John F. Kennison

WHAT IS THE FUNDAMENTAL GROUP

Abstract For well-behaved topological spaces, there are several equivalent definitions for the fundamental group. For example, we could use equivalence classes of loops under homotopy, or deck translations of the universal covering. For badly-behaved spaces, these definitions diverge. A comprehensive definition (for connected, locally-connected spaces) is given using chains of open sets from an open cover, instead of loops. In the limit we get a localic group which classifies torsors over the space.

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]).

Mathematical concepts for analyzing ambiguous situations

Abstract We examine some mathematical tools for dealing with ambiguous situations. The main tool is the use of non-standard logic with truth-values in what is called a locale. This approach is related to fuzzy set theory, which we briefly discuss. We also consider probabilistic concepts. We include specific examples and describe the way a researcher can set up a suitable locale to analyse a concrete situation.

Synthetic solution manifolds for differential equations

The space of all solutions for a first-order differential equation can be regarded as a manifold, provided we generalize the traditional notion of differential manifold. We consider two such generalizations using C∞-rings and the smooth Basel topos B. Our definition enables us to define non-standard solutions such as probabilistic ones. There is a sense in which all first-order differential equations have global solutions (possibly non-standard) satisfying given initial conditions. We also prove change of variable theorems and discuss a smoothness condition.