We show that viewing labelled transition systems as relational presheaves captures several recently studied examples. This approach takes into account possible algebraic structure on labels. Weak closure of a labelled transition system is characterised as a left (2-)adjoint to a change-of-base functor. A famous application of coalgebra theory [3] is as a pleasingly abstract setting for the theo...

We shall make a survey of the most recent results obtained in connection with the programme of investigating notable categorical equivalences for MV-algebras from a topos-theoretic perspective commenced in [3]. In [3] and [2] we generalize to a topos-theoretic setting two classical equivalences arising in the context of MV-algebras: Mundici’s equivalence [4] between the category of MV-algebras ...

We show that the operational semantics for HOPLA is in strong correspondence with its presheaf semantics [2, 1]. The proof is a fairly standard logical relations proof, exploiting the path semantics of the language [3, 4]. Strong correspondence can be proved for full HOPLA by making use of the path semantics to get a logical relations proof off the ground. We’ll use the notation J−K for the pat...

1. Elegant Reedy categories 1.1. Reedy categories. Recall that a Reedy structure on a category C is given by: a pair of identity-on-objects subcategories C− and C+ of C together with a degree function |–| : ob C → N such that: • For all non-identity σ : c→ d in C−, we have |c| > |d|; • For all non-identity δ : c→ d in C+, we have |c| < |d|. • Every map γ : c→ d of C admits a unique factorisatio...

Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined an “algebraic pattern”, which we mean ?-category equipped with a factorization system and collection of “elementary” objects. Examples that occur as such “Segal O-spaces” for pattern O include ?-categories, (?,n)-categories, ?-operads (including symmetric, non-symmetric, cyclic, modular ones),...

We define and develop two-level type theory (2LTT), a version of Martin-L\"of which combines two different theories. refer to them as the inner outer theory. In our case interest, is homotopy (HoTT) may include univalent universes higher inductive types. The traditional form validating uniqueness identity proofs (UIP). One point view on it internalised meta-theory There are motivations for 2LTT...

Abstract We define and develop two-level type theory (2LTT), a version of Martin-Löf which combines two different theories. refer to them as the ‘inner’ ‘outer’ theory. In our case interest, inner is homotopy (HoTT) may include univalent universes higher inductive types. The outer traditional form validating uniqueness identity proofs (UIP). One point view on it internalised meta-theory There a...

For a quantaloid Q, considered as a bicategory, Walters introduced categories enriched in Q. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad T over the discrete presheaf monad of the small quantaloid Q the primary data of the theory, rather than the l...

One can use next the globalization machinery developed in [S-F] to get a similar looking spectral sequence for any smooth scheme of finite type over a field. Moreover, it’s not hard to see that the resulting spectral sequence coincides with the one constructed in [F-S]. What is nice, however, with this approach is that it avoids completely the use of the paper of Bloch and Lichtenbaum [B-L], wh...