We show that the category of Poisson manifolds and Poisson maps, the category of symplectic microgroupoids and lagrangian submicrogroupoids (as morphisms), and the category of monoids and monoid morphisms in the microsymplectic category are equivalent symmetric monoidal categories.

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [S, S] . An idempotent e of this ring will split the homotopy category: [X,Y ] ∼= e[X,Y ]⊕(1−e)[X,Y ] . We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that i...

Automata over a symmetric monoidal category M are introduced, and a multi-step simulation is defined among such automata. The collection of M automata is given the structure of a 2-category on the same objects as M , in which the vertical structure is determined by groups of indistinguishable simulations. TwoM -automata are called simulation equivalent if they are connected by an isomorphism of...

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...

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...

This paper investigates the use of symmetric monoidal closed (smc) structure for representing syntax with variable binding, in particular for languages with linear aspects. In this setting, one first specifies an smc theory T , which may express binding operations, in a way reminiscent from higher-order abstract syntax (hoas). This theory generates an smc category S(T ) whose morphisms are, in ...

To provide a categorical semantics for co-intuitionistic logic, one has to face the fact, noted by Tristan Crolard, that the definition of co-exponents as adjuncts of co-products does not work in the category Set, where co-products are disjoint unions. Following the familiar construction of models of intuitionistic linear logic with exponent !, we build models of co-intuitionistic logic in symm...

The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal category V and whose morphisms are modules. Mathematics Subject Classific...

In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. Special attention is given to the category which has finite dimensional Hilbert spaces as objects, linear maps as morphisms, and the tensor product as its monoidal structure ...