are in $\mathcal{E}$ by assumption. \end{proof} Therefore, in an equippable 2-category, we can define the composite of codiscrete cofibrations $A\to B\leftarrow C$ and $C\to D\leftarrow E$ to be the codiscrete coreflection of the cofibration $A \to B +_C D\leftarrow E$. \begin{uprop} If $K$ is equippable, there is a 2-category $CodCofib(K)$, with the same objects as $K$, and with codiscrete cofibrations as 1-morphisms. Moreover, there is a [[locally fully faithful 2-functor|locally fully faithful]] identity-on-objects (pseudo) 2-functor $(-)_* K\to CodCofib(K)$ such that each 1-morphism $f_*$ has a right adjoint. Therefore, $K$ is canonically a [[2-category equipped with proarrows]] (hence the term ``equippable''). \end{uprop} \begin{proof} This is essentially (MB, 4.20). \end{proof} One can then impose additional axioms on $K$ to get good behavior of this equipment, and try to characterize the equipments arising in this way; see (MB, section 5) and (PC). \hypertarget{canonical_factorization_systems}{}\subsubsection*{{Canonical factorization systems}}\label{canonical_factorization_systems} Note that since coreflections are determined by a universal property, the composite of codiscrete cofibrations is independent of the chosen factorization system $(\mathcal{E},\mathcal{M})$. In fact, there are two different ``extreme'' ways that we might try to define an equippable factorization system; we could either \begin{enumerate}% \item Define $\mathcal{E}$ to be the class of liberal and cofaithful morphisms, or \item Define $\mathcal{E}$ to be generated by the class of codiscrete cofibrations. \end{enumerate} In the second case we mean that $\mathcal{M}$ is the class of all morphisms right orthogonal to the morphisms $A+B\to C$ such that $A\to C \leftarrow B$ is a codiscrete cofibration, and then $\mathcal{E}$ is the class of all morphisms left orthogonal to $\mathcal{M}$. This implies, of course, that $\mathcal{E}$ contains the codiscrete cofibrations. Neither of the above choices is guaranteed to produce a factorization system (since the factorizations may not exist), but if either one does, then that factorization system is automatically pre-equippable. In the first case this is obvious, since all codiscrete cofibrations are cofaithful and liberal, while in the second case, it follows since inverters and equifiers are then necessarily in $\mathcal{M}$, and anything left orthogonal to inverters and equifiers must be cofaithful and liberal. Thus, a 2-category is equippable if either of these two choices produces a factorization system for which $\mathcal{M}$ is closed under pushout and tensors with $I$. \begin{uprop} The (essentially surjective, $V$-fully faithful) factorization system is generated by the codiscrete cofibrations, and is equippable. \end{uprop} \begin{proof} It suffices to show that a $V$-functor $f\colon A\to B$ is right orthogonal to codiscrete cofibrations if and only if it is $V$-fully faithful, i.e. each morphism $A(a,a') \to B(f a, f a')$ is an isomorphism in $V$. For ``if'', it suffices to observe that $V$-fully faithful functors are right orthogonal to all essentially surjective ones, and any codiscrete cofibration is essentially surjective. For ``only if,'' suppose given $a,a'\in A$, let $X=Y=I$ be the unit $V$-category, consider the object $B(f a,f a')\in V$ as a $V$-profunctor $X \to Y$, and let $E$ be its [[collage]]. Then we have a square \begin{displaymath} \itexarray{X\sqcup Y & \overset{[a,a']}{\to} & A\\ \downarrow && \downarrow\\ E & \underset{[f a, f a']}{\to} & B} \end{displaymath} where the bottom arrow is the identity on the nontrivial hom-object $B(f a,f a')$. A lifting in this square supplies a [[section]] of $A(a,a') \to B(f a, f a')$, and uniqueness of lifting against the collage of $A(a,a')$ (also as a profunctor $I\to I$) shows that it is an inverse isomorphism; hence $f$ is $V$-fully faithful. Finally, it is straightforward to verify that $V$-fully-faithful functors are closed under pushout and tensors with $I$. \end{proof} \begin{uprop} In $V Cat$, every liberal is automatically cofaithful, and there is a pre-equippable factorization system in which $\mathcal{E}$ is the class of liberal morphisms. However, it is not equippable, even when $V=Set$. \end{uprop} \begin{proof} This is essentially (MB, 3.4). In this case $\mathcal{M}$ consists of the $V$-fully faithful morphisms which are additionally closed under [[absolute colimits]], while $\mathcal{E}$ consists of the functors which are surjective up to absolute colimits (``Cauchy dense'' functors). When $V=Set$, all absolute colimits are generated by retracts, and it is easy to construct an example of a fully faithful functor closed under retracts and a pushout of it which is no longer closed under retracts. \end{proof} An equippable 2-category with $\mathcal{E} =$ liberal cofaithfuls = liberals is called \textbf{faithfully co-conservational} in (MB). This is the only case considered there, but the proofs generalize directly to any equippable 2-category. Note that $V Cat$ is \emph{not} faithfully co-conservational, since the above factorization system is only pre-equippable: $\mathcal{M}$ is not closed under pushout. Its sub-2-category $V Cat_{cc}$ of [[Cauchy complete category|Cauchy complete]] $V$-categories is faithfully co-conservational, but this is arguably just because when restricted to $V Cat_{cc}$, the above factorization coincides with the other, better one. Thus, it seems that perhaps in general it is better to consider the factorization system generated by the codiscrete cofibrations. \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Ross Street]], ``Fibrations in bicategories'', \href{http://www.ams.org/mathscinet-getitem?mr=574662}{MR}, and \href{http://www.ams.org/mathscinet-getitem?mr=903151}{correction}. \item [[Aurelio Carboni]] and [[Scott Johnson]] and [[Ross Street]] and [[Dominic Verity]], ``Modulated bicategories'' (MB) \href{http://www.ams.org/mathscinet-getitem?mr=1285544}{MR}. \item [[Bob Rosebrugh]] and [[Richard Wood]], ``Proarrows and cofibrations'' (PC), \href{http://www.ams.org/mathscinet-getitem?mr=961365}{MR} \end{itemize} [[!redirects codiscrete cofibrations]] [[!redirects two-sided cofibration]] [[!redirects two-sided cofibrations]] [[!redirects two-sided codiscrete cofibration]] [[!redirects two-sided codiscrete cofibrations]] [[!redirects gamut]] [[!redirects gamuts]] [[!redirects conservational bicategory]] [[!redirects conservational bicategories]] [[!redirects faithfully conservational bicategory]] [[!redirects faithfully conservational bicategories]] [[!redirects co-conservational bicategory]] [[!redirects co-conservational bicategories]] [[!redirects faithfully co-conservational bicategory]] [[!redirects faithfully co-conservational bicategories]] [[!redirects modulated bicategory]] [[!redirects modulated bicategories]] [[!redirects comodulated bicategory]] [[!redirects comodulated bicategories]] \end{document}