Subobject image/preimage, completion of a partial subobject

src/categorical_algebra/CSets.jl

@@ -6,7 +6,7 @@ export ACSetTransformation, CSetTransformation,
   ACSetHomomorphismAlgorithm, BacktrackingSearch, HomomorphismQuery,
   components, force, is_natural, homomorphism, homomorphisms, is_homomorphic,
   isomorphism, isomorphisms, is_isomorphic,
-  generate_json_acset, parse_json_acset, read_json_acset, write_json_acset
+  generate_json_acset, parse_json_acset, read_json_acset, write_json_acset, hom_inv, induce_subobject
 
 using Base.Iterators: flatten
 using Base.Meta: quot
@@ -1115,6 +1115,71 @@ function common_ob(A::Subobject, B::Subobject)
   return X
 end
 
+"""
+f:A->B as a map of subobjects of A to subjects of B
+"""
+(f::ACSetTransformation)(X::SubACSet)::SubACSet = begin
+  codom(hom(X)) == dom(f) || error("Cannot apply $f to $X")
+  Subobject(codom(f); Dict(
+    [k=>f.(collect(components(X)[k])) for (k,f) in pairs(components(f))])...)
+end
+
+"""
+f:A->B as a map from A to a subobject of B
+i.e. we cast the ACSet A to its top subobject
+"""
+(f::ACSetTransformation)(X::StructACSet)::SubACSet =
+  X == dom(f) ? f(top(X)) : error("Cannot apply $f to $X")
+
+"""    hom_inv(f::ACSetTransformation,Y::Subobject)::SubACSet
+Inverse of f:A->B as a map of subobjects of B to subjects of A.
+It can be thought of as incident, but for homomorphisms.
+"""
+hom_inv(f::ACSetTransformation,Y::Subobject)::SubACSet = begin
+  codom(hom(Y)) == codom(f) || error("Cannot apply $f to $X")
+
+  Subobject(dom(f); Dict{Symbol, Vector{Int}}(
+    [k => vcat([preimage(f,y) for y in collect(components(Y)[k])]...)
+     for (k,f) in pairs(components(f))])...)
+end
+
+"""    hom_inv(f::CSetTransformation,Y::StructACSet)::SubACSet
+Inverse f:A->B as a map from subobjects of B to subobjects of A.
+Cast an ACSet to subobject, though this has a trivial answer when computing
+the preimage (it is necessarily the top subobject of A).
+"""
+hom_inv(f::CSetTransformation,Y::StructACSet)::SubACSet =
+  Y = codom(f) ? top(dom(f)) : error("Cannot apply inverse of $f to $Y")
+
+
+"""    induce_subobject(X::StructACSet{S}; vs...)
+
+Takes a partially-specified ACSet subobject and completes it so that there are
+no undefined references.
+
+For example: induce_subobject(my_wiring_diagram; Wire=[2])
+will yield a subobject of my_wiring_diagram with Wire#2, as well as the
+ports it's connected to and the boxes those ports are connected to.
+"""
+function induce_subobject(X::StructACSet{S}; vs...) where {S}
+  vs = Dict([k=>Set(get(vs, k, [])) for k in ob(S)])
+  changed = true
+  while changed
+    changed = false
+    for (k, d, cd) in zip(hom(S), dom(S), codom(S))
+      for v in X[collect(vs[d]), k]
+        if v ∉ vs[cd]
+          push!(vs[cd], v)
+          changed=true
+        end
+      end
+    end
+  end
+  return Subobject(X; Dict{Symbol,Vector{Int}}([k=>sort(collect(v))
+                                                for (k,v) in pairs(vs)])...)
+end
+
+
 # FIXME: Should these two accessors go elsewhere?
 
 @generated function all_subparts(X::StructACSet{S},

test/categorical_algebra/CSets.jl

@@ -476,6 +476,36 @@ A = Subobject(S, X=[3,4,5])
 @test ¬Subobject(ι₂) |> force == Subobject(ι₁)
 @test ~A |> force == ⊤(S) |> force
 
+# Image and preimage
+g1 = path_graph(Graph, 2)
+g2 = apex(coproduct(g1, g1))
+g3 = @acset Graph begin V=2; E=3; src=[1,1,2]; tgt=[1,2,2] end
+h = homomorphisms(g1,g2)[2] # V=[3,4], E=[2]
+ϕ = homomorphism(g2,g3; initial=(V=[1,1,2,2],))
+@test components(hom(h(g1))) == (
+  V = FinFunction([3, 4], 2, 4), E = FinFunction([2], 1, 2))
+@test hom_inv(h, Subobject(g2, V=[1])) |> force == bottom(g1) |> force
+@test hom_inv(h, Subobject(g2, V=[3])) |> force == Subobject(g1, V=[1]) |> force
+@test ϕ(h(g1)) == Subobject(g3, V=[2,2], E=[3])
+
+# Preimage
+G = path_graph(Graph, 1) ⊕ path_graph(Graph, 2) ⊕ path_graph(Graph, 3)
+H = let Loop=apex(terminal(Graph)); Loop ⊕ Loop ⊕ Loop end
+# Each path graph gets its own loop
+f = homomorphism(G, H; initial=(V=Dict([1=>1,2=>2,4=>3]),))
+for i in 1:3
+  @test is_isomorphic(dom(hom(hom_inv(f, Subobject(H, V=[i],E=[i])))),
+                      path_graph(Graph, i))
+end
+
+# Induce subobject
+e1 = Subobject(g2, V=[1,2], E=[1])
+@test induce_subobject(g2, E=[1]) |> force == e1 |> force
+@test induce_subobject(g2, V=[1]) |> force == Subobject(g2, V=[1]) |> force
+@test induce_subobject(g2, E=[2]) |> force == h(g1) |> force
+
+
+
 # Serialization
 ###############