New York Journal of Mathematics
Volume 7 (2001) 257-265


Peter Schauenburg

Turning Monoidal Categories into Strict Ones

Published: November 13, 2001
Keywords: monoidal category, strict monoidal category
Subject: 18W10

It is well-known that every monoidal category is equivalent to a strict one. We show that for categories of sets with additional structure (which we define quite formally below) it is not even necessary to change the category: The same category has a different (but isomorphic) tensor product, with which it is a strict monoidal category. The result applies to ordinary (bi)modules, where it shows that one can choose a realization of the tensor product for each pair of modules in such a way that tensor products are strictly associative. Perhaps more surprisingly, the result also applies to such nontrivially nonstrict categories as the category of modules over a quasibialgebra.

Author information

Mathematisches Institut der Universität München, Theresienstr. 39, 80333 München, Germany