Semicategories, like categories, appear as semipresheaves on the category with two objects and two morphisms.
A (small) semicategory or non-unital category consists of
a set of morphisms (or arrows);
two functions called source (or domain) and target (or codomain);
such that the following properties are satisfied:
source and target are respected by composition: and ;
composition is associative: whenever and .
One often writes , , or for the collection of morphisms ; the latter two have the advantage of making clear which category is being discussed. People also often write instead of as a short way to indicate that is an object of . Also, some people write and instead of and .
We discuss the relation of semicategories to categories. (See for instance the beginning of (Harpaz) for a quick review of basics, with an eye towards their generalization to the relation between complete Segal spaces and complete semi-Segal spaces.)
There is an evident forgetful functor
For a semi-category, def. 1, write
A semicategory is the semicategory underlying a category, hence is in the image of the functor of def. 3, precisely if every object has a neutral endomorphism, hence precisely if the composite diagonal function in
Equivalently one could use the target map instead of the source map in the formulation of prop. 1.
The diagram appearing in prop. 1 is a simple version of the univalence condition appearing in definition of a complete semi-Segal space, a semi-category object in an (infinity,1)-category. See there for more on this.
The functor of def. 3 has a left adjoint, which freely adjoins identity morphisms to a semicategory in the obvious way. It also has a right adjoint, which sends a semicategory to the category whose objects are the idempotents of and whose morphisms are the morphisms of that commute suitably with them, as described at Karoubi envelope. Indeed, the monad on Cat generated by this latter adjunction is exactly the monad for idempotent completion, also called Cauchy completion. (Note, however, that this is not a 2-monad, because the right adjoint of is not a 2-functor.)
Start with the category of metric spaces and short maps. An occasionally useful semicategory can be formed from it by considering the nonempty spaces and strictly contractive functions.
This is a semicategory, since:
The interest in this semicategory arises from the fact that all morphisms have unique fixed points, by Banach’s fixed point theorem.
The concept of semicategory has more or less evident analogs and generalizations in higher category theory.
For models of higher categories by simplicial sets, i.e. presehaves on the simplex category (such as Kan complexes, quasi-categories, weak complicial sets) the corresponding semi-category notion is obtained by discarding the degeneracy maps (which are the identity-assigning maps in the simplicial framework), i.e. by considering just presheaves on the subcategory on injective morphisms (see the discuss of at Reedy model structure for more details).
Simpson's conjecture says that every -category has a model where all composition operations are strict and only the unit laws hold up to coherent homotopies. This would mean that the -semicategory underlying any -category can always be chosen to be strict.
Semicategories and semigroups are mentioned for instance in section 2 in
Semicategories with an eye towards their generalization to semi-Segal spaces are briefly discussed at the beginning of
Structures obtained by further relaxing also the associativity law are discussed in