Profunctor naturality #22
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Vtec234
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Vtec234
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Mar 21, 2025
| @[simps!] | ||
| def comp [HasPullbacks C] [HasTerminal C] | ||
| {E B D A : C} (P : UvPoly E B) (Q : UvPoly D A) : UvPoly (compDom P Q) (compCod P Q) := | ||
| {E B D A : C} (P : UvPoly E B) (Q : UvPoly D A) : UvPoly (compDom P Q) (P @ A) := |
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compDom seems useful to have as an abbreviation that gets referred to a lot, but for the codomain I think just P @ A is ok; more definitions generally entail needing more lemmas later.
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Defines the UP of polynomial functors as a natural isomorphism of profunctors and concludes
naturality_left/naturality_rightfrom that. Along the way we build up an API for 'dependent profunctors' to express mappings such as(X, Y) ↦ ∑(g: X → A), Hom(g.left, Y)asFunctor.Sigmaof a functor expressing the second copmponent overyAwhich expresses the first component.Imports
lean4-seqfor dependent rewriting with therw!tactic.Discuss how to integrate this with partial products.
Reorganize contents in the PR in correct modules.