Partial abstraction / variable substitution+merging

[kris-brown]

VarACSets work very well with colimits, but limits are subtle. Likewise, maps out of VarACSets are very nice, but maps into VarACSets are subtle. Some more machinery for working with VarACSets is implemented here (which could potentially be upstreamed once it gets battle-tested).

Firstly, sub_vars takes an ACSet + some data (e.g. AttrVar(1) -> :x, AttrVar(3)->:abc) and "evaluates" the variables. This PR also allows this to merge variables together based on input data like [AttrVar(2), AttrVar(4)]. This is all done via a pushout.

Secondly, a notion of "partial abstraction" is needed because we want to take maps B->X and lift them to maps into the "fully abstracted" version of X¹ (call it A). This is hard for two reasons: 1.) no ACSet with concrete attribute values can map into A, 2.) an issue with needing to send an AttrVar to two distinct AttrVars in A, explained more in the docstring. This problem is addressed via applying sub_vars to a fully abstracted ACSet.

This corresponds to a factorization of B->X into B->ParAbs(X)->X (observed by @KevinArlin: see nlab and the blog for more details).

Footnotes

1. This replaces all attributes with distinct variables, even if (e.g.) all the weights of the graph edges are the same value. ↩

[codecov]

Codecov Report

Attention: 1 lines in your changes are missing coverage. Please review.

Comparison is base (8f785c7) 79.88% compared to head (60cce70) 81.22%.

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #29      +/-   ##
==========================================
+ Coverage   79.88%   81.22%   +1.33%     
==========================================
  Files          27       27              
  Lines        1700     1672      -28     
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  Hits         1358     1358              
+ Misses        342      314      -28     

Files	Coverage Δ
src/categorical_algebra/CSets.jl	95.41% <100.00%> (+7.50%)	⬆️
src/rewrite/PBPO.jl	88.05% <100.00%> (+4.72%)	⬆️
src/rewrite/Representable.jl	58.33% <100.00%> (ø)
src/schedules/Wiring.jl	82.05% <100.00%> (+2.74%)	⬆️
src/schedules/Basic.jl	68.29% <0.00%> (ø)

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