We endow the homotopy category of well generated (pretriangulated) dg categories with a tensor product satisfying universal property. The resulting monoidal structure is symmetric and closed respect to cocontinuous RHom (in sense To\"en [26]). give construction in terms localisations derived categories, making use enhanced Gabriel-Popescu theorem [21]. Given regular cardinal alpha, we define co...

A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb with a ∗-structure, conjugate-linear on the hom-sets, satisfying 〈fg, h〉 = 〈g, f∗h〉 = 〈f, hg∗〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which we call ...

For a symmetric monoidal-closed category X and any object K, the category of K-Chu spaces is small-topological over X and small cotopological over X . Its full subcategory of M-extensive K-Chu spaces is topological over X when X is Mcomplete, for any morphism class M. Often this subcategory may be presented as a full coreflective subcategory of Diers’ category of affine K-spaces. Hence, in addi...

It is known, that ordinary isomorphisms (associativity and commutativity of \times", isomorphisms for \times" unit and currying) provide a complete axiomatisation of isomorphism of types in multiplica-tive linear lambda calculus (isomorphism of objects in a free symmetric monoidal closed category). One of the reasons to consider linear isomor-phism of types instead of ordinary isomorphism was t...

In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for n-categories of the well-known stabilization theorems in homotopy theory. To explain the statement, recall that Baez-Dolan introduce the notion of k-uply monoidal n-category which is an n+ k-category having only one i...

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bire ectivity. Bire ective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterise them in terms of splitidempotent natural transformations on idA. In the special case that A is a presheaf category, we characterise...

When Gödel developed his functional interpretation, also known as the Dialectica interpretation, his aim was to prove (relative) consistency of first order arithmetic by reducing it to a quantifier-free theory with finite types. Like other functional interpretations (e.g. Kleene’s realizability interpretation and Kreisel’s modified realizability) Gödel’s Dialectica interpretation gives rise to ...

A category with biproducts is enriched over (commutative) additive monoids. A category with tensor products is enriched over scalar multiplication actions. A symmetric monoidal category with biproducts is enriched over semimodules. We show that these extensions of enrichment (e.g. from hom-sets to homsemimodules) are functorial, and use them to make precise the intuition that “compact objects a...

We study the “higher algebra” of spectral Mackey functors, which the first named author introduced in Part I of this paper. In particular, armed with our new theory of symmetric promonoidal ∞-categories and a suitable generalization of the second named author’s Day convolution, we endow the∞-category of Mackey functors with a wellbehaved symmetric monoidal structure. This makes it possible to s...

We generalize Dress and Müller’s main result in [5]. We observe that their result can be seen as a characterization of free algebras for certain monad on the category of species. This perspective allows to formulate a general exponential principle in a symmetric monoidal category. We show that for any groupoid G, the category !̂G of presheaves on the symmetric monoidal completion !G of G satisfi...