Category theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer… Expand

Quillen [17] introduced model categories as an abstract framework for homotopy theory which would apply to a wide range of mathematical settings. By all accounts this program has been a success… Expand

This work investigates a configuration of three toposes, which is regarded as describing a notion of relative computability, and deriving a modal calculus as its internal logic.Expand

In 1929 Carnap gave a paper in Prague on “Investigations in General Axiomatics”; a briefsummary was published soon after. Its subject lookssomething like early model theory, and the mainresult,… Expand

This paper is the first in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins… Expand

It is shown that a category admits a natural model of type theory if it has a class of maps that behave like the abstract fibrations in axiomatic homotopy theory: They should be stable under pullback, closed under composition and relative products, and there should be weakly orthogonal factorizations into the class.Expand