We believe that the study of the notions of universal algebra modelled in an arbitarry topos rather than in the category of sets provides a deeper understanding of the real features of the algebraic notions. [2], [3], [4], [S], [6], [7], [13], [14] are some examples of this approach. The lattice Id(L) of ideals of a lattice L (in the category of sets) is an important ingredient of the categ...

A topos of presheaves can be seen as an extension of classical set theory, where sets vary over informational states, therefore it is a powerful and expressive mathematical framework. I introduce a suitable topos of presheaves where imprecise probabilities and imprecise probabilistic reasoning can be represented. In this way we obtain a mathematical definition of imprecise-probabilistic sets. A...

Let E be a cocomplete topos. We show that if the exact completion of E is a topos then every indecomposable object in E is an atom. As a corollary we characterize the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere–Schanuel characterization of Boolean presheaf toposes and Hofstra’s characterization of the locally connected Gro...

This paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. In paper II, we studied the topos representations of the propositional language PL(S) for the case of qu...

We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes. We also investigate enrichments of these categories over the base topos; there are in fact often two distinct enrichments.

We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes. We also investigate enrichments of these categories over the base topos; there are in fact often two distinct enrichments.

This paper deenes a solution manifold and a stable submanifold for a system of diierential equations. Although we eventually work in the smooth topos, the rst two sections do not mention topos theory and should be of interest to non-topos theorists. The paper characterizes solutions in terms of barriers to growth and deenes solutions in what are called lter rings (characterized as C 1-reduced r...

We discuss two aspects of the presentation of the theory of principal ∞-bundles in an ∞-topos, introduced in [NSSa], in terms of categories of simplicial (pre)sheaves. First we show that over a cohesive site C and for G a presheaf of simplicial groups which is C-acyclic, G-principal ∞-bundles over any object in the ∞-topos over C are classified by hyper-Čech-cohomology with coefficients in G. T...

We are developing tools for working with arbitrary left-exact localizations of ?-topoi. introduce a notion higher sheaf respect to an set maps ? in ?-topos E. show that the full subcategory sheaves Sh(E,?) is ?-topos, and reflection E?Sh(E,?) localization generated by ?. The proof depends on congruence, which substitute Grothendieck topology 1-topos theory.

DNA topoisomerases (topos) and DNA polymerases (pols) are involved in many aspects of DNA metabolism such as replication reactions. We found that long chain unsaturated fatty acids such as polyunsaturated fatty acids (PUFA) (i.e., eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA)) inhibited the activities of eukaryotic pols and topos in vitro, and the inhibitory effect of conjugated fa...