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 CC (the base pre-category); 2. a set MM (star-morphisms); 3. a function arity\operatorname{arity} defined on MM (how many objects are connected by this multimorphism); 4. a function Obj m:aritymObj(C)\operatorname{Obj}_m : \operatorname{arity} m \rightarrow \operatorname{Obj} \left( C \right) defined for every mMm \in M; 5. a function (star composition) (m;f)StarComp(m;f)\left( m ; f \right) \mapsto \operatorname{StarComp} \left( m ; f \right) defined for mMm \in M and ff being an (aritym)(\operatorname{arity} m)-indexed family of morphisms of CC such that iaritym:Srcf i=Obj mi\forall i \in \operatorname{arity} m : \operatorname{Src} f_i = \operatorname{Obj}_m i (Srcf i\operatorname{Src} f_i is the source object of the morphism f if_i) such that arityStarComp(m;f)=aritym\operatorname{arity} \operatorname{StarComp} \left( m ; f \right) = \operatorname{arity} m

such that:

  1. StarComp(m;f)M\operatorname{StarComp} \left( m ; f \right) \in M;
  2. (associativiy law)
    (1)StarComp(StarComp(m;f);g)=StarComp(m;λiaritym:g if i). \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 λ\lambda indicates function abstraction.)

The meaning of the set MM is an extension of CC having as morphisms things with arbitrary (possibly infinite) indexed set Obj m\operatorname{Obj}_m of objects, not just two objects as morphims of CC 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 mm:

(2)StarComp(m;λiaritym:id Obj mi)=m. \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)