SuperdiagramsAsHeterofunctors

Definition 0.1 Superdiagrams $\Sigma_S$ are defined as heterofunctors $\mathcal F_S$ that are subject to ETAS axioms and link categorical diagrams $\Sigma_C$ (regarded as (homo)functors, which are subject to the eight ETAC axioms) in a manner similar to how groupoids are being constructed as many-object structures of linked groups with all invertible morphisms between the linked groups. Thus, in the supercategory definition–instead of a groupoid with all invertible morphisms– one replaces the linked groups by several $\Sigma_C$ 's linked by hetero-functors $\mathcal F_S$ between such categorical diagrams or categorical sequences with different structure. The heterofunctors corresponding to superdiagrams also need not be invertible (as in the case of supergroupoid structures). In this construction, one defines a supercategorical diagram in terms of the composition “ $*$

” of the heterofunctors $\mathcal F_S$ with the (homo)functors $F_C$

determined by $\Sigma_C$ , so that

$\displaystyle \mathcal F_S * F_C := \mathcal F_S (F_C);$

the right hand side of this equation is to be interpreted as a heterofunctor acting on the (homo)functor(s) $F_C$

determined by the categorical diagram, or the categorical sequence, $\Sigma_C$ .

Remark In a certain sense, the superdiagrams defined here as superfunctors resemble also the groupoid functor categories, as well as topological categories, if one regards the class of links between the different types of categorical diagrams as a meta-network or metagraph (in the sense defined by Mac Lane and Moerdijk (2000).