scholarly journals Cartesian Differential Categories as Skew Enriched Categories

2021 ◽  
Author(s):  
Richard Garner ◽  
Jean-Simon Pacaud Lemay
Keyword(s):  
Linear Logic ◽  
Usual Sense ◽  
Enriched Category ◽  
Enriched Categories ◽  
Full Structure ◽  
Monoidal Closed Category ◽  
Category Of Modules ◽  
Differential Categories ◽  
Linear Category ◽  
Open Question

AbstractWe exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

scholarly journals Remarks on triples in enriched categories

1970 ◽  
Vol 3 (3) ◽  
pp. 375-383 ◽  
Author(s):  
H. Wiesler ◽  
G. Calugareanu
Keyword(s):  
Inverse Limit ◽  
Enriched Category ◽  
Closed Category ◽  
Enriched Categories ◽  
Symmetric Monoidal Closed Category ◽  
Monoidal Closed Category

Let v be a symmetric monoidal closed category with equalizers. The v–triples T, T′, … in the enriched category A, together with suitably defined morphisms form a category v–Trip(A). The v–categories AT, AT′, … and the v–functors R: AT′ → AT which are compatible with the forgetful functors form a category V–Alg(A).In the subsequent note it is shown that V–Trip(A) is isomorphic to the dual of V–Alg(A) and that the morphisms of V–Alg(A) are inverse limit preserving V–functors.

Convenient antiderivatives for differential linear categories

2020 ◽  
Vol 30 (5) ◽  
pp. 545-569
Author(s):  
Jean-Simon Pacaud Lemay
Keyword(s):  
Natural Transformation ◽  
Linear Logic ◽  
Natural Isomorphism ◽  
Relational Model ◽  
Vector Spaces ◽  
Integration Operator ◽  
Categorical Models ◽  
Differential Categories ◽  
Linear Category ◽  
Fundamental Theorems

AbstractDifferential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation , which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove this result, we show that a differential linear category – which is a differential category with a monoidal coalgebra modality – has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold. We also show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential linear categories with antiderivatives.

Completions of Non-Symmetric Metric Spaces Via Enriched Categories

2009 ◽  
Vol 16 (1) ◽  
pp. 157-182
Author(s):  
Vincent Schmitt
Keyword(s):  
Category Theory ◽  
Metric Spaces ◽  
Enriched Category ◽  
Closed Category ◽  
Enriched Categories ◽  
Terminal Object ◽  
Base Category ◽  
Monoidal Closed Category ◽  
The One ◽  
Universal Completion

Abstract It is known from [Lawvere, Repr. Theory Appl. Categ. 1: 1–37 2002] that nonsymmetric metric spaces correspond to enrichments over the monoidal closed category [0, ∞]. We use enriched category theory and in particular a generic notion of flatness to describe various completions for these spaces. We characterise the weights of colimits commuting in the base category [0, ∞] with the conical terminal object and cotensors. Those can be interpreted in metric terms as very general filters, which we call filters of type 1. This correspondence extends the one between minimal Cauchy filters and weights which are adjoint as modules. Translating elements of enriched category theory into the metric context, one obtains a notion of convergence for filters of type 1 with a related completeness notion for spaces, for which there exists a universal completion. Another smaller class of flat presheaves is also considered both in the context of both metric spaces and preorders. (The latter being enrichments over the monoidal closed category 2.) The corresponding completion for preorders is the so-called dcpo completion.

scholarly journals Proof of a Conjecture of S. Mac Lane

1996 ◽  
Vol 3 (61) ◽  
Author(s):  
Sergei Soloviev
Keyword(s):  
Linear Logic ◽  
Sufficient Conditions ◽  
Vector Spaces ◽  
Categorical Equivalence ◽  
Closed Category ◽  
Symmetric Monoidal Closed Category ◽  
Monoidal Closed Category ◽  
Coherence Theorem ◽  
Mac Lane

Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions (only this case was considered in the original Mac Lane conjecture). Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered (together with categorical equivalence). Two derivations of the same sequent are equivalent if and only if all their interpretations in K are equal. In fact, the assignment of values (objects of K) to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the "abstract coherence theorem" found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture proves also the correctness of the modified description.

scholarly journals A quantum double construction in Rel

2012 ◽  
Vol 22 (4) ◽  
pp. 618-650 ◽  
Author(s):  
MASAHITO HASEGAWA
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Binary Relations ◽  
Monoidal Categories ◽  
Quantum Double ◽  
Closed Category ◽  
Category Of Modules ◽  
Extra Structure

We study bialgebras and Hopf algebras in the compact closed categoryRelof sets and binary relations. Various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras and Hopf algebras inRel. In particular, for any groupG, we derive a ribbon category of crossedG-sets as the category of modules of a Hopf algebra inRelthat is obtained by the quantum double construction. This category of crossedG-sets serves as a model of the braided variant of propositional linear logic.

On Köthe sequence spaces and linear logic

2002 ◽  
Vol 12 (5) ◽  
pp. 579-623 ◽  
Author(s):  
THOMAS EHRHARD
Keyword(s):  
Simple Structure ◽  
Linear Logic ◽  
Sequence Spaces ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
Cartesian Closed ◽  
Linear Category ◽  
Köthe Sequence Spaces ◽  
Locally Convex ◽  
Standard Concept

We present a category of locally convex topological vector spaces that is a model of propositional classical linear logic and is based on the standard concept of Köthe sequence spaces. In this setting, the ‘of course’ connective of linear logic has a quite simple structure of a commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting in which typed λ-calculus and differential calculus can be combined; we give a few examples of computations.

scholarly journals A Little Quantum Help for Cosmic Censorship and a Step Beyond All That

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Nikolaos Pappas
Keyword(s):  
General Relativity ◽  
Event Horizon ◽  
Crucial Role ◽  
Naked Singularity ◽  
Cosmic Censorship ◽  
Usual Sense ◽  
Naked Singularities ◽  
Open Question ◽  
Heisenberg's Uncertainty Principle ◽  
The Universe

The hypothesis of cosmic censorship (CCH) plays a crucial role in classical general relativity, namely, to ensure that naked singularities would never emerge, since it predicts that whenever a singularity is formed an event horizon would always develop around it as well, to prevent the former from interacting directly with the rest of the Universe. Should this not be so, naked singularities could eventually form, in which case phenomena beyond our understanding and ability to predict could occur, since at the vicinity of the singularity both predictability and determinism break down even at the classical (e.g., nonquantum) level. More than 40 years after it was proposed, the validity of the hypothesis remains an open question. We reconsider CCH in both its weak and strong versions, concerning point-like singularities, with respect to the provisions of Heisenberg’s uncertainty principle. We argue that the shielding of the singularities from observers at infinity by an event horizon is also quantum mechanically favored, but ultimately it seems more appropriate to accept that singularities never actually form in the usual sense; thus no naked singularity danger exists in the first place.

scholarly journals Enriched category as a model of qualia structure based on similarity judgements

2021 ◽  
Author(s):  
Naotsugu Tsuchiya ◽  
Steven Phillips ◽  
Hayato Saigo
Keyword(s):  
Category Theory ◽  
Sorites Paradox ◽  
Perceived Similarity ◽  
Enriched Category ◽  
Enriched Categories ◽  
Important Extension ◽  
Similarity Relationships ◽  
Yoneda Lemma ◽  
Similarity Judgements ◽  
Qualia Structure

Qualitative relationships between two instances of conscious experiences can be quantified through the perceived similarity. Previously, we proposed that by defining similarity relationships as arrows and conscious experiences as objects, we can define a category of qualia in the context of category theory. However, the example qualia categories we proposed were highly idealized and limited to cases where perceived similarity is binary: either present or absent without any gradation. When similarity is graded, a situation can arise where A0 is similar to A1, A1 is similar to A2, and so on, yet A0 is not similar to An, which is called the Sorites paradox. Here, we introduce enriched category theory to address this situation. Enriched categories generalize the concept of a relation between objects as a directed arrow (or morphism) in ordinary category theory to a more flexible notion, such as a measure of distance. As an alternative relation, here we propose a graded measure of perceived dissimilarity between the two objects. These measures combine in a way that addresses the Sorites paradox; even if the dissimilarity between Ai and Ai+1 is small for i = 0 … n, hence perceived as similar, the dissimilarity between A0 and An can be large, hence perceived as different. In this way, we show how dissimilarity-enriched categories of qualia resolve the Sorites paradox. We claim that enriched categories accommodate various types of conscious experiences. An important extension of this claim is the application of the Yoneda lemma in enriched category; we can characterize a quale through a collection of relationships between the quale and the other qualia up to an (enriched) isomorphism.

scholarly journals THH and Traces of Enriched Categories

2020 ◽  
Author(s):  
John D Berman
Keyword(s):  
Category Theory ◽  
Hochschild Homology ◽  
Independent Interest ◽  
Elementary Model ◽  
Enriched Category ◽  
Enriched Categories ◽  
Higher Category Theory ◽  
Model Independent ◽  
Universal Construction ◽  
Topological Hochschild Homology

Abstract We prove that topological Hochschild homology (THH) arises from a presheaf of circles on a certain combinatorial category, which gives a universal construction of THH for any enriched $\infty $-category. Our results rely crucially on an elementary, model-independent framework for enriched higher-category theory, which may be of independent interest. For those interested only in enriched category theory, read Sections 1.3 and 2.

scholarly journals A linear category of polynomial diagrams

2013 ◽  
Vol 24 (1) ◽  
Author(s):  
PIERRE HYVERNAT
Keyword(s):  
Tensor Product ◽  
Linear Logic ◽  
Structure Tensor ◽  
Multiplicative Structure ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Categorical Model ◽  
Cartesian Closed ◽  
Linear Category ◽  
Intuitionistic Linear Logic

We present a categorical model for intuitionistic linear logic in which objects are polynomial diagrams and morphisms aresimulation diagrams. The multiplicative structure (tensor product and its adjoint) can be defined in any locally cartesian closed category, but the additive (product and coproduct) and exponential (-comonoid comonad) structures require additional properties and are only developed in the categorySet, where the objects and morphisms have natural interpretations in terms of games, simulation and strategies.

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