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G. S. H. Cruttwell

Bio: G. S. H. Cruttwell is an academic researcher from Mount Allison University. The author has contributed to research in topics: Differential geometry & Tangent. The author has an hindex of 8, co-authored 23 publications receiving 197 citations. Previous affiliations of G. S. H. Cruttwell include Dalhousie University & University of Calgary.

Papers
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Journal ArticleDOI
TL;DR: It is shown 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.
Abstract: 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.

66 citations

01 Jan 2010
TL;DR: In this paper, a unied frame-work is proposed for generalized multicategories in a way that unies all previous examples, while at the same time simplifying and clarifying the theory.
Abstract: Notions of generalized multicategory have been dened in numerous con- texts throughout the literature, and include such diverse examples as symmetric multi- categories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are dened as the \lax algebras" or \Kleisli monoids" rela- tive to a \monad" on a bicategory. However, the meanings of these words dier from author to author, as do the specic bicategories considered. We propose a unied frame- work: by working with monads on double categories and related structures (rather than bicategories), one can dene generalized multicategories in a way that unies all previous examples, while at the same time simplifying and clarifying much of the theory.

44 citations

Posted Content
TL;DR: A direct axiomatization of a category with a reverse derivative operation is given, in a similar style to that given by Cartesian differential categories for a forward derivative, to show that these linear maps form an additively enriched category with dagger biproducts.
Abstract: The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.

27 citations

Posted Content
TL;DR: Tangent bundles generalize the notion of smooth vector bundles in classical differential geometry as discussed by the authors and provide an abstract setting for differential geometry by axiomatizing key aspects of the subject which allow the basic theory of these geometric settings to be captured.
Abstract: Tangent categories are categories equipped with a tangent functor: an endofunctor with certain natural transformations which make it behave like the tangent bundle functor on the category of smooth manifolds. They provide an abstract setting for differential geometry by axiomatizing key aspects of the subject which allow the basic theory of these geometric settings to be captured. Importantly, they have models not only in classical differential geometry and its extensions, but also in algebraic geometry, combinatorics, computer science, and physics. This paper develops the theory of "differential bundles" for such categories, considers their relation to "differential objects", and develops the theory of fibrations of tangent categories. Differential bundles generalize the notion of smooth vector bundles in classical differential geometry. However, the definition departs from the standard one in several significant ways: in general, there is no scalar multiplication in the fibres of these bundles, and in general these bundles need not be locally trivial. To understand how these differential bundles relate to differential objects, which are the generalization of vector spaces in smooth manifolds, requires some careful handling of the behaviour of pullbacks with respect to the tangent functor. This is captured by "transverse" and "display" systems for tangent categories, which leads one into the fibrational theory of tangent categories. A key example of a tangent fibration is provided by the "display" differential bundles of a tangent category with a display system. Strikingly, in such examples the fibres are Cartesian differential categories demonstrating a -- not unexpected -- tight connection between the theory of these categories and that of tangent categories.

19 citations

Posted Content
TL;DR: In this paper, the Bianchi identities, Bianchi identity for curvature and torsion, almost complex structure, and parallel transport of tangent connections are investigated. But their definition and structure in the abstract setting is not investigated.
Abstract: Connections are an important tool of differential geometry This paper investigates their definition and structure in the abstract setting of tangent categories At this level of abstraction we derive several classically important results about connections, including the Bianchi identities, identities for curvature and torsion, almost complex structure, and parallel transport

16 citations


Cited by
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01 Jan 2005
TL;DR: A1 Regular and Cartesian Closed Categories A2 Toposes - Basic Theory A3 Allegories A4 Geometric Morphisms - Basic theory B1 Fibrations and Indexed Categories B2 Internal and Locally Internal Categories B3 Toposes over a base B4 BTop/S as a 2-category as discussed by the authors.
Abstract: A1 Regular and Cartesian Closed Categories A2 Toposes - Basic Theory A3 Allegories A4 Geometric Morphisms - Basic Theory B1 Fibrations and Indexed Categories B2 Internal and Locally Internal Categories B3 Toposes over a base B4 BTop/S as a 2-Category

346 citations

Book
Emily Riehl1
26 May 2014
TL;DR: A sampling of 2-categorical aspects of quasi-category theory can be found in this article, with a brief tour of Reedy category theory and a brief discussion of the relation between Reedy and enriched homotopy theory.
Abstract: Part I. Derived Functors and Homotopy (Co)limits: 1. All concepts are Kan extensions 2. Derived functors via deformations 3. Basic concepts of enriched category theory 4. The unreasonably effective (co)bar construction 5. Homotopy limits and colimits: the theory 6. Homotopy limits and colimits: the practice Part II. Enriched Homotopy Theory: 7. Weighted limits and colimits 8. Categorical tools for homotopy (co)limit computations 9. Weighted homotopy limits and colimits 10. Derived enrichment Part III. Model Categories and Weak Factorization Systems: 11. Weak factorization systems in model categories 12. Algebraic perspectives on the small object argument 13. Enriched factorizations and enriched lifting properties 14. A brief tour of Reedy category theory Part IV. Quasi-Categories: 15. Preliminaries on quasi-categories 16. Simplicial categories and homotopy coherence 17. Isomorphisms in quasi-categories 18. A sampling of 2-categorical aspects of quasi-category theory.

202 citations

Journal ArticleDOI
TL;DR: The theory of enriched ∞-categories as mentioned in this paper is a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal ∞ -category, and it is useful even when an enriched category comes from a model category (as is often the case in examples of interest).

137 citations

MonographDOI
14 Jan 2022
TL;DR: The theory of ∞-categories was developed in this article in a model-independent fashion using the axiomatic framework of an ∞cosmos, the universe in which ∞categories live as objects.
Abstract: The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

61 citations