An -category is like a 2-category, but with two types of 1-morphism, one of which we think of as “stricter” than the other. The stricter morphisms are called tight and the less strict ones are called loose.
Let denote the category whose objects are functors that are fully faithful and injective on objects, and whose morphisms are commutative squares (a full subcategory of the arrow category of Cat). We call the objects of , for the nonce, full embeddings. Then is cartesian closed, complete and cocomplete, hence a Benabou cosmos.
A strict -category is a category enriched over . Therefore, between every two objects, an -category has an object , hence a full embedding . The objects of are called tight morphisms , and the objects of are called loose morphisms .
Since full embeddings are injective on objects, “being tight” is a property of a loose morphism. (This would still be true in the “up to unique isomorphism” sense even if we did not ask for injectivity on objects, but when dealing with strict things, it is easier to keep them as strict as possible.) And since full embeddings are fully faithful, the 2-cells between two tight morphisms are the same whether we regard them as tight or as loose.
For any -category , the objects, tight morphisms, and 2-cells form a strict 2-category , and the objects, loose morphisms, and 2-cells form a strict 2-category . There is an obvious strict 2-functor
K_\tau \to K_\lambda
which is the identity on objects, strictly faithful on 1-morphisms, and locally fully faithful. Since can be recovered from this 2-functor, an equivalent definition of a strict -category is as a strict 2-functor with these properties.
Probably the best “fully weak” version of -categories is obtained by redefining to consist of fully faithful functors, with squares that commute up to specified isomorphism, and then by considering -enriched bicategories rather than enriched categories. Such a thing would be equivalent to an identity-on-objects and locally-fully-faithful pseudofunctor between bicategories.
One could consider semi-strict versions as well, in which (for example) the tight morphisms form a strict 2-category.
Any proarrow equipment is an -category (perhaps weak, perhaps semi-strict).
In fact, this can be generalized to any -monad on an -category.
Any 2-category gives rise to two -categories:
itself becomes an -category in the usual way. Its tight morphisms are just the morphisms in the underlying ordinary category , while its loose morphisms are simply functors between the loose parts (the codomains of the full embeddings).
The general machinery of enriched category theory applied to gives us a notion of weighted limit. Note first that an -enriched diagram in an -category is a diagram of morphisms in which some are required to be tight, and others are not (but could “accidentally” be tight).
In general, a weighted limit of such a diagram in a (strict) -category is a weighted (strict) 2-limit in its 2-category of loose morphisms, with the property that certain specified projections from the limit object are tight and “jointly detect tightness”, in the sense that a morphism into the limit is tight if and only if its composites with all of the specified projections are tight. Details and examples can be found in (LS).
One of the most important things about -categories is that they allow us to define the classes of rigged limits, which are the -weighted limits that are created by the forgetful functors from the various -categories of algebras and strict/pseudo/lax/colax morphisms over a 2-monad (or an -monad).