An $\mathcal{M}$-category is a category with two classes of morphisms: tight and loose.
A categorification of the concept of $\mathcal{M}$-category to $2$-categories is the concept of $\mathcal{F}$-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 $2$-morphisms, and an $\mathcal{F}$-category becomes an $\mathcal{M}$-category upon forgetting its nonidentity $2$-cells.
Let $\mathcal{M}$ 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 $Set^\to$, where the reflector preserves products; as a result, $\mathcal{M}$ is complete, cocomplete cartesian closed category. (In fact, $\mathcal{M}$ is a Grothendieck quasitopos, as discussed here.)
A (locally small) $\mathcal{M}$-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 $\mathcal{M}$-category consists of the following data: * a class of objects; * for each pair $x, y$ of objects, a set $L(x,y)$ of loose morphisms; * for each pair $x, y$ of objects, a subset $T(x,y)$ of $L(x,y)$ of tight morphisms; * for each object $x$, a tight identity morphism $id_x$; and * for each triple $x, y, z$ of objects, a binary operation composition $\circ_{x,y,z}$ from $L(y,z)$ and $L(x,y)$ to $L(x,z)$ such that: * $f \circ g$ is tight whenever $f, g$ are; * $\id_y \circ f = f = f \circ \id_x$ whenever this makes sense; and * $(f \circ g) \circ h = f \circ (g \circ h)$ whenever this makes sense.
Equivalently, an $\mathcal{M}$-category is a category of objects and loose morphisms together with a wide subcategory of 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 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
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
Every $\dagger$-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.
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).
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 ‘isomorphism’ while a tight isomorphism is called a ‘global isometry’.
We can make Met (the category of metric spaces) into an $\mathcal{M}$-category is several ways, with short maps contained in Lipschitz maps 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.)
Any strict category is an $\mathcal{M}$-category with equalities as the tight morphisms. (Thus the wide subcategory of tight morphisms is skeletal.) In particular, the category of sets (or any category) in material set theory is an $\mathcal{M}$-category.
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.
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 $a$.
Similarly, quotient objects form an $\mathcal{M}$-category.
The 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 $C$.
$\mathcal{M}$ 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.
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, this includes all examples.
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.
More generally, any strict $\mathcal{F}$-category can be made into an $\mathcal{M}$-category by forgetting the non-identity 2-cells.
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 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 equivalent $\mathcal{M}$-categories with posetal tight categories (in the usual sense of equivalence for enriched categories) have 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.
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 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 this post by Peter May.
$\mathcal{M}$-categories are mentioned as $Subset$-categories (thinking of $\mathcal{M}$ as the category of subset inclusions) in
We discussed them in the n-Forum as part of a discussion of the category of Banach spaces: