Seely’s paper Locally cartesian closed categories and type theory contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-Löf type theories with Π,Σ, and extensional identity types. However, Seely’s proof relies on the problematic assumption that substitution in types can be interpreted by pullback...

It was already known that the category of T 0 topological spaces is not itself cartesian closed, but can be embedded into the cartesian closed categories FIL of lter spaces and EQU of equilogical spaces where the latter embeds into the cartesian closed category ASSM of assemblies over algebraic lattices. Here, we rst clarify the notion of lter space|there are at least three versions FIL a FIL b...

We generalize the standard construction of realizability models (specifically, of categories of assemblies) to a wide class of computability structures, broad enough to embrace models of computation such as labelled transition systems and process algebras. We consider a general notion of simulation between such computability structures, and show how these simulations correspond precisely to cer...

1 is unique (1 = 11 = 1, so any left identity equals any right identity). A morphism takes 1 to the identity of φX (since φ(x) = φ(1x) = φ(1)φ(x)), but φ(1) = 1 is an extra property that ought to be satisfied by morphisms; its kernel is the sub-algebra φ(1); zero object is { 1 }; not Cartesian-closed (the terminal and initial objects are the same, yet not all groups are isomorphic). A quasi-gro...

There have traditionally been two approaches to modelling environments, one by use of nite products in Cartesian closed categories , the other by use of the base categories of indexed categories with structure. Recently, there have been more general deenitions along both of these lines: the rst generalising from Cartesian to symmetric premonoidal categories, the second generalising from indexed...

There are two main approaches to obtaining “topological” cartesian-closed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed — for example, the category of sequential spaces. Under the other, one generalises the notion of space — for example, to Scott’s notion of equilogical space. In this paper we show that the two appr...

In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environments in call-by-value programming languages. There have been two more general definitions along bot...

In the context of semi-abelian categories, several additional conditions have been considered in order to obtain a closer group-like behavior. Among them are locally algebraic cartesian closedness and algebraic coherence. The recent notion of S-protomodular categories, whose main examples are the category of monoids and, more generally, categories of monoids with operations and Jónsson-Tarski v...

Using standard analysis only, we present an extension •R of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesimals may be also useful in infinite dimensional Differential Geometry, e.g. to study space...