nLab premulticategory




A premulticategory is to a multicategory as a premonoidal category is to a monoidal category.


A definition of premulticategory is obtained by starting from a definition of multicategory in terms of the binary composition operations

i:C(y 1,,y i,,y n;z)×C(x 1,,x m;y i)C(y 1,,y i1,x 1,,x n,y i+1,,y n;z) \circ_i : C(y_1,\dots,y_i, \dots,y_n; z) \times C(x_1,\dots,x_m; y_i) \to C(y_1,\dots,y_{i-1},x_1,\dots,x_n,y_{i+1},\dots,y_n;z)

and removing the “parallel associativity” or “commutativity” which says that composing hC(y 1,,y n;z)h\in C(y_1,\dots,y_n;z) with two morphisms fC(x 1,,x n;y i)f\in C(x_1,\dots,x_n ; y_i) and gC(w 1,,w p;y j)g\in C(w_1,\dots,w_p; y_j) can be done in either order.

A morphism ff is central if for any gg and hh as above, the two methods of composition commute: (hf)g=(hg)f(h \circ f) \circ g = (h \circ g) \circ f. So a premulticategory is a multicategory precisely if all morphisms are central.

Relation to premonoidal categories

A premulticategory that has all tensor products and units, in a usual multicategorical sense, is equivalent to a premonoidal category.


  • Sam Staton and Paul Blain Levy, Universal properties of impure programming languages. POPL ‘13: Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages, 2013, doi

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