ENH: Data structure for discrete categories.

A trivial, yet useful, special case.

src/categorical_algebra/Categories.jl

@@ -40,21 +40,13 @@ The basic operations available in any category are: [`dom`](@ref),
 """
 abstract type Cat{Ob,Hom} end
 
-""" Look up object in category.
-
-```julia
-ob(C::Cat{Ob,Hom}, x)::Ob where {Ob,Hom}
-```
+""" Coerce or look up object in category.
 """
-function ob end
-
-""" Look up morphism in category.
+ob(::Cat, x) = x
 
-```julia
-hom(C::Cat{Ob,Hom}, f)::Hom where {Ob,Hom}
-```
+""" Coerce or look up morphism in category.
 """
-function hom end
+hom(::Cat, f) = f
 
 """ Domain of morphism in category.
 """
@@ -89,8 +81,6 @@ Ob(::TypeCat{T}) where T = TypeSet{T}()
 # FIXME: This isn't practical because types are often too tight.
 #ob(::TypeCat{Ob,Hom}, x) where {Ob,Hom} = convert(Ob, x)
 #hom(::TypeCat{Ob,Hom}, f) where {Ob,Hom} = convert(Hom, f)
-ob(::TypeCat, x) = x
-hom(::TypeCat, f) = f
 
 # Functors
 ##########

src/categorical_algebra/FinCats.jl

@@ -59,6 +59,29 @@ is_free(C::FinCat) = isempty(equations(C))
 
 Ob(C::FinCat{Int}) = FinSet(length(ob_generators(C)))
 
+# Discrete categories
+#####################
+
+""" Discrete category on a finite set.
+
+The only morphisms in a discrete category are the identities, which in this
+implementation are identified with the objects.
+"""
+@auto_hash_equals struct DiscreteCat{Ob,S<:FinSet{<:Any,Ob}} <: FinCat{Ob,Ob}
+  set::S
+end
+
+FinCat(s::FinSet) = DiscreteCat(s)
+
+ob_generators(C::DiscreteCat) = C.set
+hom_generators(C::DiscreteCat) = ()
+
+dom(C::DiscreteCat{T}, f) where T = f::T
+codom(C::DiscreteCat{T}, f) where T = f::T
+id(C::DiscreteCat{T}, x) where T = x::T
+compose(C::DiscreteCat{T}, f, g) where T = (f::T == g::T) ? f :
+  error("Nontrivial composite in discrete category: $f != $g")
+
 # Categories on graphs
 ######################
 

test/categorical_algebra/FinCats.jl

@@ -4,6 +4,16 @@ using Test
 using Catlab, Catlab.Theories, Catlab.CategoricalAlgebra, Catlab.Graphs
 using Catlab.Graphs.BasicGraphs: TheoryGraph
 
+# Discrete categories
+#####################
+
+C = FinCat(FinSet(3))
+@test C isa FinCat{Int,Int}
+@test collect(ob_generators(C)) == 1:3
+@test isempty(hom_generators(C))
+@test (dom(C, 1), codom(C, 1)) == (1, 1)
+@test (id(C, 2), compose(C, 2, 2)) == (2, 2)
+
 # Categories on graphs
 ######################
 
@@ -143,14 +153,14 @@ G = FinDomFunctor(TheoryGraph, g)
 @test α⋅σ == FinTransformation(F, G, V=FinFunction([1,2,2]), E=FinFunction([2,4]))
 
 # Pullback data migration by pre-whiskering.
-ιV = FinFunctor([:V], [], FinCat(Graph(1)), FinCat(TheoryGraph))
+ιV = FinFunctor([:V], [], FinCat(FinSet(1)), FinCat(TheoryGraph))
 αV = ιV * α
 @test ob_map(dom(αV), 1) == ob_map(F, :V)
 @test ob_map(codom(αV), 1) == ob_map(G, :V)
 @test component(αV, 1) == component(α, :V)
 
 # Post-whiskering and horizontal composition.
-ιE = FinFunctor([:E], [], FinCat(Graph(1)), FinCat(TheoryGraph))
+ιE = FinFunctor([:E], [], FinCat(FinSet(1)), FinCat(TheoryGraph))
 ϕ = FinTransformation([:src], ιE, ιV)
 @test is_natural(ϕ)
 @test component(ϕ*F, 1) == hom_map(F, :src)