nLab
category with star-morphisms

Remark

By a pre-category I mean a category without identities (a semicategory).

Definition

A pre-category with star-morphisms consists of

1. a pre-category $C$ (the base pre-category);
2. a set $M$ (star-morphisms);
3. a function $arity$ defined on $M$ (how many objects are connected by this multimorphism);
4. a function ${Obj}_m:aritym\to Obj\left(C\right)$ defined for every $m\in M$;
5. a function (star composition) $(m;f)\mapsto StarComp(m;f)$ defined for $m\in M$ and $f$ being an $(aritym)$-indexed family of morphisms of $C$ such that $\forall i\in aritym:Src{f}_i={Obj}_{m}i$ ($Src{f}_i$ is the source object of the morphism $f_i$) such that $arityStarComp(m;f)=aritym$

such that:

1. $StarComp(m;f)\in M$;

2. (associativiy law)

   (1)$$StarComp(StarComp(m;f);g)=StarComp(m;\lambda i\in aritym:g_i\circ f_i).$$\operatorname{StarComp} \left( \operatorname{StarComp} \left( m ; f \right) ; g \right) = \operatorname{StarComp} \left( m ; \lambda i \in \operatorname{arity} m : g_i \circ f_i \right) .

(Here $\lambda$ indicates function abstraction.)

The meaning of the set $M$ is an extension of $C$ having as morphisms things with arbitrary (possibly infinite) indexed set ${Obj}_m$ of objects, not just two objects as morphims of $C$ have only source and destination.

Definition

A category with star-morphisms is a pre-category with star-morphisms whose base pre-category is a category and the following equality (the law of composition with identities) holds for every multimorphism $m$:

(2)$$StarComp(m;\lambda i\in aritym:{id}_{{Obj}_{m}i})=m.$$\operatorname{StarComp} \left( m ; \lambda i \in \operatorname{arity}m : \operatorname{id}_{\operatorname{Obj}_m i} \right) = m.

See also abrupt categories.

Categories with star-morphisms and abrupt categories arise in Victor Porton's research on cross-composition products.