Topos Of Sets Research Articles

Categorical Nonstandard Analysis

In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while we make use of an endofunctor U on a topos of sets S together with a natural transformation υ, instead of the terms as “standard”, “internal”, or “external”. Moreover, we propose a general notion of a space called U-space, and the category USpace whose objects are U-spaces and morphisms are functions called U-spatial morphisms. The category USpace, which is shown to be Cartesian closed, gives a unified viewpoint toward topological and coarse geometric structure. It will also be useful to further study symmetries/asymmetries of the systems with infinite degrees of freedom, such as quantum fields.

Applications of Non-Standard analysis in Topoi to Mathematical Neurosciences and Artificial Intelligence: Infons, Energons, Receptons (I)

The purpose of this paper is to promote new methods in mathematical modeling inspired by neuroscience—that is consciousness and subconsciousness—with an eye toward artificial intelligence as parts of the global brain. As a mathematical model, we propose topoi and their non-standard enlargements as models, due to the fact that their logic corresponds well to human thinking. For this reason, we built non-standard analysis in a special class of topoi; before now, this existed only in the topos of sets (A. Robinson). Then, we arrive at the pseudo-particles from the title and to a new axiomatics denoted by Intuitionistic Internal Set Theory (IIST); a class of models for it is provided, namely, non-standard enlargements of the previous topoi. We also consider the genetic–epigenetic interplay with a mathematical introduction consisting of a study of the Yang–Baxter equations with new mathematical results.

The principal bundles over an inverse semigroup

This paper is a contribution to the development of the theory of representations of inverse semigroups in toposes. It continues the work initiated by Funk and Hofstra (Theory Appl Categ 24(7):117–147, 2010). For the topos of sets, we show that torsion-free functors on Loganathan’s category L(S) of an inverse semigroup S are equivalent to a special class of non-strict representations of S, which we call connected. We show that the latter representations form a proper coreflective subcategory of the category of all non-strict representations of S. We describe the correspondence between directed and pullback preserving functors on L(S) and transitive and effective representations of S, as well as between filtered such functors and universal representations introduced by Lawson, Margolis and Steinberg. We propose a definition of a universal representation, or, equivalently, an S-torsor, of an inverse semigroup S in the topos of sheaves \({\mathsf {Sh}}(X)\) on a topological space X. We prove that the category of filtered functors from L(S) to the topos \({\mathsf {Sh}}(X)\) is equivalent to the category of universal representations of S in \({\mathsf {Sh}}(X)\). We finally propose a definition of an inverse semigroup action in an arbitrary Grothendieck topos, which arises from a functor on L(S).

Regular functors and relative realisability categories

The relative realisability toposes introduced by Awodey, Birkedal and Scott in Awodey et al. (2002) satisfy a universal property involving regular functors to other categories. We use this universal property to define what relative realisability categories are when they are based on categories other than the topos of sets. This paper explains the property and gives a construction for relative realisability categories that works for arbitrary base Heyting categories. The universal property also provides some new geometric morphisms to relative realisability toposes.

Cartesian monads on toposes

We consider two related questions: ‘When can a cartesian functor between toposes be factored as an inverse image functor followed by a direct image functor?’ and ‘Does every cartesian monad on a topos ∄ arise from a geometric morphism with codomain ∄?’. The connection between the two questions is that the answer to the first, in the particular case when both the domain and codomain of the functor are the topos of sets, is ‘if and only if it carries a monad structure’. We investigate the class of cartesian monads on the topos of sets, providing a new proof that they correspond bijectively to strongly zero-dimensional locales in this topos. By combining two old results on strong functors, we show that any cartesian monad on an arbitrary topos has a canonical extension to an indexed monad; this suffices to extend the arguments employed in the topos of sets to an arbitrary Boolean base topos, but we also show how they fail in non-Boolean toposes. For such toposes, the answer to the second question therefore remains unknown.

Toposes in Logic and Logic in Toposes

The purpose of this paper is to justify the claim that Topos theory and Logic (the latter interpreted in a wide enough sense to include Model theory and Set theory) may interact to the advantage of both fields. Once the necessity of utilizing toposes (other than the topos of Sets) becomes apparent, workers in Topos theory try to make this task as easy as possible by employing a variety of methods which, in the last instance, find their justification in metatheorems from Logic. Some concrete instances of this assertion will be given in the form of simple proofs that certain theorems of Algebra hold in any (Grothendieck) topos, in order to illustrate the various techniques that are used. In the other direction, Topos theory can also be a useful tool in Logic. Examples of this are independence proofs in (classical as well as intuitionistic) Set theory, as well as transfer methods in the presence of a sheaf representation theorem, the latter applied, in particular, to model theoretic properties of certain theories.

Quotients of decidable objects in a topos

In (5), Peter Freyd recently raised the question of whether every Grothendieck topos could be obtained from the topos of sets by means of the two constructions of taking sheaves on a locale and of taking continuous actions of a topological group (i.e. the topos-theoretic analogues of the set-theorists' techniques of forcing extensions and permutation models). He showed that these two constructions do suffice to within epsilon; provided we allow ourselves the freedom to take exponential varieties (4) (which do not change the internal logic of the topos) we can obtain every Grothendieck topos in this way.

Separable algebraic closure in a topos

In this paper we construct the separable algebraic closure of any field in a topos 8. (A separable closure is a field extension in which every separable polynomial splits and which is generated by the roots of these polynomials. P is a separable polynomial if P and its derivative P’ are relatively prime.) The separable closure lives in a new topos, over b, and has the universal property which makes it a spectrum in the sense of Cole, see [l], [9] and [4, Theorem 6.581. Recall that if 8 is the topos of Sets there are two ways to construct this spectrum. One is to use the &ale topos which is the home of the generic separable closure of an arbitrary commutative ring in Sets (see [3,12]). As shown by Hakim [3, pp. 77-841, this construction can be applied within any Grothendieck topos by setting up the &ale topology (object by object on the defining site of the topos). Alternatively, we can describe the generic separable closure of a field as a field extension in the topos of continuous G-Sets where G is the profinite Galois group. In this paper, we generalize the profinite Galois group approach. We feel that the value of this paper lies not so much in the alternative, more internal, construction of the separable closure but in the topos theoretic structure we develop along the way. We define a profinite group and more generally profinite groupoid, and profinite category in any topos. If I-is a profinite category in 8 then we construct the topos gr of continuous r-actions. We generalize the profinite Galois group of the algebraic closure of a field. In general it is a connected profinite groupoid. As shown in Section 3, below, it is Morita equivalent to a profinite group precisely when the separable closure of K in 8 can be constructed within the topos 8 itself. For example, any field in Sets has an algebraic closure in Sets so the profinite Galois groupoid is effectively equivalent to a profinite group. When there does exist a separable closure of K in 8 itself then the Galois groupoid is equivalent to the profinite group constructed in [6].

Galois theory and theaters of action in a topos

Constructing a Galois theory for geometric fields in a topos presents certain difficulties. If K E F is a Galoisian field extension in a topos 5Z, then one can construct the internal group of K-automorphisms of F but it might be trivial even when K f F (see Example 6.6). In the topos of presheaves over a category C, a Galois group appears to be a presheaf over Cop. There are similar difficulties with splitting a polynomial over a field K in a topos. If we are splitting x2 + 1 over K in the topos of Sets, then we usually use a dichotomous procedure: if x2+ 1 is irreducible we construct K[i], if .r* + 1 is reducible we use K. This procedure is problematic for topoi since irreducibility is not a coherent condition (i.e., not geometric in the sense of [3]). A procedure for splitting polynomials is given in Section 3. See Example 6.3 also. Of course the field containing the added roots lives in a new topos, and this is an instance of the Cole Spectrum (see [l, 3,7]).