We give a series of glueing constructions for categorical models of fragments of linear logic. Specifically, we consider the glueing of (i) symmetric monoidal closed categories (models of Multiplicative Intuitionistic Linear Logic), (ii) symmetric monoidal adjunctions (for interpreting the modality !) and (iii) -autonomous categories (models of Multiplicative Linear Logic); the glueing construc...

This paper establishes a relation between the notion of a codifferential category and the more classic theory of Kähler differentials in commutative algebra. A codifferential category is an additive symmetric monoidal category with a monad T , which is furthermore an algebra modality, i.e. a natural assignment of an associative algebra structure to each object of the form T (C). Finally, a codi...

The Grothendieck construction is a process to form a single category from a diagram of small categories. In this paper, we extend the definition of the Grothendieck construction to diagrams of small categories enriched over a symmetric monoidal category satisfying certain conditions. Symmetric monoidal categories satisfying the conditions in this paper include the category of k-modules over a c...

Abstract We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind enriched category. The base for enrichment is category commutative monoids—or in straightforward generalisation, modules over rig k . However, tensor product on this not usual one, but rather warping it by certain monoidal comonad Q Thus sense, skew sense Szlachányi. Our first main result ...

There are many interesting situations in which algebraic structure can be described by operads [1, 12, 13, 14, 17, 20, 27, 32, 33, 34, 35]. Let (C,⊗, k) be a symmetric monoidal closed category (Section 2) with all small limits and colimits. It is possible to define two types of operads (Definition 6.1) in this setting, as well as algebras and modules over these operads. One type, called Σ-opera...

We prove that any rigid additive symmetric monoidal category can be mapped to a abelian in universal way. This yields novel approach Grothendieck's standard conjecture D and Voevodsky's smash nilpotence conjecture.

Recent work on gravity in two dimensions has a natural generalization to four dimensions. 1. Basic definitions 1.1 The (symmetric monoidal) two-category (Gravity)d+1 has objects: compact oriented d-manifolds, with • morphisms V0 → V1 : (d+ 1)-manifolds W with ∂W ∼= V op 0 t V1, and • diffeomorphisms W̃ →W as two-morphisms. The category Mor(V0, V1) with cobordisms from V0 to V1 as objects and dif...

It is proved that MacLane’s coherence results for monoidal and symmetric monoidal categories can be extended to some other categories with multiplication; namely, to relevant, affine and cartesian categories. All results are formulated in terms of natural transformations equipped with “graphs” (g-natural transformations) and corresponding morphism theorems are given as consequences. Using these...

We compute the spectrum of category derived Mackey functors (in sense Kaledin) for all finite groups. find that this space captures precisely top and bottom layers (i.e. height infinity zero parts) equivariant stable homotopy category. Due to truncation chromatic information, we are able obtain a complete description groups, despite our incomplete knowledge topology From different point view, s...