Proposition
(Full sub-strict-symmetric-premonoidal-category).
C
P
is a full sub-strict-symmetric-premonoidal category of
D
Q.
We have a functor
F :
P
Q given by the pushout diagram. We will first show
that this functor is monic.
For any strict symmetric premonoidal category
P
let
P
be the strict symmetric premonoidal category given by adjoining a new object
and making the tensors strict.
Let lift be the functor
P
P
.
We can find a strict symmetric premonoidal functor test such that:
lift : | LGraph(C,P) | ![]() |
LGraph(C,P)![]() |
![]() |
![]() |
||
test : | State(LGraph(Mix(C,P))) | ![]() |
LGraph(C,P)![]() |
such that test(S) = .
Then:
LGraph(C,P) | ![]() |
P |
![]() |
![]() |
|
State(LGraph(Mix(C,P))) |
![]() ![]() ![]() |
P![]() |
and since we defined
Q
as a pushout, this means we have a unique symmetric premonoidal functor
test :
Q
P
making the pushout diagram commute. In particular, we have
F ; test
= lift, and since lift is monic, so is F.
For fullness, for any f : F(X)
F(Y)
in Q
we can find
(since [[_]]Q is epi) a graph
G : X
Y
such that:
Since test(F(X))
we have
S
X, so X is in
LGraph(C,P),
and similarly for Y. Since
LGraph(C,P)
State(LGraph(Mix(C,P)))
is a full subcategory,
G is in
LGraph(C,P)
and so [[G]]P is in P.
By the pushout diagram, F[[G]]P =
f, and so
P is a full sub-strict-symmetric-premonoidal-category of
Q.