definition of freyd multicategory (with specified central morphisms)

sketch.org

+* Freyd Multicategories as Generalized Multicategories
+
+(Based on Discussion with Mike Shulman about Premonoidal Categories)
+
+First, there is a double category of Subset Matrices whose objects are
+sets, vertical arrows are functions, horizontal arrows $R : A -/-> B$
+are for each a,b a set $R_l(a,b)$ with a specified subset $R_t(a,b)
+\subset R_l(a,b)$. A monoid in this double category is an M-category
+or a pure-effectful category which has a set of objects, for each pair
+of objects a set of "loose arrows" which are "possibly effectful", a
+subset of "tight arrows" which are "pure" such that any composable
+string of tight arrows is tight (including empty string, i.e. the
+identity).
+
+There is a "funny cartesian" monoidal product called □ and defined by
+generators and relations as follows. $C □ D$ has as objects pairs
+(c,d) of objects. Arrows are generated by for each arrow $f : c -> c'$
+and object d in D, there is an arrow $f □ id$ and vice-versa for
+arrows in D. Subject to the following equations:
+    (f □ id) o (f' □ id) = (f o f' □ id)
+    (id □ g) o (id □ g') = (id □ g o g')
+    (p □ id) o (id □ g) = (id □ g) o (p □ id)
+    (id □ q) o (f □ id) = (f □ id) o (id □ q)
+where p,q are tight morphisms.
+The monoids with respect to □ are strict premonoidal categories with a
+specified subset of the central morphisms.
+Restricted to tight arrows, it is the cartesian produt.
+
+There should be a free monoid monad T on M-category, and then "freyd
+multicategories" should be T-multicategories.
+
+These have a set C0 of objects, for every list of objects G and object
+A a set of loose morphisms C_l(G;A) with a specified subset of tight
+morphisms