Papers
The formal theory of tangentads PART II
Theory and Applications of Categories
SUBMITTED
SUBMITTED
Marcello Lanfranchi
Tangent category theory is a well-established categorical framework for differential geometry. A long list of fundamental geometric constructions, such as the tangent bundle functor, vector fields, Euclidean spaces, and vector bundles have been successfully generalized and internalized within tangent categories. Over the past decade, the theory has also been extended in several directions, yielding concepts such as tangent monads, tangent fibrations, tangent restriction categories, and reverse tangent categories. It is natural to wonder how these new flavours of the theory interact with the geometric constructions. How does a tangent monad or a tangent fibration lift to the tangent category of differential bundles of a tangent category? What is the correct notion of connections for a tangent restriction category? In previous work, we introduced tangentads, a unifying framework that generalizes many tangent-like notions, and developed a formal theory of vector fields for tangentads. In this paper, we extend this formal theory to three further fundamental constructions. These are differential objects, which generalize Euclidean spaces, differential bundles, which represent vector bundles in tangent category theory, and connections on differential bundles, which are the analogue of Koszul connections. These notions are introduced in the general theory of tangentads via appropriate universal properties. We then extend some of the main results of tangent category theory, including the equivalence between differential objects and differential bundles over the terminal object, and show that connections admit well-defined notions of covariant derivative, curvature, and torsion. Finally, we construct connections using PIE limits and apply our framework to several concrete instances of tangentads.
Local categories: a new framework for partiality
Theoretical Computer Science
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Marcello Lanfranchi, Jean-Simon Pacaud Lemay
Restriction categories provide a categorical framework for partiality. In this paper, we introduce three new categorical theories for partiality: local categories, partial categories, and inclusion categories. The objects of a local category are partially accessible resources, and morphisms are processes between these resources. In a partial category, partiality is addressed via two operators, restriction and contraction, which control the domain of definition of a morphism. Finally, an inclusion category is a category equipped with a family of monics which axiomatize the inclusions between sets. The main result of this paper shows that restriction categories are -equivalent to local categories, that partial categories are -equivalent to inclusion categories, and that both restriction/local categories are -equivalent to bounded partial/inclusion categories. Our result offers four equivalent ways to describe partiality: on morphisms, via restriction categories; on objects, with local categories; operationally, with partial categories; and via inclusions, with inclusion categories. We also translate several key concepts from restriction category theory to the local category context, which allows us to show that various special kinds of restriction categories, such as inverse categories, are -equivalent to their analogous kind of local categories. In particular, the equivalence between inverse (restriction) categories and inverse local categories is a generalization of the celebrated Ehresmann-Schein-Nambooripad theorem for inverse semigroups.
The formal theory of tangentads PART I
Journal of Pure and Applied Algebra
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SUBMITTED
Marcello Lanfranchi
Tangent categories offer a categorical context for differential geometry, by categorifying geometric notions like the tangent bundle functor, vector fields, Euclidean spaces, vector bundles, connections, etc. In the last decade, the theory has been extended in new directions, providing concepts such as tangent monads, tangent fibrations, tangent restriction categories, reverse tangent categories and many more. It is natural to wonder how these new flavours interact with the geometric constructions offered by the theory. How does a tangent monad or a tangent fibration lift to the tangent category of vector fields of a tangent category? What is the correct notion of vector bundles for a tangent restriction category? We answer these questions by adopting the formal approach of tangentads. Introduced in our previous work, tangentads provide a unifying context for capturing the different flavours of the theory and for extending constructions like the Grothendieck construction or the equivalence between split restriction categories and M-categories, to the tangent-categorical context. In this paper, we construct the formal notion of vector fields for tangentads, by isolating the correct universal property enjoyed by vector fields in ordinary tangent categories. We show that vector fields form a Lie algebra and a 2-monad and show how to construct vector fields using PIE limits. Finally, we compute vector fields for some examples of tangentads. In a forthcoming paper, we extend the theory to other constructions: differential objects, differential bundles, and connections.
Representable tangent structures for affine schemes
Theory and Applications of Categories
ACCEPTED
ACCEPTED
Marcello Lanfranchi, Jean-Simon Pacaud Lemay
The category of affine schemes is a tangent category whose tangent bundle functor is induced by Kähler differentials, providing a direct link between algebraic geometry and tangent category theory. Moreover, this tangent bundle functor is represented by the ring of dual numbers. How special is this tangent structure? Are there any other (non-trivial) tangent structure on the category of affine schemes? In this paper, we characterize the representable tangent structures on the category of affine schemes. To this end, we introduce a useful tool, the notion of tangentoids, which are precisely the objects in a monoidal category that induce a tangent structure via tensoring. Furthermore, coexponentiable tangentoids induce tangent structures on the opposite category. As such, we first prove that tangentoids in the category of commutative unital algebras are equivalent to commutative associative solid non-unital algebras, that is, commutative associative non-unital algebras whose multiplication is an isomorphism. From there, we explain how representable tangent structures on affine schemes correspond to finitely generated projective commutative associative solid non-unital algebras. In particular, for affine schemes over a principal ideal domain, we show that there are precisely two representable tangent structures: the trivial one and the one given by Kähler differentials.
Tangentads: a formal approach to tangent categories
Higher Structures
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Marcello Lanfranchi
Tangent category theory is a well-established categorical context for differential geometry. In a previous paper, a formal approach was adopted to provide a genuine Grothendieck construction in the context of tangent categories by introducing tangentads. A tangentad is to a tangent category as a formal monad is to a monad of a category. In this paper, we discuss the formal notion of tangentads, construct a -comonad structure on the -functor of tangentads, and introduce Cartesian, adjunctable, and representable tangentads. We also reinterpret the subtangent structure with negatives of a tangent structure as a right Kan extension. Furthermore, we present numerous examples of tangentads, such as tangent (split) restriction categories, tangent fibrations, tangent monads, display tangent categories, and infinitesimal objects. Finally, we employ the formal approach to prove that every tangent monad admits the construction of algebras, provided the underlying monad does, and show that tangent split restriction categories are 2-equivalent to tangent 2-categories.
Pullbacks in tangent categories and tangent display maps
Applied Categorical Structures
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SUBMITTED
Geoffrey Cruttwell, Marcello Lanfranchi
In differential geometry, the existence of pullbacks is a delicate matter, since the category of smooth manifolds does not admit all of them. When pullbacks are required, often submersions are employed as an ideal class of maps which behaves well under this operation and the tangent bundle functor. This issue is reflected in tangent category theory, which aims to axiomatize the tangent bundle functor of differential geometry categorically. Key constructions such as connections, tangent fibrations, or reverse tangent categories require one to work with pullbacks preserved by the tangent bundle functor. In previous work, this issue has been left as a technicality and solved by introducing extra structure to carry around. This paper gives an alternative to this by focusing on a special class of maps in a tangent category called tangent display maps; such maps are well-behaved with respect to pullbacks and applications of the tangent functor. We develop some of the general theory of such maps, show how using them can simplify previous work in tangent categories, and show that in the tangent category of smooth manifolds, they are the same as the submersions. Finally, we consider a subclass of tangent display maps to define open subobjects in any tangent category, allowing one to build a canonical split restriction tangent category in which the original one naturally embeds.
The Grothendieck construction in the context of tangent categories
Mathematical Structures in Computer Science
PUBLISHED
PUBLISHED
Marcello Lanfranchi
The Grothendieck construction establishes an equivalence between fibrations, a.k.a. fibred categories, and indexed categories, and is one of the fundamental results of category theory. Cockett and Cruttwell introduced the notion of fibrations into the context of tangent categories and proved that the fibres of a tangent fibration inherit a tangent structure from the total tangent category. The main goal of this paper is to provide a Grothendieck construction for tangent fibrations. Our first attempt will focus on providing a correspondence between tangent fibrations and indexed tangent categories, which are collections of tangent categories and tangent morphisms indexed by the objects and morphisms of a base tangent category. We will show that this construction inverts Cockett and Cruttwell's result but it does not provide a full equivalence between these two concepts. In order to understand how to define a genuine Grothendieck equivalence in the context of tangent categories, inspired by Street's formal approach to monad theory we introduce a new concept: tangent objects. We show that tangent fibrations arise as tangent objects of a suitable -category and we employ this characterization to lift the Grothendieck construction between fibrations and indexed categories to a genuine Grothendieck equivalence between tangent fibrations and tangent indexed categories.
The differential bundles of the geometric tangent category of an operad
Applied Categorical Structures
PUBLISHED
PUBLISHED
Marcello Lanfranchi
Affine schemes can be understood as objects of the opposite of the category of commutative and unital algebras. Similarly, P-affine schemes can be defined as objects of the opposite of the category of algebras over an operad P. An example is the opposite of the category of associative algebras. The category of operadic schemes of an operad carries a canonical tangent structure. This paper aims to initiate the study of the geometry of operadic affine schemes via this tangent category. For example, we expect the tangent structure over the opposite of the category of associative algebras to describe algebraic non-commutative geometry. In order to initiate such a program, the first step is to classify differential bundles, which are the analogs of vector bundles for differential geometry. In this paper, we prove that the tangent category of affine schemes of the enveloping operad P over a P-affine scheme is precisely the slice tangent category over of P-affine schemes. We are going to employ this result to show that differential bundles over a P-affine scheme are precisely 2-modules in the operadic sense.
The Rosický Tangent Categories of Algebras over an Operad
Higher Structures
PUBLISHED
PUBLISHED
Sacha Ikonicoff, Marcello Lanfranchi, Jean-Simon Pacaud Lemay
Tangent categories provide a categorical axiomatization of the tangent bundle. There are many interesting examples and applications of tangent categories in a variety of areas such as differential geometry, algebraic geometry, algebra, and even computer science. The purpose of this paper is to expand the theory of tangent categories in a new direction: the theory of operads. The main result of this paper is that both the category of algebras of an operad and its opposite category are tangent categories. The tangent bundle for the category of algebras is given by the semi-direct product, while the tangent bundle for the opposite category of algebras is constructed using the module of Kähler differentials, and these tangent bundles are in fact adjoints of one another. To prove these results, we first prove that the category of algebras of a coCartesian differential monad is a tangent category. We then show that the monad associated to any operad is a coCartesian differential monad. This also implies that we can construct Cartesian differential categories from operads. Therefore, operads provide a bountiful source of examples of tangent categories and Cartesian differential categories, which both recaptures previously known examples and also yield new interesting examples. We also discuss how certain basic tangent category notions recapture well-known concepts in the theory of operads.