We study the monoidal structure of the standard strictification functor st : Bicat → 2Cat. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the 2-category whose objects are 2-categories.

This paper extends the Day Re ection Theorem to skew monoidal categories. We also provide conditions under which a skew monoidal structure can be lifted to the category of Eilenberg-Moore algebras for a comonad.

Astract. Within the Geometry of Interaction (GoI) paradigm, we present a setting that enables qualitative differences between classical and quantum processes to be explored. The key construction is the physical interpreta-tion/realization of the traced monoidal categories of finite-dimensional vector spaces with tensor product as monoidal structure and of finite sets and relations with Cartesia...

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...

This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, state-based modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of standard string diagrams for monoidal categories, for representing and manipulating component diagra...

Hopf algebras are closely related to monoidal categories. More precise, k-Hopf algebras can be characterized as those algebras whose category of finite dimensional representations is an autonomous monoidal category such that the forgetful functor to k-vectorspaces is a strict monoidal functor. This result is known as the Tannaka reconstruction theorem (for Hopf algebras). Because of the importa...

In this paper we prove the strong standard completeness of interval-valued monoidal t-norm based logic (IVMTL) and some of its extensions. For other extensions we show that they are not strong standard complete. We also give a local deduction theorem for IVMTL and other extensions of interval-valued monoidal logic. Similar results are obtained for interval-valued fuzzy logics expanded with Baaz...

Joyal and Street note in their paper on braided monoidal categories [10] that the 2–category V–Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. What is meant by “based upon” here will be made more clear in the present paper. The exception that they mention is the case in which V is symmetric, which leads to V–Ca...

Author's Note. When this manuscript was submitted in January 1972, the editor asked that it be expanded to study the relation of operads to clubs. The author found this too daunting a task at a busy time and the manuscript was never published. Reading through the manuscript now, more than thirty years later, elicits two strong impressions. First, the treatment is very complete: the only item no...

Joyal and Street note in their paper on braided monoidal categories [9] that the 2–category V –Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V . The exception that they mention is the case in which V is symmetric, which leads to V –Cat being symmetric as well. The symmetry in V –Cat is based upon the symmetry of...