A categorical and Łukasiewicz-Topos framework for Łukasiewicz Algebraic Logic models of nonlinear dynamics in complex functional systems such as neural networks, genomes and cell interactomes is proposed. Łukasiewicz Algebraic Logic models of genetic networks and signaling pathways in cells are formulated in terms of nonlinear dynamic systems with n-state components that allow for the generaliz...

In our paper [1] we have introduced the basic concepts and facts for scientific problem solving by help of mathematical machines, i.e. by logical reasoning about programming of these machines. These fundamental concepts were: category, cartesian closed category, diagram and limit, topos and elementary topos, but the most important was the concept of basic types. Basic types actually form the st...

This paper establishes a new limitative relation between the polymorphic lambda calculus and the kind of higher order type theory which is em bodied in the logic of toposes It is shown that any embedding in a topos of the cartesian closed category of closed types of a model of the poly morphic lambda calculus must place the poly morphic types well away from the powertypes of the topos in the se...

The class of functors known as discrete Conduch e brations forms a common generalization of discrete brations and discrete oppbrations, and shares many of the formal properties of these two classes. F. Lamarche 7] conjectured that, for any small category B, the category DCF=B of discrete Conduch e brations over B should be a topos. In this note we show that, although for suitable categories B t...

Category theory has two unexpected links to constructivism: First, why is topos logic so close to intuitionistic logic? The paper argues that in part the resemblance is superficial, in part it is due to selective attention, and in part topos theory is objectively tied to the motives for later intuitionistic logic little related to Brouwer’s own stated motives. Second, why is so much of general ...

The category of Set-valued presheaves on a small category B is a topos. Replacing Set by a bicategory S whose objects are sets and morphisms are spans, relations, or partial maps, we consider a category Lax(B,S) of S-valued lax functors on B. When S = Span, the resulting category is equivalent to Cat/B, and hence, is rarely even cartesian closed. Restricting this equivalence gives rise to expon...