### 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 $latex {C}&fg=000000$ (the base pre-category);
2. a set $latex {M}&fg=000000$ (star-morphisms);
3. a function “$latex {\mathrm{arity}}&fg=000000$” defined on $latex {M}&fg=000000$ (how many objects are connected by this multimorphism);
4. a function $latex {\mathrm{Obj}_m : \mathrm{arity} m \rightarrow \mathrm{Obj} \left( C \right)}&fg=000000$ defined for every $latex {m \in M}&fg=000000$;
5. a function (star composition) $latex {\left( m ; f \right) \mapsto \mathrm{StarComp} \left( m ; f \right)}&fg=000000$ defined for $latex {m \in M}&fg=000000$ and $latex {f}&fg=000000$ being an $latex {(\mathrm{arity} m)}&fg=000000$-indexed family of morphisms of $latex {C}&fg=000000$ such that $latex {\forall i \in \mathrm{arity} m : \mathrm{Src} f_i = \mathrm{Obj}_m i}&fg=000000$ ($latex {\mathrm{Src} f_i}&fg=000000$ is the source object of the morphism $latex {f_i}&fg=000000$) such that $latex {\mathrm{arity} \mathrm{StarComp} \left( m ; f \right) = \mathrm{arity} m}&fg=000000$

such that it holds:

1. $latex {\mathrm{StarComp} \left( m ; f \right) \in M}&fg=000000$;
2. (associativiy law)

$latex \displaystyle \mathrm{StarComp} \left( \mathrm{StarComp} \left( m ; f \right) ; g \right) = \mathrm{StarComp} \left( m ; \lambda i \in \mathrm{arity} m : g_i \circ f_i \right) . &fg=000000$

(Here by definition $latex {\lambda x \in D : F \left( x \right) = \left\{ \left( x ; F \left( x \right) \right) \hspace{0.5em} | \hspace{0.5em} x \in D \right\}}&fg=000000$.)

The meaning of the set $latex {M}&fg=000000$ is an extension of $latex {C}&fg=000000$ having as morphisms things with arbitrary (possibly infinite) indexed set $latex {\ensuremath{\mathrm{Obj}}_m}&fg=000000$ of objects, not just two objects as morphims of $latex {C}&fg=000000$ 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 $latex {m}&fg=000000$:

$latex \displaystyle \ensuremath{\mathrm{StarComp}} \left( m ; \lambda i \in \ensuremath{\mathrm{arity}}m : \ensuremath{\mathrm{id}}_{\ensuremath{\mathrm{Obj}}_m i} \right) = m. &fg=000000$