This note is about work in progress on the topic of “internal type theory” where we investigate the internal formalization of the categorical metatheory of constructive type theory in (an extension of) itself. The basic notion is that of a category with families, a categorical notion of model of dependent type theory. We discuss how to formalize the notion of category with families inside type ...

Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following facts: (1) For any given positive integer N , there are N (mutually non-isomorphic) projective complex K3 surfaces such that their Picard groups are not isomorphic but their transcendental lattices are Hodge isometric, or equivale...

This paper is a part of the series [KL]; however, it can be read independently of the first two parts. In [D3], Drinfeld proved the existence of an equivalence between a tensor category of representations of a quantum group over C[[ ro]] and a tensor category of representations of an undeformed enveloping algebra over C[[ro]] , in which the associativity constraints are given by the Knizhnik-Za...

Given an algebraic theory which can be described by a (possibly symmetric) operad P , we propose a definition of the weakening (or categorification) of the theory, in which equations that hold strictly for P -algebras hold only up to coherent isomorphism. This generalizes the theories of monoidal categories and symmetric monoidal categories, and several related notions defined in the literature...

A functor G : C → D is said to preserve limits of a diagram D : I → C if it sends any limiting cone from x to D to a limiting cone from G(x) to G◦D. When G preserves limits of a diagram D this entails directly that there is an isomorphism G(lim ←−ID) ∼= lim ←−I(G ◦D) between objects. In general, such an isomorphism alone is not sufficient to ensure that G preserves limits. This paper shows how,...

In the subgraph querying problem, given a query graph and a dataset of graphs, we want to locate which graphs in the dataset contain the query and/or find all its occurrences. Over the years, numerous methods, fragmented in 2 categories, were proposed to tackle the problem. In the first category, methods follow the filter-then-verify (FTV) paradigm where an index is used to filter out graphs th...

We describe a fundamental additive functor Efund on the orbit category of a group. We prove that any isomorphism conjecture valid for Efund also holds for all additive functors, like K-theory, (topological) Hochschild or cyclic homology, etc. Finally, we reduce this universal isomorphism conjecture to K-theoretic ones, at the price of introducing some coefficients.

We provide a categorical interpretation of a well-known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras. As a consequence, we deduce that the Serre functor of a finite dimensional triangular algebra A has always a lift, up to shift, to a product of suitably defined reflection functors in the category of ...

Given a smooth projective varietyX , we denote byD(X) the bounded derived category of coherent sheaves D(Coh(X)). All varieties we consider below are over the complex numbers. A result of Rouquier, [Ro] Théoréme 4.18, asserts that if X and Y are smooth projective varieties with D(X) ' D(Y ) (as linear triangulated categories), then there is an isomorphism of algebraic groups Aut(X)× Pic(X) ' Au...

The object of this note is to state certain theorems, whose proofs together with related results will appear elsewhere. The theorems are mainly concerned with asymptotic enumeration of the isomorphism classes of finite rings or finite-dimensional algebras lying in various naturally-defined categories. Two results concern enumeration of subalgebras or subrings in a given algebra or ring. All the...