]> Sub-Object Hierarchies

Sub-Object Hierarchies

In orthodox category theory, there are objects as well as morphisms; since each object is essentially synonymous with an identity morphism (on that object), it is possible to elide the objects themselves and use identity morphisms as tokens for them. However, to faithfully represent some orthodox categories, it remains necessary to tag each morphism with the identities (as ciphers for objects) it comes from and goes to: for example, the category Set (whose objects are sets, morphisms are functions – tagged in the manner under discussion – under clean composition, i.e. f&on;g is only defined if every output of g is an input of f) requires us to distinguish (A:f:) and (B:f:) whenever A and B are distinct sets subsuming (:x←x:f), the set of f's outputs; yet, as relations, (A:f:) and (B:f:) are identical in this case. (A similar complication arises with transformations between functors.) With such tagging, two morphisms may package the same underlying function (or whatever) yet be deemed distinct by the category: it will only compose two morphisms if the end-tag of the first matches the start-tag of the second.

While you are welcome to use such packaging to simulate an orthodox category, and this will work, I chose not to impose such a constraint on categoric binary operators.

End-relations of follow

For a given categoric binary operator, *, the relation follow(*) on its operands encodes what * can combine; follow(*) relates a to b iff a*b is defined. In general, for any relation r, I define end-relations

= (r: x←y; (:z←z:r&on;{y}) subsumes (:z←z:r&on;{x}) :)
= (: x←y; ({x}&on;r:z←z:) subsumes ({y}&on;r:z←z:) :r)

on the right and left values (respectively) of r; the former relates x to y if x is a right value and r relates z to x implies r relates z to y (whence, in particular, y is also a right value). Each is manifestly transitive and reflexive (because subsumes is); hence intersecting each with its reverse yields an equivalence. The end-equivalences of follow(*) and the end-relations that yield them will encode the objects of our category and a containment hierarchy among them.

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