Definition 0.1. Superdiagrams ΣS are defined as heterofunctors ℱS that are subject to
ETAS axioms and link categorical diagrams ΣC (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 ΣC’s linked by
hetero-functors ℱS 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 ℱS with the (homo)functors
FC determined by ΣC, so that
the right hand side of this equation is to be interpreted as a heterofunctor acting on the
(homo)functor(s) FC determined by the categorical diagram, or the categorical sequence,
ΣC.
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).
