category with star-morphisms

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

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:

- $\operatorname{StarComp} \left( m ; f \right) \in M$;
- (
*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.

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

See also abrupt categories.

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

Revised on June 15, 2012 00:34:35
by Toby Bartels
(98.19.38.0)