From cf6370fc5a667023ec20ec3cb20caf4926bdf803 Mon Sep 17 00:00:00 2001
From: Evan Patterson <evan@epatters.org>
Date: Mon, 29 Nov 2021 21:38:23 -0800
Subject: [PATCH] ENH: Colimits of finite sets whose elements are meaningfully
 named.

---
 src/categorical_algebra/FinSets.jl      | 63 ++++++++++++++++++-------
 src/categorical_algebra/FreeDiagrams.jl |  4 +-
 2 files changed, 47 insertions(+), 20 deletions(-)

diff --git a/src/categorical_algebra/FinSets.jl b/src/categorical_algebra/FinSets.jl
index 58ff2d134..f77398250 100644
--- a/src/categorical_algebra/FinSets.jl
+++ b/src/categorical_algebra/FinSets.jl
@@ -1123,42 +1123,69 @@ end
 #--------------------
 
 """ Compute colimit of finite sets whose elements are meaningfully named.
+
+This situation seems to be mathematically uninteresting but is practically
+important. The colimit is computed by reduction to the skeleton of **FinSet**
+(`FinSet{Int}`) and the names are assigned afterwards, following some reasonable
+conventions and add tags where necessary to avoid name clashes.
 """
 struct NamedColimit <: ColimitAlgorithm end
 
-function colimit(::Type{<:Tuple{<:FinSet{S,T},<:FinFunction}}, d) where
-    {S, T<:Union{Symbol,AbstractString}}
+function colimit(::Type{<:Tuple{<:FinSet{<:Any,T},<:FinFunction}}, d) where
+    {T <: Union{Symbol,AbstractString}}
   colimit(d, NamedColimit())
 end
 
-function colimit(span::Multispan{<:FinSet{S,T}}, ::NamedColimit) where {S,T}
-  X = skeletize(apex(span), index=false)
-  feet_skel = map(skeletize, feet(span))
-  legs_skel = map((f, Y) -> skeletize(f, X, Y), legs(span), feet_skel)
-  span_skel = Multispan(dom(X), legs_skel)
-  colim_skel = colimit(span_skel)
+function colimit(d::FixedShapeFreeDiagram{<:FinSet{<:Any,T},Hom},
+                 alg::NamedColimit) where {T,Hom}
+  # Reducing to the case of bipartite free diagrams is a bit lazy, but at least
+  # using `SpecializeColimit` below should avoid some gross inefficiencies.
+  colimit(BipartiteFreeDiagram{FinSet{<:Any,T},Hom}(d), alg)
+end
+function colimit(d::BipartiteFreeDiagram{<:FinSet{<:Any,T}}, ::NamedColimit) where T
+  # Compute colimit of diagram in the skeleton of FinSet (`FinSet{Int}`).
+  # Note: no performance would be gained by using `DisjointSets{T}` from
+  # DataStructures.jl because it is just a wrapper around `IntDisjointSets` that
+  # internally builds the very same indices that we use below.
+  sets₁_skel = map(set -> skeletize(set, index=false), ob₁(d))
+  sets₂_skel = map(set -> skeletize(set, index=true), ob₂(d))
+  funcs = map(edges(d)) do e
+    skeletize(hom(d,e), sets₁_skel[src(d,e)], sets₂_skel[tgt(d,e)])
+  end
+  d_skel = BipartiteFreeDiagram{FinSetInt,eltype(funcs)}()
+  add_vertices₁!(d_skel, nv₁(d), ob₁=dom.(sets₁_skel))
+  add_vertices₂!(d_skel, nv₂(d), ob₂=dom.(sets₂_skel))
+  add_edges!(d_skel, src(d), tgt(d), hom=funcs)
+  colim_skel = colimit(d_skel, SpecializeColimit())
 
+  # Assign elements/names to the colimit set.
   elems = Vector{T}(undef, length(apex(colim_skel)))
-  for (ι, Y) in zip(colim_skel, feet_skel)
+  for (ι, Y) in zip(colim_skel, sets₂_skel)
     for i in dom(Y)
       elems[ι(i)] = Y(i)
     end
   end
-  ι = compose(first(span_skel), first(colim_skel))
-  for i in dom(X)
-    elems[ι(i)] = X(i)
+  # The vector should already be filled, but to reduce arbitrariness we prefer
+  # names from the layer 1 sets whenever possible. For example, when computing a
+  # pushout, we prefer names from the apex of cospan to names from the feet.
+  for (u, X) in zip(vertices₁(d_skel), sets₁_skel)
+    e = first(incident(d_skel, u, :src))
+    f, ι = hom(d_skel, e), legs(colim_skel)[tgt(d_skel, e)]
+    for i in dom(X)
+      elems[ι(f(i))] = X(i)
+    end
   end
+  # Eliminate clashes in provisional list of names.
   unique_by_tagging!(elems)
 
-  Z = FinSet(elems)
-  ιs = map(colim_skel, feet_skel) do ι, Y
-    FinFunction(Dict(Y(i) => elems[ι(i)] for i in dom(Y)), Z)
+  ιs = map(colim_skel, sets₂_skel) do ι, Y
+    FinFunction(Dict(Y(i) => elems[ι(i)] for i in dom(Y)), FinSet(elems))
   end
-  Colimit(span, Multicospan(Z, ιs))
+  Colimit(d, Multicospan(FinSet(elems), ιs))
 end
 
-function skeletize(set::FinSet; index::Bool=true)
-  # FIXME: Should support `unique_index`.
+function skeletize(set::FinSet; index::Bool=false)
+  # FIXME: We should support `unique_index` and it should be used here.
   FinDomFunction(collect(set), set, index=index)
 end
 function skeletize(f::FinFunction, X, Y)
diff --git a/src/categorical_algebra/FreeDiagrams.jl b/src/categorical_algebra/FreeDiagrams.jl
index 8623f35bf..abb5de1ec 100644
--- a/src/categorical_algebra/FreeDiagrams.jl
+++ b/src/categorical_algebra/FreeDiagrams.jl
@@ -458,7 +458,7 @@ function BipartiteFreeDiagram{Ob,Hom}(F::Functor{<:FinCat{Int}};
   end
   return d
 end
-BipartiteFreeDiagram(F::Functor{<:FinCat{Int},<:TypeCat{Ob,Hom}}; kw...) where {Ob,Hom} =
+BipartiteFreeDiagram(F::Functor{<:FinCat{Int},<:Cat{Ob,Hom}}; kw...) where {Ob,Hom} =
   BipartiteFreeDiagram{Ob,Hom}(F; kw...)
 
 # Free diagrams
@@ -551,7 +551,7 @@ function FreeDiagram{Ob,Hom}(F::Functor{<:FinCat{Int}}) where {Ob,Hom}
   diagram[:hom] = collect_hom(F)
   diagram
 end
-FreeDiagram(F::Functor{<:FinCat{Int},<:TypeCat{Ob,Hom}}) where {Ob,Hom} =
+FreeDiagram(F::Functor{<:FinCat{Int},<:Cat{Ob,Hom}}) where {Ob,Hom} =
   FreeDiagram{Ob,Hom}(F)
 
 (::Type{BFD})(diagram::FreeDiagram; kw...) where BFD <: BipartiteFreeDiagram =