PlanetPhysics/Superdiagrams As Heterofunctors

Superdiagrams ${\mathrm{\Sigma }}_S$ are defined as heterofunctors Failed to parse (unknown function "\F"): {\displaystyle \F_S} that are subject to [[../ETACAxioms/|ETAS axioms]] and link [[../CategoricalDiagramsDefinedByFunctors/|categorical diagrams]] ${\mathrm{\Sigma }}_C$ (regarded as (homo[[../TrivialGroupoid/|)functors]], which are subject to the eight [[../ETACAxioms/|ETAC axioms]]) in a manner similar to how [[../EquivalenceRelation/|groupoids]] are being constructed as many-object structures of linked [[../TrivialGroupoid/|groups]] with all invertible [[../TrivialGroupoid/|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 ${\mathrm{\Sigma }}_C$'s linked by hetero-functors Failed to parse (unknown function "\F"): {\displaystyle \F_S} between such categorical diagrams or [[../HomologicalSequence2/|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 [[../TrivialGroupoid/|diagram]] in terms of the [[../Cod/|composition]] "$*$" of the heterofunctors Failed to parse (unknown function "\F"): {\displaystyle \F_S} with the (homo)functors $F_C$ determined by ${\mathrm{\Sigma }}_C$, so that Failed to parse (unknown function "\F"): {\displaystyle \F_S * F_C := \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, ${\mathrm{\Sigma }}_C$.

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