We develop a 2-categorical theory for recursively defined domains. In particular, we generalise the traditional approach based on order-theoretic structures to categorytheoretic ones. A motivation for this development is the need of a domain theory for concurrency, with an account of bisimulation. Indeed, the leading examples throughout the paper are provided by recursively defined presheaf mod...

Along the way paved by the recent concurrent game semantics for process algebra CCS and π-calculus, based on the basic characteristics of the actor computational model and the very reductive semantics for actors, we establish a fully abstract concurrent game semantics for actors by borrowing the algebraic structure from CCS. This semantics can both be seen as an innocent presheaf semantics, and...

In analogy with the relation between closure operators in presheaf toposes and Grothendieck topologies, we identify the structure in a category with finite limits that corresponds to universal closure operators in its regular and exact completions. The study of separated objects in exact completions will then allow us to give conceptual proofs of local cartesian closure of different categories ...

We define a variety of notions of cubical sets, based on sites organized using substructural algebraic theories presenting PRO(P)s or Lawvere theories. We prove that all our sites are test categories in the sense of Grothendieck, meaning that the corresponding presheaf categories of cubical sets model classical homotopy theory. We delineate exactly which ones are even strict test categories, me...

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...

We propose a new research program which provides an approach to consciousness and the t problem ofdeep reality. Our model is based on category theory. The need to t relate l t local l behavior to global behavior has convinced us early on that a good model l for conscious entities had to be found in the notion of sheaf. With t this t i formulation, f l ti , every presheaf represents a brain (orr...

We develop a 2-categorical theory for recursively defined domains. In particular, we generalise the traditional approach based on order-theoretic structures to categorytheoretic ones. A motivation for this development is the need of a domain theory for concurrency, with an account of bisimulation. Indeed, the leading examples throughout the paper are provided by recursively defined presheaf mod...

We introduce a new type of local and microlocal asymptotic analysis in algebras of generalized functions, based on the presheaf properties of those algebras and on the properties of their elements with respect to a regularizing parameter. Contrary to the more classical frequential analysis based on the Fourier transform, we can describe a singular asymptotic spectrum which has good properties w...

We develop some of the foundations of affinoid pre-adic spaces without Noetherian or finiteness hypotheses. We give explicit examples of non-adic affinoid pre-adic spaces, and also a new condition ensuring that the structure presheaf on Spa(R,R) is a sheaf. This condition can be used to give a new proof that the spectrum of a perfectoid algebra is an adic space.

Article history: Received 7 June 2014 Available online 16 October 2014 Communicated by G. Rosolini MSC: 03B15; 03G30 We give an account of the basic combinatorial structure underlying the notion of type dependency. We do so by considering the category of generalised algebraic theories in the sense of Cartmell, and exhibiting it as the category of algebras for a monad on a presheaf category. The...