Anders Kock

Strong functors and monoidal monads

In [4] we proved that a commutative monad on a symmetric monoidal closed category carries the structure of a symmetric monoidal monad ([4], Theorem 3.2). We here prove the converse, so that, taken together, we have: there is a 1-1 correspondence between commutative monads and symmetric monoidal monads (Theorem 2.3 below). The main computational work needed consists in constructing an equivalence between possible strengths 8tA,B: A c~ B -+ A T ~ B T

Monads for which Structures are Adjoint to Units

Abstract We analyse the 2-dimensional categorical algebra underlying the process of completing categories, or posets. The algebra explains why and how completeness of a category is describable in monad theoretic terms, and why the limit formation for freely completed categories admits a further adjoint.

Doctrines in categorical logic

Publisher Summary This chapter presents category-theoretic methods in logic; the focus is on model theory and set theory. It is organized by increasing richness of the doctrines involved. These doctrines are categorical analogs of fragments of logical theories that have sufficient category-theoretic structure for their models to be described as functors. The introduction of the categorical notion of algebraic theory led to a systematic theory of relative interpretations of one equational theory into another, as well as a theory about the categories (or varieties) of algebras for these, and their relationship. This progress springs from having a presentation-invariant notion of equational (or algebraic) theory. The chapter also deals with doctrines of equational, Cartesian, finitary coherent, and infinitary coherent logic. Higher order logic and set theory are also discussed in the chapter.

Universal projective geometry via topos theory

The idea of this article is that linear algebra and projective geometry over a local commutative ring is equivalent to intuitionistic linear algebra and intuitionistic pure projective geometry over a field (at least in so far as caherenr sentences are concerned; this term will be explained). By pure projective geometry is meant a formulation of synthetic geometry in terms of the predicates “incidence” and “equality” alone, in contrast to formulations which include a special apartncss predicate. Heyting’s intuitionistic projective geometry, for instance, has such an apartness predicate o, and it is also necessary to have such w if one wants to formulate a reasonable synthetic theory for the projective geometry over a local ring. There are probably other and better reasons for looking for the “projective geometry over a local ring” than the one which gave rise to the present research: Study’s transfer principle. It says that whatever is true in plane projective geometry over the ring of dual numbers D = R[E] (with E’ = 0), can be reinterpreted as a theorem about the set of lines in euclidean 3-spL.:e. see e.g. [C;). This set (which is a 4-dimensional manifold) is thereby made into a Hjelmslev plune; Hjelmslev used this fact. Of course, the value of the transfer principle is that one gets, or hopes to get, theorems for the projective plane over D ;~y “analogy” with the well-known projective geometry over R. We are stud;% ig the meta-mathematics of that “analogy”.

Fibre bundles in general categories

Abstract A notion of fibre bundle is described, which makes sense in any category with finite inverse limits. An assumption is that some of the structural maps that occur are descent maps: this is the categorical aspect of the notion of glueing objects together out of local data. Categories of fibre bundles are proved to be equivalent to certain categories of groupoid actions. Some applications in locale theory are indicated.

Differential forms as infinitesimal cochains

Abstract In the context of synthetic differential geometry (SDG), we provide, for any manifold, a homotopy equivalence between its de Rham complex, and a complex of infinitesimal singular cochains. The equivalence takes wedge product of forms to cup product of singular cochains.

Oracle Inequalities for Convex Loss Functions with Nonlinear Targets

This article considers penalized empirical loss minimization of convex loss functions with unknown target functions. Using the elastic net penalty, of which the Least Absolute Shrinkage and Selection Operator (Lasso) is a special case, we establish a finite sample oracle inequality which bounds the loss of our estimator from above with high probability. If the unknown target is linear, this inequality also provides an upper bound of the estimation error of the estimated parameter vector. Next, we use the non-asymptotic results to show that the excess loss of our estimator is asymptotically of the same order as that of the oracle. If the target is linear, we give sufficient conditions for consistency of the estimated parameter vector. We briefly discuss how a thresholded version of our estimator can be used to perform consistent variable selection. We give two examples of loss functions covered by our framework.

Synthetic geometry of manifolds

co-) frame, 187 action on 1-monads, 46 active aspect, 59 ad, 168 adjoint action, 176 admitting path integration, 197 ad∇, 173 affine Bianchi identity, 221 affine combination, 20, 34, 298 affine connections, 52 affine scheme, 282 affine space, 298 Ak(E), 237 algebra connection, 69 algebraic commutator, 206, 230 algebroid, 160, 182, 184 Ambrose-Singer Theorem, 195, 202 an-holonomic distribution, 78 anchor, 82, 183 annular, 84 annular k-jet, 237 anti-derivative, 106 as if, 7, 25 atlas, 38 atom, 9 average value property, 271 axis-parallel rectangle, 300 base point, 40 basis, 286 Bianchi Identity, 133, 174, 216 bundle, 41, 66, 274, 279 bundle connection, 66 bundle theoretic differential operator, 244 Burgers vector, 54 cancellation principles, 24, 263 cancelling universally quantified ds, 23 canonical affine connection, 60 canonical framing, 48 cartesian closed, 273 central reflection, 45 chain rule, 32 Christoffel symbols, 55 C∞(ξ ),C∞(M), 245 Clairaut’s Theorem, 32 classical cotangent, 143, 146 classifier, 25 closed, 106 closed 1-form, 50 codiscrete groupoid, 166 combinatorial differential form, 92 comlete integral, 196 commutator, 157, 205, 206 complex numbers, 288 conformal, 263, 271 conformal matrix, 263 conjugate affine connection, 54 connected, 292 connection element, 175 connection form, 190 connection in a groupoid, 171 constant differential form, 99 constant groupoid, 168 construction site framing, 48 constructive matematics, 280 contracting jet, 84 contravariant determination of ∼, 42 convention, 28 coordinate n-tuple, 48 coordinate chart, 38

COMBINATORIAL NOTIONS RELATING TO PRINCIPAL FIBRE BUNDLES

This article aims at clarifying the relationship between principal fibre bundles and groupoids, and along with it, the relationship between connections in the bundle or groupoid, and the associated connection forms. These notions are essentially due to E. Cartan [2] and to Ehresmann [4], [5], who in them saw some of the fundamental aspects of differential geometry, cf. also [IO]. To this end, we introduce the notion of pregroupoid over a base ‘space’ B (‘space’ may mean either ‘topological space’, ‘smooth manifold’, or ‘object in a topos (;5’ ‘, and accordingly for ‘map’ (or ‘operation’ or ‘law’)). We use the word ‘set’ synonymously with ‘space’, and give some standard comments for this abuse below. Formally, a pregroupoid over B is a set E-B over B equipped with a partially defined ternary operation A, satisfying certain equations. In essence, a pregroupoid over B is the same as a principal fibre bundle, or torsor, over B, but whereas for a torsor, a group has to be given in advance, a pregroupoid canonically creates its own group. Also, by a dual construction, a pregroupoid creates a groupoid over B. Identifying the pregroupoid with a principal bundle H, this groupoid is Ehresmann’s HH- ‘, [4]. In the context of differential geometry, a typical example of a pregroupoid is the bundle E of orthonormal frames on a Riemannian manifold B; for x,y,z such frames, with x and z being frames at the same point of B, L(x, y,z) is the frame (at the same point as y) which has the same coordinates in terms of y as z does in terms of x. To describe the relationship between connections and connection forms, we need to assume that the base ‘space’ B comes equipped with a reflexive symmetric ‘neighbour’ relation. Except for the two trivial extreme cases, topological spaces do not carry any natural relation of this kind, nor do smooth manifolds. However, for the latter, the method of synthetic differential geometry (cf. e.g. [7]) becomes available: essentially, it consists in embedding the category Mf of smooth manifolds into a suitable ‘well-adapted’ topos (5’. When viewed in 6, any smooth manifold does

Every étendue comes from a local equivalence relation

Abstract We first prove that, under suitable connectedness assumptions, the equivariant sheaves for a local equivalence relation on a space (or a locale) form an etendue topos. Our main result is that conversely, every etendue can be obtained in this way.