Synthetic Differential Geometry Research Articles

Perception-Dominant Control Types for Human/Machine Systems

We explore a novel approach to complex domain modelling by emphasising primitives based on perception. The usual approach either focuses on actors or cognition associated with tokens that convey information. In related research, we have examined using effects and/or outcomes as primitives, and influences as the generator of those outcomes via categoric functors. That approach (influences, effects) has advantages: it leverages what is known and supports the expanded logics we use, where we want to anticipate and engineer possible futures. But it has weaknesses when placed in a dynamic human-machine system where what is perceived or assumed matters more than what is known. The work reported here builds on previous advances in type specification and reasoning to ‘move the primitives forward’ more toward situation encounter and away from situation understanding. The goal is in the context of shared human-machine systems where: • reaction times are shorter than the traditional ingestion/comprehension/response loop can support; • situations that are too complex or dynamic for current comprehension by any means; • there simply is insufficient knowledge about governing situations for the comprehension model to support action; and/or, • the many machine/human and system/system interfaces that are incapable of conveying the needed insights; that is, the communication channels choke the information or influence flows. While the approach is motivated by the above unfriendly conditions, we expect significant benefits. We will explore these but engineer toward a federated decision paradigm where decisions by local human, machine or synthesis are not whole-situation-aware, but that collectively ‘swarm’ locally across the larger system to be more effective, ‘wiser’ than a convention paradigm may produce. The supposed implementation strategy will be through extending an existing ‘playbooks as code’ project whose goals are to advise on local action by modelling and gaming complex system dynamics. A sponsoring context is ‘grey zone’ competition that avoids armed conflict, but that can segue to a mixed system course of action advisory. The general context is a costly ‘blue swan’ risk in large commercial and government enterprises. The method will focus on patterns and relationships in synthetic categories used to model type transitions within topological models of system influence. One may say this is applied intuitionistic type theory, following mechanisms generally described by synthetic differential geometry. In this context, the motivating supposition of this study is that information-carrying influence channels are best modelled in our challenging domain as perceived types rather than understood types.

THE UNITY AND IDENTITY OF DECIDABLE OBJECTS AND DOUBLE-NEGATION SHEAVES

AbstractLet ${\cal E}$ be a topos, ${\rm{Dec}}\left( {\cal E} \right) \to {\cal E}$ be the full subcategory of decidable objects, and ${{\cal E}_{\neg \,\,\neg }} \to {\cal E}$ be the full subcategory of double-negation sheaves. We give sufficient conditions for the existence of a Unity and Identity ${\cal E} \to {\cal S}$ for the two subcategories of ${\cal E}$ above, making them Adjointly Opposite. Typical examples of such ${\cal E}$ include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

A Synthetic Version of Lie’s Second Theorem

We formulate and prove a generalisation of Lie’s second theorem that integrates homomorphisms between formal group laws to homomorphisms between Lie groups. Firstly we generalise classical Lie theory by replacing groups with groupoids. Secondly we include groupoids whose underlying spaces are not smooth manifolds. The main intended application is when we replace the category of smooth manifolds with a well-adapted model of synthetic differential geometry. In addition we provide an axiomatic system that provides all the abstract structure that is required to prove Lie’s second theorem.

An embedding theorem for tangent categories

Tangent categories were introduced by Rosický as a categorical setting for differential structures in algebra and geometry; in recent work of Cockett, Crutwell and others, they have also been applied to the study of differential structure in computer science. In this paper, we prove that every tangent category admits an embedding into a representable tangent category—one whose tangent structure is given by exponentiating by a free-standing tangent vector, as in, for example, any well-adapted model of Kock and Lawvere's synthetic differential geometry. The key step in our proof uses a coherence theorem for tangent categories due to Leung to exhibit tangent categories as a certain kind of enriched category.

Higher-dimensional general Jacobi identities I

It was shown by the author [International Journal of Theoretical Physics 36 (1997), 1099-1131] in synthetic differential geometry that what is called the general Jacobi identity obtaining in microcubes underlies the Jacobi identity of vector fields. It is well known in the theory of Lie algebras that a plethora of higher-dimensional generalizations of the Jacobi identity hold, though it is usually established not as a direct derivation from the axioms of Lie algebras but by making an appeal to the so-called Poincare-Birkhoff-Witt theorem. The general Jacobi identity was rediscovered by Kirill Mackenzie in the second decade of this century [Geometric Methods in Physics, 357-366, Birkhauser/Springer 2013]. The principal objective in this paper is to investigate a four-dimensional generalization of the general Jacobi identity in detail. In a subsequent paper we will propose a uniform method for establishing a bevy of higher-dimensional generalizations of the Jacobi identity under a single umbrella.

Infinitesimal Structure of Singularities

Some important problems of general relativity, such as the quantisation of gravity or classical singularity problems, crucially depend on geometry on very small scales. The so-called synthetic differential geometry—a categorical counterpart of the standard differential geometry—provides a tool to penetrate infinitesimally small portions of space-time. We use this tool to show that on any “infinitesimal neighbourhood” the components of the curvature tensor are themselves infinitesimal, and construct a simplified model in which the curvature singularity disappears, owing to this effect. However, one pays a price for this result. Using topoi as a generalisation of spaces requires a weakening of arithmetic (the existence of infinitesimals) and of logic (to the intuitionistic logic). Is this too high a price to pay for acquiring a new method of solving unsolved problems in physics? Without trying, we shall never know the answer.

Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry

We extend some fundamental definitions and constructions in the established generalisation of Lie theory involving Lie groupoids by reformulating them in terms of groupoids internal to a well-adapted model of synthetic differential geometry. In particular we define internal counterparts of the definitions of source path and source simply connected groupoid and the integration of $A$-paths. The main results of this paper show that if a classical Hausdorff Lie groupoid satisfies one of the classical connectedness conditions it also satisfies its internal counterpart.

Poisson Algebras for Non-Linear Field Theories in the Cahiers Topos

We develop an approach to construct Poisson algebras for non-linear scalar field theories that is based on the Cahiers topos model for synthetic differential geometry. In this framework the solution space of the field equation carries a natural smooth structure and, following Zuckerman's ideas, we can endow it with a presymplectic current. We formulate the Hamiltonian vector field equation in this setting and show that it selects a family of observables which forms a Poisson algebra. Our approach provides a clean splitting between geometric and algebraic aspects of the construction of a Poisson algebra, which are sufficient to guarantee existence, and analytical aspects that are crucial to analyze its properties.

Synthetic differential geometry of Chen's iterated integrals

Chen's iterated integrals are treated within synthetic differential geometry. The main result is that iterated integrals produce a subcomplex of the de Rham complex on the free path space as well as based path spaces.

Internal Diagrams and Archetypal Reasoning in Category Theory

We can regard operations that discard information, like specializing to a particular case or dropping the intermediate steps of a proof, as projections, and operations that reconstruct information as liftings. By working with several projections in parallel we can make sense of statements like “Set is the archetypal Cartesian Closed Category”, which means that proofs about CCCs can be done in the “archetypal language” and then lifted to proofs in the general setting. The method works even when our archetypal language is diagrammatical, has potential ambiguities, is not completely formalized, and does not have semantics for all terms. We illustrate the method with an example from hyperdoctrines and another from synthetic differential geometry.

The Baker-Campbell-Hausdorff formula and the Zassenhaus formula in synthetic differential geometry

After the torch of Anders Kock (6), we will establish the Baker-Campbell- Hausdor formula as well as the Zassenhaus formula in the theory of Lie groups.

Differential Structure, Tangent Structure, and SDG

In 1984, J. Rosický gave an abstract presentation of the structure associated to tangent bundle functors in differential and algebraic geometry. By slightly generalizing this notion, we show that tangent structure is also fundamentally related to the more recently introduced Cartesian differential categories. In particular, tangent structure of a trivial bundle is precisely the same as Cartesian differential structure. We also provide a general result which shows how tangent structure arises from the manifold completion (in the sense of M. Grandis) of a differential restriction category. This construction includes all standard atlas-based constructions from differential geometry. Furthermore, we tighten the relationship, which Rosický had noted, between representable tangent structure and synthetic differential geometry, showing how such settings can be developed from a system of infinitesimal objects. We also show how infinitesimal objects give rise to dual tangent structure. Taken together, these results show that tangent structures appropriately span a very wide range of definitions, from the syntactic and structural differentials arising in computer science and combinatorics, through the concrete manifolds of algebraic and differential geometry, and finally to the abstract definitions of synthetic differential geometry.

Fermat reals: infinetesimals without logic

We review the theory of Fermat reals and Fermat extensions, a relatively new theory of nilpotent infinitesimals which does not need any background in Mathematical Logic. We focus on some differences from Nonstandard Analysis and Synthetic Differential Geometry using the viewpoint of intuitive interpretation and applicability in Physics. Finally, we state some open problems. 2000 Mathematics Subject Classification: Primary 14B10; Secondary 26E30, 37P20, 03H10

Topological and algebraic structures on the ring of Fermat reals

The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring by using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non-standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented.

An introduction to C-infinity schemes and C-infinity algebraic geometry

This is a survey of the author's paper arXiv:1001.0023 on Algebraic Geometry over If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a C-infinity ring. That is, for each smooth function f : R^n --> R there is an n-fold operation \Phi_f : C^\infty(X)^n --> C^\infty(X) acting by \Phi_f: (c_1,...,c_n) |--> f(c_1,...,c_n), and these operations \Phi_f satisfy many natural identities. Thus, C^\infty(X) actually has a far richer structure than the obvious R-algebra structure. We explain a version of algebraic geometry in which rings or algebras are replaced by rings. As schemes are the basic objects in algebraic the new basic objects are C-infinity schemes, a category of geometric objects generalizing manifolds, and whose morphisms generalize smooth maps. We also discuss C-infinity including Deligne-Mumford stacks, a 2-category of geometric objects generalizing orbifolds. We study quasicoherent and coherent sheaves on schemes and stacks, and orbifold strata of Deligne-Mumford stacks. This enables us to use the tools of algebraic geometry in differential and to describe singular spaces such as moduli spaces occurring in differential geometric problems. Many of these ideas are not new: rings and schemes have long been part of synthetic differential geometry. But we develop them in new directions. In a new book, surveyed in arXiv:1206.4207 and at greater length in arXiv:1208.4948, the author uses algebraic geometry to develop a theory of differential geometry, which studies d-manifolds and d-orbifolds, derived versions of smooth manifolds and orbifolds. D-orbifolds will have applications in symplectic as the geometric structure on moduli spaces of J-holomorphic curves.

Affine connections, midpoint formation, and point reflection

We describe some differential-geometric structures in combinatorial terms: namely affine connections and their torsion and curvature, and we show that torsion free affine connections may equivalently be presented in terms of some simpler combinatorial structure: midpoint formation, and point reflection (geodesic symmetry). The method employed is that of synthetic differential geometry, which is briefly explained.

Cubical Version of Combinatorial Differential Forms

The theory of combinatorial differential forms is usually presented in simplicial terms. We present here a cubical version; it depends on the possibility of forming affine combinations of mutual neighbour points in a manifold, in the context of synthetic differential geometry.

General Jacobi Identity Revisited Again

Synthetic differential geometry occupies a unique position in topos-theoretic physics. Nevertheless it has appeared somewhat too conceptual to physicists in general, partly because it has appeared to lack computational aspects. Its computational facets are really concerned with computation of the quasi-colimit of a finite diagram of infinitesimal spaces, or equivalently, with computation of the limit of a finite diagram of Weil algebras. Indeed we have been forced to do a highly involved computation of the above kind by hand in our previous papers (Nishimura, H. in Int. J. Theor. Phys. 36:1099–1131, 1997 and Nishimura, H. in Int. J. Theor. Phys. 38:2163–2174, 1999). The principal objective in this paper is to show that Gro bner bases techniques provide us with means that relegate such computations to computers.

The Meaning of Category Theory for 21st Century Philosophy

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy.

Ordinary differential equations and their exponentials

Abstract In the context of Synthetic Differential Geometry, we discuss vector fields/ordinary differential equations as actions; in particular, we exploit function space formation (exponential spaces) in the category of actions.