Turning Monoidal Categories into Strict Ones
نویسنده
چکیده
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.
منابع مشابه
Strictification of Categories Weakly Enriched in Symmetric Monoidal Categories
We offer two proofs that categories weakly enriched over symmetric monoidal categories can be strictified to categories enriched in permutative categories. This is a ”many 0-cells” version of the strictification of bimonoidal categories to strict ones.
متن کاملLax Operad Actions and Coherence for Monoidal N -categories, A∞ Rings and Modules
We establish a general coherence theorem for lax operad actions on an n-category which implies that an n-category with such an action is lax equivalent to one with a strict action. This includes familiar coherence results (e.g. for symmetric monoidal categories) and many new ones. In particular, any braided monoidal n-category is lax equivalent to a strict braided monoidal n-category. We also o...
متن کاملAssociahedral categories, particles and Morse functor
Every smooth manifold contains particles which propagate. These form objects and morphisms of a category equipped with a functor to the category of Abelian groups, turning this into a 0 + 1 topological field theory. We investigate the algebraic structure of this category, intimately related to the structure of Stasheff’s polytops, introducing the notion of associahedral categories. An associahe...
متن کاملOn Monoidal Equivalences and Ann-equivalences
The equivalence between a monoidal category and a strict one has been proved by some authors such as Nguyen Duy Thuan [8], Christian Kassel [2], Peter Schauenburg [7]. In this paper, we show another proof of the problem by constructing a strict monoidal category M(C) consisting of M -functors and M morphisms of a category C and we prove C is equivalent to it. The proof is based on a basic chara...
متن کاملCoherence for Categorified Operadic Theories
Given an algebraic theory which can be described by a (possibly symmetric) operad P , we propose a definition of the weakening (or categorification) of the theory, in which equations that hold strictly for P -algebras hold only up to coherent isomorphism. This generalizes the theories of monoidal categories and symmetric monoidal categories, and several related notions defined in the literature...
متن کامل