C
Given a symmetric monoidal category C: with a full subcategory (not necessarily symmetric monoidal) TC:
C
a partial trace is a natural transformation:
U A, Y U A]
C[X, Y]
satisfying certain axioms (given below, since they are much simpler to present graphically than equationally).
As an example, the categories
Cpo
Cpo
(where the inclusion maps D to D)
have a partial trace given by fixed points.
As another example, given a signature with a subset of sorts defined to be traceable we can let C be the category of flow graphs over that signature where any cycle through the graph must pass through at least one edge labelled with a traceable sort, and TC be the full subcategory where the objects are traceable sorts. These categories have a partial trace given by forming a feedback loop:
We shall provide a different axiomatization of traces than that provided by Joyal, Street and Verity, but we shall show below that in the case when C and TC are the same category, we recover their definition. Note that these axioms are all graph isomorphisms, and indeed are a complete axiomatization for graph isomorphism.
Naturality in X
(for
f :
X
X
and
g :
X U A
Y U A
in C):
| = | ||
| f; TrX Y A (g) | = |
TrX Y A
(f
idU A;
g)
|
Naturality in Y
(for
f :
Y
Y
and
g :
X U A
Y U A
in C):
| = | ||
| TrX Y A (g); f | = |
TrX Y A
(g;
f
idU A)
|
Yanking:
| = | ||
| TrU(A) U(A) A (symmU(A) U(A)) | = | idU(A) |
Symmetry sliding
(for
f :
X U(A) U(B)
Y U(B) U(A)):
| = | ||
|
TrX Y A
(Tr(X U(A)) (Y U(A)) B
(f;
(idY
symmU(B) U(A)))
| = |
TrX Y B
(Tr(X U(B)) (Y U(B)) A
((idX
symmU(B) U(A));
f)
|
Strength
(for
f :
V
W
and
g :
X U A
Y U A
in C):
| = | ||
|
f
TrX Y A (g)
| = |
Tr(V X) (W Y) A
(f
g)
|
For those readers familiar with Joyal, Street and Verity's axiomatization of a trace, the above axioms are obviously derivable from theirs. Moreover, the following lemmas show that we can derive their axiomatization, so in the case where TC and C coincide, our axiomatization is the same as theirs.
Lemma (Parameterized dinaturality)
(for
f :
Z U(B)
U(A)
and
g :
X U(A)
Y U(B)
in C):
| = | ||
|
Tr(X Z) Y A
((idX
symmZ U(A));
(g
idZ);
(idY
(symmU(B) Z);
f)))
| = |
Tr(X Z) Y B
((idX
f);
g)
|
U(A)
and
g :
X U(A)
Y U(B)
in C):
| = | ||
|
TrX Y A
(g;
(idY
f))
| = |
TrX Y B
((idX
f);
g)
|
Proof.
Let Z be I in the above.
Lemma (Trace respects unit)
(for
f :
I
U A
and
g :
X U A
Y
in C):
| = | ||
|
TrX Y A
(g;
idY
f)
| = |
(idX
f);
g
|
U(C)
U(A)
and
g :
X U(A)
Y U(B) U(C)
in C):
| = | ||
|
TrX Y A
(g;
idY
f)
| = |
TrX Y C
(TrX Y B
((idX
f);
g))
|
W
and
g :
X U A
Y U A
in C):
| = | ||
|
TrX Y A (g)
f
| = |
Tr(X V) (Y W) A
((idX
symmV U(A));
(g
f);
(idY
symmU(A) W))
|