For an ideal I of a preadditive category A, we study when the canonical functor C : A → A/I is local. We prove that there exists a largest full subcategory C of A for which the canonical functor C : C → C/I is local. Under this condition, the functor C turns out to be a weak equivalence between C and C/I. If A is additive (with splitting idempotents), then C is additive (with splitting idempote...

Steve Awodey (CMU): “Sketch of the homotopy interpretation of intensional type theory” Abstract: As a tutorial of sorts, I will outline the homotopy interpretation of intensional type theory and survey some of the recent results by various people. As a tutorial of sorts, I will outline the homotopy interpretation of intensional type theory and survey some of the recent results by various people...

Extending Pirashvili’s work, we characterize small additive categories A such that the inclusion in the category of functors from A to abelian groups of the full subcategory of analytic functors induces an isomorphism between extension groups.

This is essentially the talk I gave on definable additive categories; I define these categories, say where they came from, describe some of what is around them and then point out the 2-category which

We prove that the 2-category of skeletally small abelian categories with exact monoidal structures is anti-equivalent to fp-hom-closed definable additive satisfying an exactness criterion. For a fixed finitely accessible category C products and structure appropriate assumptions, we provide bijections between subcategories C, Serre tensor-ideals Cfp-mod closed subsets Ziegler-type topology. prea...

The quantum-event / prime ideal in a category/ noncommutative-point alternative to classical-event / commutative prime ideal/ point is suggested. Ideals in additive categories, prime spectra and representation of quivers are considered as mathematical tools appropriate to model quantum mechanics. The space-time framework is to be reconstructed from the spectrum of the path category of a quiver....

We prove that the 2-category of skeletally small abelian categories with exact monoidal structures is anti-equivalent to fp-hom-closed definable additive satisfying an exactness criterion. For a fixed finitely accessible category C products and structure appropriate assumptions, we provide bijections between subcategories C, Serre tensor-ideals Cfp-mod closed subsets Ziegler-type topology. prea...

Abstract. We present a categorical model for intuitionistic linear logic where objects are polynomial diagrams and morphisms are simulation diagrams. The multiplicative structure (tensor product and its adjoint) can be defined in any locally cartesian closed category, whereas the additive (product and coproduct) and exponential (⊗-comonoid comonad) structures require additional properties and a...

A denotational semantics of quantum Turing machines having a quantum control is defined in the dagger compact closed category of finite dimensional Hilbert spaces. Using the Moore-Penrose generalized inverse, a new additive trace is introduced on the restriction of this category to isometries, which trace is carried over to directed quantum Turing machines as monoidal automata. The JoyalStreet-...

We show that the homotopy category of complexes K(B) over any finitely accessible additive category B is locally well generated. That is, any localizing subcategory L in K(B) which is generated by a set is well generated in the sense of Neeman. We also show that K(B) itself being well generated is equivalent to B being pure semisimple, a concept which naturally generalizes right pure semisimpli...