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\begin{document}

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\section*{M-category}

\hypertarget{categories}{}\section*{{$\mathcal{M}$-categories}}\label{categories}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{def}{Definitions}\dotfill \pageref*{def} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An $\mathcal{M}$-category is a [[category]] with two classes of [[morphisms]]: \emph{tight} and \emph{loose}.

A [[categorification]] of the concept of $\mathcal{M}$-category to $2$-[[2-category|categories]] is the concept of $\mathcal{F}$-[[F-category|category]]. (There are other possible categorifications, such as locally $\mathcal{M}$-enriched $2$-categories, or locally $\mathcal{M}$-enriched $\mathcal{F}$-categories.) In particular, every $\mathcal{M}$-category is an $\mathcal{F}$-category with only identity $2$-[[2-morphism|morphisms]], and an $\mathcal{F}$-category becomes an $\mathcal{M}$-category upon forgetting its nonidentity $2$-cells.

\hypertarget{def}{}\subsection*{{Definitions}}\label{def}

 Let $\mathcal{M}$ (or $Mono$) be the category whose [[objects]] are [[injections]] ([[monomorphisms]] in the [[category of sets]]) and whose [[morphisms]] are [[commutative squares]]. $\mathcal{M}$ is a [[reflective subcategory]] of the [[arrow category]] $Set^\to$ (the [[Sierpinski topos]]), where the reflector [[product-preserving functor|preserves products]]; as a result, $\mathcal{M}$ is [[complete category|complete]], [[cocomplete category|cocomplete]] [[cartesian closed category]]. (In fact, $\mathcal{M}$ is a Grothendieck [[quasitopos]], as discussed \href{free+cartesian+category#Subset}{here}: it is the category of [[separated presheaves]] for the [[double negation topology]] on $Set^\to$.)

A ([[locally small category|locally small]]) \textbf{$\mathcal{M}$-category} is simply a [[category enriched]] over $\mathcal{M}$. In other words, where the hom-objects are pairs of sets $(T, L)$ where $T \subseteq L$.

In more detail, an \textbf{$\mathcal{M}$-category} consists of the following data:

\begin{itemize}%
\item a [[class]] of \textbf{[[objects]]};


\item for each pair $x, y$ of objects, a [[set]] $L(x,y)$ of \textbf{loose [[morphisms]]};


\item for each pair $x, y$ of objects, a [[subset]] $T(x,y)$ of $L(x,y)$ of \textbf{tight morphisms};


\item for each object $x$, a tight \textbf{[[identity morphism]]} $id_x$; and


\item for each triple $x, y, z$ of objects, a [[binary operation]] \textbf{[[composition]]} $\circ_{x,y,z}$ from $L(y,z)$ and $L(x,y)$ to $L(x,z)$ such that:

\begin{itemize}%
\item $f \circ g$ is tight whenever $f, g$ are;
\item $\id_y \circ f = f = f \circ \id_x$ whenever this makes sense; and
\item $(f \circ g) \circ h = f \circ (g \circ h)$ whenever this makes sense.

\end{itemize}


\end{itemize}
Equivalently, an \textbf{$\mathcal{M}$-category} is a [[category]] of objects and \textbf{loose} morphisms together with a [[wide subcategory]] of \textbf{tight} morphisms.

Note that the ``underlying ordinary category'' of an $\mathcal{M}$-category, in the usual sense that that phrase is used in [[enriched category]] theory, is its category of \emph{tight} morphisms. This is because the underlying ordinary category is induced by a monoidal change-of-base functor $\hom(I, -) \colon \mathcal{M} \to Set$ represented by the monoidal unit $I = (1, 1)$, where we have

\begin{displaymath}
\hom_{\mathcal{M}}((1, 1), (T, L)) \cong T.
\end{displaymath}
Nevertheless, it is frequently also useful to think of the category of loose morphisms as a sort of ``underlying ordinary category'' of an $\mathcal{M}$-category. This is effected by the monoidal change-of-base functor $\hom_{\mathcal{M}}((0, 1), -) \colon \mathcal{M} \to Set$, where we have

\begin{displaymath}
\hom_{\mathcal{M}}((0, 1), (T, L)) \cong L.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Every $\dagger$-[[dagger-category|category]] is an $\mathcal{M}$-category in which the tight morphisms are the [[unitary isomorphisms]]. In particular, [[Hilbert spaces]] form an $\mathcal{M}$-category with [[unitary operators]] as tight morphisms.


\item A more interesting way to make Hilbert spaces into an $\mathcal{M}$-category [[Hilb]] uses (as in the $\dagger$-category $Hilb$) all [[bounded linear operators]] as loose morphisms but only [[short linear operators]] (those with norm at most $1$) as tight morphisms. This gives the same tight isomorphisms as the $\dagger$-category $Hilb$ (but also has non-invertible tight morphisms).


\item Similarly, [[Ban]] (the category of [[Banach spaces]]) is an $\mathcal{M}$-category with all bounded linear operators as loose morphisms but only short linear operators as tight morphisms. In [[functional analysis]], a loose isomorphism in $Ban$ is traditionally called an `[[norm isomorphism|isomorphism]]' while a tight isomorphism is called a `[[global isometry]]'.


\item We can make [[Met]] (the category of [[metric spaces]]) into an $\mathcal{M}$-category is several ways, with [[short maps]] contained in [[Lipschitz map]]s contained in [[uniformly continuous maps]] contained in [[continuous maps]]; we can also take Lipschitz maps contained in [[bounded maps]]. (For linear operators between Banach spaces, the continuous and bounded operators are the same and are already Lipschitz.)


\item Any [[strict category]] is an $\mathcal{M}$-category with [[equality|equalities]] as the tight morphisms. (Thus the wide subcategory of tight morphisms is [[skeletal category|skeletal]].) In particular, the [[category of sets]] (or any category) in [[material set theory]] is an $\mathcal{M}$-category.


\item A more interesting way to make [[material set theory]]'s [[category of sets]] into an $\mathcal{M}$-category has all [[subset]] inclusions as tight morphisms. Again the tight isomorphisms are simply the equalities.


\item Given any category $C$ and object $a$ of $C$, the [[subobjects]] of $a$ form an $\mathcal{M}$-category whose category of loose morphisms is the [[full subcategory]] of $C$ on the subobjects of $a$ and whose category of tight morphisms is the [[subobject poset]] of $a$.


\item Similarly, [[quotient objects]] form an $\mathcal{M}$-category.


\item The \textbf{[[core]]} of every category $C$ is an $\mathcal{M}$-category whose loose morphisms are the morphisms of $C$ and whose tight morphisms are the [[isomorphisms]] of $C$.


\item $\mathcal{M}$ itself is an $\mathcal{M}$-category whose objects are sets equipped with a specified subset, whose tight morphisms are functions which map the subset into each other, and whose loose morphisms are arbitrary functions.


\item Given any [[faithful functor]] $F\colon C \to D$, we may make $C$ into an $\mathcal{M}$-category whose tight morphisms are the original morphisms of $C$ and whose loose morphisms from $x$ to $y$ are the $D$-morphisms from $F(x)$ to $F(y)$. This includes all of the examples above; up to [[equivalence of categories|equivalence]], this includes all examples.


\item Let $T$ be a strict 2-monad on a strict 2-category. Then the strict $T$-algebras form an $\mathcal{M}$-category $T \mathrm{Alg}$ where tight morphisms are the strict algebra morphisms and the loose morphisms are the pseudo algebra morphsims. This is just a decategorified version of the $\mathcal{F}$-category of $T$-algebras, which also includes the 2-cells between algebras.


\item More generally, any strict $\mathcal{F}$-category can be made into an $\mathcal{M}$-category by forgetting the non-identity 2-cells.



\end{itemize}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item Any $\mathcal{M}$-category whose tight morphisms form a [[preorder]] can be made into a [[strict category]] in a canonical way: declare two objects to be equal if they are \emph{tightly} isomorphic. This is an unusual sort of strict category in that its ``equality predicate'' on objects may not be literal equality (even in a foundational system where the latter makes sense). Many strict categories that arise in practice underlie $\mathcal{M}$-categories, such as the category of sets in material set theory.

Note that two \emph{equivalent} $\mathcal{M}$-categories with posetal tight categories (in the usual sense of equivalence for enriched categories) have \emph{isomorphic} underlying strict categories (in the appropriate sense, i.e. making use of the stipulated equality predicate on objects to define ``isomorphism''). In this way, some examples which may seem on the surface to be [[evil]], by referring to an isomorphism of categories, can alternatively be described non-evilly by recognizing the presence of a neglected $\mathcal{M}$-enrichment.


\item For instance, let $G = Gal(E/F)$ be the [[Galois group]] of a finite [[Galois extension]] $E/F$. Then there is an $\mathcal{M}$-category whose objects are intermediate [[fields]] $F\subset K\subset E$, whose loose maps are arbitrary field homomorphisms that fix $F$ pointwise, and whose tight maps are those which commute with the inclusions into $E$. There is also an $\mathcal{M}$-category whose objects are [[orbits]] $G/H$, whose loose maps are arbitrary maps of $G$-sets, and whose tight maps are those which commute with the quotient maps from $G$. The fundamental theorem of classical [[Galois theory]] says that these two $\mathcal{M}$-categories are equivalent \emph{as $\mathcal{M}$-categories}.

This is a stronger statement than saying that their underlying strict categories, as above, are isomorphic, which is yet stronger than saying that their underlying categories of loose maps are equivalent. See \href{http://article.gmane.org/gmane.science.mathematics.categories/5877}{this post} by [[Peter May]].



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

$\mathcal{M}$-categories are mentioned as $Subset$-categories (thinking of $\mathcal{M}$ as the category of [[subset]] inclusions) in

\begin{itemize}%
\item [[John Power]] (2002), Premonoidal categories as categories with algebraic structure, Theoretical Computer Science 278, \href{http://www.inf.ed.ac.uk/publications/online/0413.pdf}{pdf}.

\end{itemize}
We discussed them in the [[nlabmeta:n-Forum]] as part of a discussion of the category of [[Banach spaces]]:

\begin{itemize}%
\item [[Mark Meckes]] et al (2011), Banach space, n-Forum, (\href{https://nforum.ncatlab.org/discussion/3289/banach-space/}{discussion link}).

\end{itemize}
[[!redirects M-category]] [[!redirects M-categories]] [[!redirects Subset-category]] [[!redirects Subset-categories]]



\end{document}