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.

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.

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 Integral categories and calculus categories

2018 ◽  
Vol 29 (2) ◽  
pp. 243-308 ◽  
Author(s):  
J. R. B. COCKETT ◽  
J.-S. LEMAY
Keyword(s):  
Monoidal Category ◽  
Natural Transformation ◽  
Integral Transformation ◽  
Parallel Development ◽  
Differential Structure ◽  
Symmetric Monoidal Category ◽  
Abstract Setting ◽  
Integral Structure ◽  
Differential Categories ◽  
Fundamental Theorems

Differential categories are now an established abstract setting for differentiation. However, not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, sA: !A → !A ⊗ A, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorems is called a calculus category.Modifying an approach to antiderivatives by T. Ehrhard, we define having antiderivatives as the demand that a certain natural transformation, K: !A → !A, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category.When the coalgebra modality is monoidal, it is natural to demand an extra coherence between integration and the coalgebra modality. In the presence of this extra coherence, we show that a calculus category with a monoidal coalgebra modality has its integral transformation given by antiderivatives and, thus, that the integral structure is uniquely determined by the differential structure.The paper finishes by providing a suite of separating examples. Examples of differential categories, integral categories and calculus categories based on both monoidal and (mere) coalgebra modalities are presented. In addition, differential categories which are not integral categories are discussed and vice versa.

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 Categories as Mathematical Models

2018 ◽  
Author(s):  
David I. Spivak
Keyword(s):  
Dynamical Systems ◽  
Mathematical Modelling ◽  
Mathematical Models ◽  
Category Theory ◽  
Vector Spaces ◽  
Mathematical Objects ◽  
Working Definition ◽  
Modelling Framework ◽  
Categorical Models ◽  
Definition Of

Category theory is presented as a mathematical modelling framework that highlights the relationships between objects, rather than the objects in themselves. A working definition of model is given, and several examples of mathematical objects, such as vector spaces, groups, and dynamical systems, are considered as categorical models.

The shuffle Hopf algebra and noncommutative full completeness

1998 ◽  
Vol 63 (4) ◽  
pp. 1413-1436 ◽  
Author(s):  
R. F. Blute ◽  
P. J. Scott
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Natural Extension ◽  
Additive Group ◽  
Completeness Theorem ◽  
Vector Spaces ◽  
Invariance Condition ◽  
Shuffle Algebra ◽  
Topological Vector Spaces

AbstractWe present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL + MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces.This paper is a natural extension of the authors' previous work, “Linear Läuchli Semantics”, where a similar theorem is obtained for the commutative logic MLL + MIX. In that paper, we interpret proofs as dinaturals which are invariant under certain actions of the additive group of integers. Here we also present a simplification of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras in this paper corresponds to the passage from commutative to noncommutative logic. However, in our noncommutative setting, one must still keep the invariance condition on dinaturals.

scholarly journals Cartesian Difference Categories

2020 ◽  
pp. 57-76
Author(s):  
Mario Alvarez-Picallo ◽  
Jean-Simon Pacaud Lemay
Keyword(s):  
Finite Differences ◽  
Tangent Bundle ◽  
Differential Calculus ◽  
Directional Derivative ◽  
The Other ◽  
Smooth Functions ◽  
Kleisli Category ◽  
Categorical Models ◽  
Action Models ◽  
Differential Categories

AbstractCartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $$\lambda $$ λ -calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation such as the calculus of finite differences or the Boolean differential calculus. On the other hand, change action models have been shown to capture these examples as well as more “exotic” examples of differentiation. However, change action models are very general and do not share the nice properties of a Cartesian differential category. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.

scholarly journals On geometry of interaction for polarized linear logic

2017 ◽  
Vol 28 (10) ◽  
pp. 1639-1694
Author(s):  
MASAHIRO HAMANO ◽  
PHILIP SCOTT
Keyword(s):  
Proof Theory ◽  
Monoidal Category ◽  
Linear Logic ◽  
Cut Elimination ◽  
List Type ◽  
Monoidal Categories ◽  
Categorical Structure ◽  
Categorical Models ◽  
Focusing Property ◽  
Special Case

We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multi-points to various categorical models of GoI. Multi-points are shown to play an essential role in semantically characterizing the dynamics of proof networks in polarized proof theory. For example, they permit us to characterize the key feature of polarization, focusing, as well as being fundamental to our construction of concrete polarized GoI models.Our approach to polarized GoI involves following two independent studies, based on different categorical perspectives of GoI: (i)Inspired by the work of Abramsky, Haghverdi and Scott, a polarized GoI situation is defined in which multi-points are added to a traced monoidal category equipped with a reflexive object U. Using this framework, categorical versions of Girard's execution formula are defined, as well as the GoI interpretation of MLLP proofs. Running the execution formula is shown to characterize the focusing property (and thus polarities) as well as the dynamics of cut elimination.(ii)The Int construction of Joyal–Street–Verity is another fundamental categorical structure for modelling GoI. Here, we investigate it in a multi-pointed setting. Our presentation yields a compact version of Hamano–Scott's polarized categories, and thus denotational models of MLLP. These arise from a contravariant duality between monoidal categories of positive and negative objects, along with an appropriate bimodule structure (representing ‘non-focused proofs’) between them.Finally, as a special case of (ii) above, a compact model of MLLP is also presented based on Rel (the category of sets and relations) equipped with multi-points.

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