The well-known Lawvere category [0,∞] of extended real positive numbers comes with a monoidal closed structure where the tensor product is the sum. But [0,∞] has another such structure, given by multiplication, which is *-autonomous and a CL-algebra (linked with classical linear logic). Normed sets, with a norm in [0,∞], inherit thus two symmetric monoidal closed structures, and categories enri...

Lyubashenko has described enriched 2–categories as categories enriched over V–Cat, the 2–category of categories enriched over a symmetric monoidal V. This construction is the strict analogue for V–functors in V–Cat of Brian Day’s probicategories for V–modules in V–Mod. Here I generalize the strict version to enriched n–categories for k–fold monoidal V. The latter is defined as by Balteanu, Fied...

In this paper we discuss a formal foundation of systems engineering based on category theory. The main difference with other categorical approaches is the choice of the structure of the base category (symmetric monoidal or compact closed) which is, on the one hand, much better adapted to current modeling tools and languages (e.g. SysML), and on the other hand is canonically associated to a logi...

One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2-categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, the funny tensor product of categories. In fact we exhibit for any higher categorical structure definable ...

In this paper we show that to a unital associative algebra object (resp. co-unital coassociative co-algebra object) of any abelian monoidal category (C,⊗) endowed with a symmetric 2-trace, i.e. an F ∈ Fun(C,Vec) satisfying some natural trace-like conditions, one can attach a cyclic (resp. cocyclic) module, and therefore speak of the (co)cyclic homology of the (co)algebra “with coefficients in F...

For a connected pasting scheme G, under reasonable assumptions on the underlying category, the category of C-colored G-props admits a cofibrantly generated model category structure. In this paper, we show that, if G is closed under shrinking internal edges, then this model structure on G-props satisfies a (weaker version) of left properness. Connected pasting schemes satisfying this property in...

We present an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. We first identify a particular class of functors – which we call ‘verification functors’ – between traced symmetric monoidal categories and subcategories of Preord (the category of preordered sets and monotone mappings). We then give an abstract definition of Hoa...

We reconstruct Milner’s [1] category of abstract binding bigraphs Bbg(K) over a signature K as the free (or initial) symmetric monoidal closed (smc) category S(TK) generated by a derived theory TK. The morphisms of S(TK) are essentially proof nets from the Intuitionistic Multiplicative fragment (imll) of Linear Logic [2]. Formally, we construct a faithful, essentially injective on objects funct...

Lyubashenko has described enriched 2–categories as categories enriched over V–Cat, the 2–category of categories enriched over a symmetric monoidal V. Here I generalize this to a k–fold monoidal V. The latter is defined as by Balteanu, Fiedorowicz, Schwänzl and Vogt but with the addition of making visible the coherent associators α. The symmetric case can easily be recovered. The introduction of...