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June 12, 2012 / porton

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.


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