diff --git a/Project.toml b/Project.toml
index a17f94a..c07a9a0 100644
--- a/Project.toml
+++ b/Project.toml
@@ -10,6 +10,7 @@ Catlab = "134e5e36-593f-5add-ad60-77f754baafbe"
 DataStructures = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
 MLStyle = "d8e11817-5142-5d16-987a-aa16d5891078"
 Reexport = "189a3867-3050-52da-a836-e630ba90ab69"
+StructEquality = "6ec83bb0-ed9f-11e9-3b4c-2b04cb4e219c"
 SymbolicUtils = "d1185830-fcd6-423d-90d6-eec64667417b"
 Unicode = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"
 
@@ -18,6 +19,7 @@ ACSets = "0.2"
 Catlab = "0.15, 0.16"
 DataStructures = "0.18.13"
 MLStyle = "0.4.17"
+StructEquality = "2.1.0"
 SymbolicUtils = "3.1.2"
 Unicode = "1.6"
 Reexport = "1.2.2"
diff --git a/src/ThDEC.jl b/src/ThDEC.jl
index 6b16e41..c913cd8 100644
--- a/src/ThDEC.jl
+++ b/src/ThDEC.jl
@@ -1,6 +1,7 @@
 module ThDEC
 
 using MLStyle
+using StructEquality
 
 import Base: +, -, *
 
@@ -8,22 +9,46 @@ struct SortError <: Exception
     message::String
 end
 
+@struct_hash_equal struct Space
+    name::Symbol
+    dim::Int
+end
+export Space
+
+dim(s::Space) = s.dim
+Base.nameof(s::Space) = s.name
+
+struct SpaceLookup
+    default::Space
+    named::Dict{Symbol,Space}
+end
+
 @data Sort begin
     Scalar()
-    Form(dim::Int, isdual::Bool)
-    VField(isdual::Bool)
+    Form(dim::Int, isdual::Bool, space::Space)
+    VField(isdual::Bool, space::Space)
 end
 export Sort, Scalar, Form, VField
 
-const SORT_LOOKUP = Dict(
-    :Form0 => Form(0, false),
-    :Form1 => Form(1, false),
-    :Form2 => Form(2, false),
-    :DualForm0 => Form(0, true),
-    :DualForm1 => Form(1, true),
-    :DualForm2 => Form(2, true),
-    :Constant => Scalar()
-)
+function fromexpr(lookup::SpaceLookup, e, ::Type{Sort})
+    (name, spacename) = @match e begin
+        name::Symbol => (name, nothing)
+        :($name{$spacename}) => (name, spacename)
+    end
+    space = @match spacename begin
+        ::Nothing => lookup.default
+        name::Symbol => lookup.named[name]
+    end
+    @match name begin
+        :Form0 => Form(0, false, space)
+        :Form1 => Form(1, false, space)
+        :Form2 => Form(2, false, space)
+        :DualForm0 => Form(0, true, space)
+        :DualForm1 => Form(1, true, space)
+        :DualForm2 => Form(2, true, space)
+        :Constant => Scalar()
+    end
+end
 
 function Base.nameof(s::Scalar)
     :Constant
@@ -31,15 +56,19 @@ end
 
 function Base.nameof(f::Form)
     dual = isdual(f) ? "Dual" : ""
-    Symbol("$(dual)Form$(dim(f))")
+    formname = Symbol("$(dual)Form$(dim(f))")
+    Expr(:curly, formname, dim(space(f)))
 end
 
 const VF = VField
 
 dim(ω::Form) = ω.dim
 isdual(ω::Form) = ω.isdual
+space(ω::Form) = ω.space
+export space
 
 isdual(v::VField) = v.isdual
+space(v::VField) = v.space
 
 # convenience functions
 PrimalForm(i::Int) = Form(i, false)
@@ -58,20 +87,26 @@ export DualVF
 show_duality(ω::Form) = isdual(ω) ? "dual" : "primal"
 
 function Base.show(io::IO, ω::Form)
-    print(io, isdual(ω) ? "DualForm($(dim(ω)))" : "PrimalForm($(dim(ω)))")
+    print(io, isdual(ω) ? "DualForm($(dim(ω))) on $(space(ω))" : "PrimalForm($(dim(ω))) on $(space(ω))")
 end
 
+# TODO: VField
 @nospecialize
 function +(s1::Sort, s2::Sort)
     @match (s1, s2) begin
         (Scalar(), Scalar()) => Scalar()
-        (Scalar(), Form(i, isdual)) ||
-            (Form(i, isdual), Scalar()) => Form(i, isdual)
-        (Form(i1, isdual1), Form(i2, isdual2)) =>
-            if (i1 == i2) && (isdual1 == isdual2)
+        (Scalar(), Form(i, isdual, space)) ||
+            (Form(i, isdual, space), Scalar()) => Form(i, isdual, space)
+        (Form(i1, isdual1, space1), Form(i2, isdual2, space2)) =>
+            if (i1 == i2) && (isdual1 == isdual2) && (space1 == space2)
                 Form(i1, isdual1)
             else
-                throw(SortError("Cannot add two forms of different dimensions/dualities: $((i1,isdual1)) and $((i2,isdual2))"))
+                throw(SortError(
+                    """
+                    Can not add two forms of different dimensions/dualities/spaces:
+                    $((i1,isdual1,space1)) and $((i2,isdual2,space2))
+                    """)
+                )
             end
     end
 end
@@ -84,13 +119,15 @@ end
 # Negation is always valid
 -(s::Sort) = s
 
+# TODO: VField
 @nospecialize
 function *(s1::Sort, s2::Sort)
     @match (s1, s2) begin
         (Scalar(), Scalar()) => Scalar()
-        (Scalar(), Form(i, isdual)) ||
-            (Form(i, isdual), Scalar()) => Form(i, isdual)
-        (Form(_, _), Form(_, _)) => throw(SortError("Cannot scalar multiply a form with a form. Maybe try `∧`??"))
+        (Scalar(), Form(i, isdual, space)) ||
+            (Form(i, isdual, space), Scalar()) => Form(i, isdual)
+        (Form(_, _, _), Form(_, _, _)) =>
+            throw(SortError("Cannot scalar multiply a form with a form. Maybe try `∧`??"))
     end
 end
 
@@ -100,17 +137,20 @@ function as_sub(n::Int)
     join(map(d -> SUBSCRIPT_DIGIT_0 + d, digits(n)))
 end
 
+# TODO: VField
 @nospecialize
 function ∧(s1::Sort, s2::Sort)
     @match (s1, s2) begin
-        (Form(i, isdual), Scalar()) || (Scalar(), Form(i, isdual)) => Form(i, isdual)
-        (Form(i1, isdual), Form(i2, isdual)) =>
-            if i1 + i2 <= 2
-                Form(i1 + i2, isdual)
+        (Form(i, isdual, space), Scalar()) || (Scalar(), Form(i, isdual, space)) =>
+            Form(i, isdual, space)
+        (Form(i1, isdual1, space1), Form(i2, isdual2, space2)) => begin
+            (isdual1 == isdual2) && (space1 == space2) || throw(SortError("Can only take a wedge product of two forms of the same duality on the same space"))
+            if i1 + i2 <= dim(space)
+                Form(i1 + i2, isdual, space)
             else
-                throw(SortError("Can only take a wedge product when the dimensions of the forms add to less than 2: tried to wedge product $i1 and $i2"))
+                throw(SortError("Can only take a wedge product when the dimensions of the forms add to less than n, where n = $(dim(space)) is the dimension of the ambient space: tried to wedge product $i1 and $i2"))
             end
-        _ => throw(SortError("Can only take a wedge product of two forms of the same duality"))
+        end
     end
 end
 
@@ -125,11 +165,11 @@ end
 function d(s::Sort)
     @match s begin
         Scalar() => throw(SortError("Cannot take exterior derivative of a scalar"))
-        Form(i, isdual) =>
-            if i <= 1
-                Form(i + 1, isdual)
+        Form(i, isdual, space) =>
+            if i < dim(space)
+                Form(i + 1, isdual, space)
             else
-                throw(SortError("Cannot take exterior derivative of a n-form for n >= 1"))
+                throw(SortError("Cannot take exterior derivative of a k-form for k >= n, where n = $(dim(space)) is the dimension of its ambient space"))
             end
     end
 end
@@ -142,21 +182,21 @@ end
 function ★(s::Sort)
     @match s begin
         Scalar() => throw(SortError("Cannot take Hodge star of a scalar"))
-        Form(i, isdual) => Form(2 - i, !isdual)
         VF(isdual) => throw(SortError("Cannot take the Hodge star of a vector field"))
+        Form(i, isdual, space) => Form(dim(space) - i, !isdual, space)
     end
 end
 
 function Base.nameof(::typeof(★), s)
     inv = isdual(s) ? "⁻¹" : ""
-    Symbol("★$(as_sub(isdual(s) ? 2 - dim(s) : dim(s)))$(inv)")
+    Symbol("★$(as_sub(isdual(s) ? dim(space(s)) - dim(s) : dim(s)))$(inv)")
 end
 
 @nospecialize
 function ι(s1::Sort, s2::Sort)
     @match (s1, s2) begin
-        (VF(true), Form(i, true)) => PrimalForm() # wrong
-        (VF(true), Form(i, false)) => DualForm()
+        (VF(true, space), Form(i, true, space)) => PrimalForm() # wrong
+        (VF(true, space), Form(i, false, space)) => DualForm()
         _ => throw(SortError("Can only define the discrete interior product on:
                 PrimalVF, DualForm(i)
                 DualVF(), PrimalForm(i)
@@ -168,7 +208,7 @@ end
 function ♯(s::Sort)
     @match s begin
         Scalar() => PrimalVF()
-        Form(1, isdual) => VF(isdual)
+        Form(1, isdual, space) => VF(isdual, space)
         _ => throw(SortError("Can only take ♯ to 1-forms"))
     end
 end
@@ -182,7 +222,7 @@ end
 
 function ♭(s::Sort)
     @match s begin
-        VF(true) => PrimalForm(1)
+        VF(true, space) => PrimalForm(1, false, space)
         _ => throw(SortError("Can only apply ♭ to dual vector fields"))
     end
 end
@@ -196,7 +236,7 @@ end
 
 function ♭♯(s::Sort)
     @match s begin
-        Form(i, isdual) => Form(i, !isdual)
+        Form(i, isdual, space) => Form(i, !isdual, space)
         _ => throw(SortError("♭♯ is only defined on forms."))
     end
 end
@@ -204,7 +244,7 @@ end
 # Δ = ★d⋆d, but we check signature here to throw a more helpful error
 function Δ(s::Sort)
     @match s begin
-        Form(0, isdual) => Form(0, isdual)
+        Form(0, isdual, space) => Form(0, isdual, space)
         _ => throw(SortError("Δ is not defined for $s"))
     end
 end
diff --git a/src/decasymbolic.jl b/src/decasymbolic.jl
index 75e8b62..de291ac 100644
--- a/src/decasymbolic.jl
+++ b/src/decasymbolic.jl
@@ -20,50 +20,56 @@ abstract type DECType <: Number end
 """
 i: dimension: 0,1,2, etc.
 d: duality: true = dual, false = primal
+s: name of the space (a symbol)
+n: dimension of the space
 """
-struct FormT{i,d} <: DECType end
+struct FormT{i,d,s,n} <: DECType end
+export FormT
 
-struct VFieldT{d} <: DECType end
+struct VFieldT{d,s,n} <: DECType end
+export VFieldT
 
 dim(::Type{<:FormT{d}}) where {d} = d
 isdual(::Type{FormT{i,d}}) where {i,d} = d
 
 # convenience functions
-const PrimalFormT{i} = FormT{i,false}
+const PrimalFormT{i,s,n} = FormT{i,false,s,n}
 export PrimalFormT
 
-const DualFormT{i} = FormT{i,true}
+const DualFormT{i,s,n} = FormT{i,true,s,n}
 export DualFormT
 
-const PrimalVFT = VFieldT{false}
+const PrimalVFT{s,n} = VFieldT{false,s,n}
 export PrimalVFT
 
-const DualVFT = VFieldT{true}
+const DualVFT{s,n} = VFieldT{true,s,n}
 export DualVFT
 
-# convert Real to DecType 
-Sort(::Type{<:Real}) = Scalar()
+# ##
+# 
+# ##
 
-# convert Real to ThDEC
+Sort(::Type{<:Real}) = Scalar()
 Sort(::Real) = Scalar()
 
-# convert DECType to ThDEC
-Sort(::Type{FormT{i,d}}) where {i,d} = Form(i, d)
+# convert Real to DecType 
+function Sort(::Type{FormT{i,d,s,n}}) where {i,d,s,n}
+    Form(i, d, Space(s, n))
+end
 
-# convert DECType to ThDEC
-Sort(::Type{VFieldT{d}}) where {d} = VField(d)
+function Sort(::Type{VFieldT{d,s,n}}) where {d,s,n}
+    VField(d, Space(s, n))
+end
 
 Sort(::BasicSymbolic{T}) where {T} = Sort(T)
 
-# convert Form to DECType
-Number(f::Form) = FormT{dim(f), isdual(f)}
-
-# convert VField to DECType
-Number(v::VField) = VFieldT{isdual(v)}
-
 # convert number to real
 Number(s::Scalar) = Real
 
+Number(f::Form) = FormT{dim(f),isdual(f), nameof(space(f)), dim(space(f))}
+
+Number(v::VField) = VFieldT{isdual(v), nameof(space(v)), dim(space(v))}
+
 # for every unary operator in our theory, take a BasicSymbolic type, convert its type parameter to a Sort in our theory, and return a term
 unop_dec = [:∂ₜ, :d, :★, :♯, :♭, :-]
 for unop in unop_dec