# Categories with star-morphisms, a generalization of categories

In my research aroused a new kind of structures which I call categories with star-morphisms. In this blog post I define categories with star-morphisms. For sample usages of star categories see this draft article.

Definition 1 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 “ ${\mathrm{arity}}$” defined on ${M}$ (how many objects are connected by this multimorphism);
4. a function ${\mathrm{Obj}_m : \mathrm{arity} m \rightarrow \mathrm{Obj} \left( C \right)}$ defined for every ${m \in M}$;
5. a function (star composition) ${\left( m ; f \right) \mapsto \mathrm{StarComp} \left( m ; f \right)}$ defined for ${m \in M}$ and ${f}$ being an ${(\mathrm{arity} m)}$-indexed family of morphisms of ${C}$ such that ${\forall i \in \mathrm{arity} m : \mathrm{Src} f_i = \mathrm{Obj}_m i}$ ( ${\mathrm{Src} f_i}$ is the source object of the morphism ${f_i}$) such that ${\mathrm{arity} \mathrm{StarComp} \left( m ; f \right) = \mathrm{arity} m}$

such that it holds:

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

(Here by definition ${\lambda x \in D : F \left( x \right) = \left\{ \left( x ; F \left( x \right) \right) \hspace{0.5em} | \hspace{0.5em} x \in D \right\}}$.)

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

Definition 2 A star category is a star pre-category whose base is a category and the following equality (the law of composition with identity) holds for every multimorphism ${m}$: $\displaystyle \ensuremath{\mathrm{StarComp}} \left( m ; \lambda i \in \ensuremath{\mathrm{arity}}m : \ensuremath{\mathrm{id}}_{\ensuremath{\mathrm{Obj}}_m i} \right) = m.$