Premonoidal categories and state transformers

The graphical presentation of symmetric premonoidal categories uses extra control arcs to model control dependencies. This can be seen as a simple form of state transformer where each graph has one extra incoming arc and one extra outgoing arc.

Given a strict symmetric monoidal category C with a chosen object S, the state transformer category of C, State(C) is given by:

Objects are those of C.
The homset State(C)[X, Y] is C[X S, Y S].

The identity-on-objects inclusion C

State(C) maps f to f

id_S.

C State(C) is a strict symmetric premonoidal category with premonoidal structure given by:

f g is f g.
f g is (id symm) ; (f g) ; (id symm).

The state transformer construction is well-known, and is given by Power and Robinson as one of the canonical examples of a symmetric premonoidal category. We shall now show that state transformers are canonical strict symmetric premonoidal categories by showing that for any strict symmetric premonoidal category C

P we can find a strict symmetric monoidal category D such that:

C		P

D		State(D)

that is C P is a full sub-strict-symmetric-premonoidal-category of D State(D).

The rest of this section shows how to construct D, and show that it has the required properties.

Let LGraph(_C) be the subcategory of Graph(_V,_C) category of linear flow graphs, that is ones where:

All nodes are labelled from _C.
Each edge occurs exactly once either as an outgoing edge of the graph, or an incoming edge of a node.

Note in particular that since we cannot duplicate or discard edges, this means this category has no diagonal or terminal. In fact, we can adapt the previous proof to show that this is the initial strict symmetric monoidal category over

_C.

Similarly, let LGraph(_C,_P) be the subcategory of Graph(_V,_C,_P) where:

All nodes are labelled from _C or _P.
Each edge occurs exactly once either as an outgoing edge of the graph, or an incoming edge of a node.

We can adapt the previous proof to show that this is the initial strict symmetric premonoidal category over

_P with centre LGraph(

_C).

Given two signatures _C and _P with the same sorts, define Mix(_C,_P) to be the signature with:

Sorts those of _C plus a new sort S.
Constructors the disjoint union of those of _C and those of _P.
c : X Y in Mix(_C,_P) whenever c : X Y in _C.
c : XS YS in Mix(_C,_P) whenever c : X Y in _P.

We then have:

LGraph(_C)		LGraph(_C,_P)

LGraph(Mix(_C,_P))		State(LGraph(Mix(_C,_P)))

where the full subcategories are taken by only including objects which do not include the new sort S.

We can regard a monoidal category C as a signature with:

Sorts are objects.
Constructors f : X₁,...,X_m Y₁,...,Y_n are morphisms f : X₁...X_m Y₁...Y_n.

and similarly for premonoidal categories. Any category is trivially a category over itself, and so we have a unique symmetric premonoidal functor:

[[_]]_C :	LGraph(C)		C
[[_]]_P :	LGraph(C,P)		P

such that:

[[ ]]_C = c [[ ]]_P = d and so we can define D

Q as a pushout in the category of strict symmetric premonoidal categories:

[[_]]_C :	LGraph(C)	C

[[_]]_D :	LGraph(Mix(C,P))	D

[[_]]_P : LGraph(C,P) P

[[_]]_Q : State(LGraph(Mix(C,P))) Q

Proposition (Full sub-strict-symmetric-premonoidal-category). C P is a full sub-strict-symmetric-premonoidal category of D Q.

Proof.

Proposition (Q is State(D)). Q is isomorphic to State(D).

Proof.

Thus for any strict symmetric premonoidal category C P we can find an strict symmetric monoidal category D such that:

C		P

D		State(D)

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