# 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 $\operatorname{arity}$ defined on $M$ (how many objects are connected by this multimorphism); 4. a function $\operatorname{Obj}_m : \operatorname{arity} m \rightarrow \operatorname{Obj} \left( C \right)$ defined for every $m \in M$; 5. a function (star composition) $\left( m ; f \right) \mapsto \operatorname{StarComp} \left( m ; f \right)$ defined for $m \in M$ and $f$ being an $(\operatorname{arity} m)$-indexed family of morphisms of $C$ such that $\forall i \in \operatorname{arity} m : \operatorname{Src} f_i = \operatorname{Obj}_m i$ ($\operatorname{Src} f_i$ is the source object of the morphism $f_i$) such that $\operatorname{arity} \operatorname{StarComp} \left( m ; f \right) = \operatorname{arity} m$

such that:

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

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