Tangent categories provide an axiomatic framework for understanding various tangent bundles and differential operations that occur in differential geometry, algebraic geometry, abstract homotopy theory, and computer science. Previous work has shown that one can formulate and prove a wide variety of definitions and results from differential geometry in an arbitrary tangent category, including generalizations of vector fields and their Lie bracket, vector bundles, and connections.In this paper we investigate differential and sector forms in tangent categories. We show that sector forms in any tangent category have a rich structure: they form a symmetric cosimplicial object. This appears to be a new result in differential geometry, even for smooth manifolds. In the category of smooth manifolds, the resulting complex of sector forms has a subcomplex isomorphic to the de Rham complex of differential forms, which may be identified with alternating sector forms. Further, the symmetric cosimplicial structure on sector forms arises naturally through a new equational presentation of symmetric cosimplicial objects, which we develop herein.

Certain axiomatic notions of affine space over a ring and convex space over a preordered ring are examples of the notion of T -algebra for an algebraic theory T in the sense of Lawvere. Herein we study the notion of commutant for Lawvere theories that was defined by Wraith and generalizes the notion of centralizer clone. We focus on the Lawvere theory of left R-affine spaces for a ring or rig R, proving that this theory can be described as a commutant of the theory of pointed right R-modules. Further, we show that for a wide class of rigs R that includes all rings, these theories are commutants of one another in the full finitary theory of R in the category of sets. We define left R-convex spaces for a preordered ring R as left affine spaces over the positive part R + of R. We show that for any firmly archimedean preordered algebra R over the dyadic rationals, the theories of left R-convex spaces and pointed right R +modules are commutants of one another within the full finitary theory of R + in the category of sets. Applied to the ring of real numbers R, this result shows that the connection between convex spaces and pointed R + -modules that is implicit in the integral representation of probability measures is a perfect 'duality' of algebraic theories. * The author gratefully acknowledges financial support in the form of an AARMS Postdoctoral Fellowship,

We define a monad M on a category of measurable bornological sets, and we show how this monad gives rise to a theory of vector-valued integration that is related to the notion of Pettis integral. We show that an algebra X of this monad is a bornological locally convex vector space endowed with operations which associate vectors \int f dm in X to incoming maps f:T --> X and measures m on T. We prove that a Banach space is an M-algebra as soon as it has a Pettis integral for each incoming bounded weakly-measurable function. It follows that all separable Banach spaces, and all reflexive Banach spaces, are M-algebras.Comment: shortened, e.g. by citing references regarding basic lemmas; made changes to ordering of some lemmas and section

We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having an affine structure on a manifold is equivalent to having a flat torsion-free connection on its tangent bundle. This equivalence allows us to define a category of affine objects associated to a tangent category and we show that the resulting category is also a tangent category, as are several related categories. As a consequence of some of these ideas we also give two new characterizations of flat torsion-free connections.We also consider 2-categorical structure associated to the category of tangent categories and demonstrate that assignment of the tangent category of affine objects to a tangent category induces a 2-comonad.

Integral categories were recently developed as a counterpart to differential categories. In particular, integral categories come equipped with an integration operator, known as an integral transformation, whose axioms generalize the basic integration identities from calculus such as integration by parts. However, the literature on integral categories contains no example that captures integration of arbitrary smooth functions: the closest are examples involving integration of polynomial functions. This paper fills in this gap by developing an example of an integral category whose integral transformation operates on smooth 1-forms. We also provide an alternative viewpoint on the differential structure of this key example, investigate derivations and coderelictions in this context, and prove that free C ∞ -rings are Rota-Baxter algebras.

A locally small category E is totally distributive (as defined by Rosebrugh-Wood) if there exists a string of adjoint functors t -| c -| y, where y : E --> E^ is the Yoneda embedding. Saying that E is lex totally distributive if, moreover, the left adjoint t preserves finite limits, we show that the lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes, studied by Johnstone and Joyal. We characterize the totally distributive categories with a small set of generators as exactly the essential subtoposes of presheaf toposes, studied by Kelly-Lawvere and Kennett-Riehl-Roy-Zaks.Comment: Now includes extended result: The lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes; Made changes to abstract and intro to reflect the enhanced result; Changed formatting of diagram

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