We prove the universal property of the infinitary exact completion of a category with weak small limits. As an application, we slightly weaken the conditions characterizing essential localizations of varieties (in particular, of module categories) and of presheaf categories. Introduction An essential localization is a reflective subcategory such that the reflector has a left adjoint. In [12], R...

The semantics of name-passing calculi is often defined employing coalgebraic models over presheaf categories. This elegant theory lacks finiteness properties, hence it is not apt to implementation. Coalgebras over named sets, called history-dependent automata, are better suited for the purpose due to locality of names. A theory of behavioural functors for named sets is still lacking: the semant...

In this paper we provide a syntax for the cubical set model of type theory [3]. We start by defining a heterogeneous equality as a logical relation in an extended context (section 1.1). This can be seen as a different presentation of parametricity for dependent types [1]. We investigate the higher dimensional structure induced by the logical relation in section 1.2. The relation defined so far ...

We show that universes of fibrations in various models of homotopy type theory have an essentially global character: they cannot be described in the internal language of the presheaf topos from which the model is constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct ...

Language is contextual and sheaf theory provides a high level mathematical framework to model contextuality. We show how sheaf theory can model the contextual nature of natural language and how gluing can be used to provide a global semantics for a discourse by putting together the local logical semantics of each sentence within the discourse. We introduce a presheaf structure corresponding to ...

Abstract It is well established that equational algebraic theories and the monads they generate can be used to encode computational effects. An important insight of Power Shkaravska comodels an theory $\mathbb{T}$ – i.e., models in opposite category $\mathcal{S}\mathrm{et}^{\mathrm{op}}$ provide a suitable environment for evaluating effects encoded by . As already noted Shkaravska, taking yield...

The notion of a joint system, as captured by the monoidal (a.k.a. tensor) product, is fundamental to compositional, process-theoretic approach physical theories. Promonoidal categories generalise replacing functors normally used form systems with profunctors. Intuitively, this allows formation which may not always give system again, but instead generalised given presheaf. This extra freedom giv...

For small involutive and integral quantaloids Q it is shown that covariant presheaves on symmetric Q-categories are equivalent to certain subalgebras of a speci c monad on the category of symmetric Q-categories. This construction is related to a weakening of the subobject classi er axiom which does not require the classi cation of all subalgebras, but only guarantees that classi able subalgebra...

In mathematics and computer science, we often encounter structures which are faithfully encoded by a graphical calculus of the following sort. The basic data of the structure are depicted as certain diagrams; the basic operations of the structure act by glueing together these diagrams along certain parts of their boundaries; and the axioms of the structure are just those necessary to ensure tha...

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 [3, 4] is as a pleasingly abstract setting for the theory o...