This document discusses comonoids in the context of categorical logic and the Dialectica construction. It provides motivation for studying comonoids by explaining their connection to linear logic and modalities. It then gives an overview of the Dialectica construction, explaining that it produces categories with symmetric monoidal closed structure that can model linear logic. The document discusses how in these Dialectica categories, the ! operator is a cofree comonad and objects with a comonoid structure in their second coordinate define a comonad. It considers taking alternative approaches to studying comonoids in Dialectica categories. In conclusions, it notes applications of comonoids to programming languages and differentiation.
Talk at the Melbourne Logic Seminar
on Shawn Standefer invitation
Dialectica categories for the Lambek calculus
Valeria de Paiva
The Topos Institute, Berkeley, CA
Abstract:
Dialectica categorical models of the Lambek Calculus were first presented in the Amsterdam Colloquium a long time ago. Following Lambek's lead, we approached the Lambek Calculus from the perspective of Linear Logic and adapted the Dialectica categorical models for Linear Logic to Lambek's non-commutative calculus. The old work took for granted the syntax of the Lambek calculus and only discussed the exciting possibilities of new models for the modalities that Linear Logic introduced. Many years later we find that the work on dialectica models of the Lambek calculus is still interesting and that it might inform some of the most recent work on the relationship between Categorial Grammars and notions of Distributional Semantics.
Thus we revisited the old work, making sure that the syntax details that were sketchy on the first version got completed and verified, using automated tools such as Agda and Ott. Ultimately we are interested in the applicability of the original systems to their intended uses in the construction of semantics of Natural Language. But before we can discuss it, we need to make sure that the mathematical properties that make the Lambek calculus attractive are all properly modeled and this is the main aim of this paper.
We recall the Lambek calculus with its Curry-Howard isomorphic term assignment system. We extend it with a $\kappa$ modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality $!$, as Girard did, re-introducing weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present algebraic semantics and categorical semantics, proved sound and complete for the whole system. Finally, we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before.
In this talk, logically distributive categories are introduced to provide a sound and complete semantics to multi-sorted, first-order, intuitionistic-based logical theories. The peculiar aspect is that no universe is required to interpret terms, making the semantics really point-free.
Dialectica Categories Surprising Application: mapping cardinal invariantsValeria de Paiva
Talk at 2nd Set Theory and General Topology Week in Salvador, Bahia, Brazil, March 2012
Abstract: Goethe famously said that "Mathematicians are like Frenchmen:
whatever you say to them they translate into their own language and forthwith it is something entirely different." True. Even more true of category theorists. Following this great tradition of appropriating other people's work, I want to tell you how I learned about "cardinalities of the continuum" from Blass and Morgan and da Silva
and how I want to rock their boat, just a little, in the direction of my kind of mathematics.
Talk at the Melbourne Logic Seminar
on Shawn Standefer invitation
Dialectica categories for the Lambek calculus
Valeria de Paiva
The Topos Institute, Berkeley, CA
Abstract:
Dialectica categorical models of the Lambek Calculus were first presented in the Amsterdam Colloquium a long time ago. Following Lambek's lead, we approached the Lambek Calculus from the perspective of Linear Logic and adapted the Dialectica categorical models for Linear Logic to Lambek's non-commutative calculus. The old work took for granted the syntax of the Lambek calculus and only discussed the exciting possibilities of new models for the modalities that Linear Logic introduced. Many years later we find that the work on dialectica models of the Lambek calculus is still interesting and that it might inform some of the most recent work on the relationship between Categorial Grammars and notions of Distributional Semantics.
Thus we revisited the old work, making sure that the syntax details that were sketchy on the first version got completed and verified, using automated tools such as Agda and Ott. Ultimately we are interested in the applicability of the original systems to their intended uses in the construction of semantics of Natural Language. But before we can discuss it, we need to make sure that the mathematical properties that make the Lambek calculus attractive are all properly modeled and this is the main aim of this paper.
We recall the Lambek calculus with its Curry-Howard isomorphic term assignment system. We extend it with a $\kappa$ modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality $!$, as Girard did, re-introducing weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present algebraic semantics and categorical semantics, proved sound and complete for the whole system. Finally, we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before.
In this talk, logically distributive categories are introduced to provide a sound and complete semantics to multi-sorted, first-order, intuitionistic-based logical theories. The peculiar aspect is that no universe is required to interpret terms, making the semantics really point-free.
Dialectica Categories Surprising Application: mapping cardinal invariantsValeria de Paiva
Talk at 2nd Set Theory and General Topology Week in Salvador, Bahia, Brazil, March 2012
Abstract: Goethe famously said that "Mathematicians are like Frenchmen:
whatever you say to them they translate into their own language and forthwith it is something entirely different." True. Even more true of category theorists. Following this great tradition of appropriating other people's work, I want to tell you how I learned about "cardinalities of the continuum" from Blass and Morgan and da Silva
and how I want to rock their boat, just a little, in the direction of my kind of mathematics.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
4. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Dialectica construction?
This is the main subject of this talk, we will define it properly
soon. But think about taking two objects of C, U and X and a
relation between them, α : U × X → 2, and imagine a neat notion
of morphism between such objects (U, X, α).
Meanwhile, think about having a cartesian closed category C and
applying the dialectica construction to it.
We obtain the dialectica category Dial(C) with some interesting
logical properties.
Then we want to see what are comonoids in Dial(C) and in C and
what they can tell us.
4 / 30
6. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Categorical logic
Elegant mathematics will of itself tell a tale, and one
with the merit of simplicity. This may carry philosophi-
cal weight. But that cannot be guaranteed: in the end
one cannot escape the need to form a judgement of sig-
nificance.
Martin Hyland, Proof Theory in the Abstract, 2002. (Kleisli in the picture)
6 / 30
7. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Constructive reasoning
Why: Reasoning principles that are safer
if I ask you whether “is there an x such that P(x)?”
I’m happier with the answer “yes, x0”, than with an answer
“yes, for all x it is not the case that not P(x)”
want reasoning to be as precise and safe as possible
How: constructive reasoning as much as possible, classical if
need be, but tell me where and why
Today: monoids, comonoids and (co)monads in categorical
logic
7 / 30
9. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Basic Case of Categorical logic
Attach lambda-terms to Natural Deduction proofs and think of
them as morphisms in a Cartesian closed category:
Γ, x : A ` t : B
Γ ` λx : A.t : A → B
(→ I)
Γ ` t : A → B Γ ` u : A
Γ ` tu : B
(→ E)
Logical implication (→) is the internal-hom (or exponential object)
and conjunction is the product × in the cartesian closed category.
The categorical adjunction corresponds to the Deduction theorem
A × B → C iff A → (B → C)
9 / 30
10. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Comonoids in CCCs
For each object A in C we have an identity morphism idA : A → A.
When C is a CCC, we have a diagonal map ∆ : A → A × A and
co-unit maps ρ : A → A × 1 and λ : A → 1 × A with commuting
diagrams.
so every object is a comonoid with respect to the product.
Thus in cartesian closed categories comonoids are not that
interesting, as every object is a comonoid, naturally.
This comonoid structure corresponds to the satisfaction of the
structural rules of weakening and contraction:
Γ ` B
Γ, A ` B
Γ, A, A ` B
Γ, A ` B
10 / 30
11. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Linear Logic and Modalities
Girard’s idea: remove contraction and weakening from usual
rules of logic, so logic becomes resource-aware.
This is GOOD! easier to model features of the world
But the logic is too weak: to get back the expressive power of
usual logic use a modality, written as ! ( read as ”of course!”)
Contraction and weakening available only for !A
formulas/objects.
! or bang is a unary operator over a linear basis whose rules
correspond to the (or necessity) S4 modality
It was realized very early on that this modality should be a
comonad categorically. Why? Which kind of comonad?
11 / 30
12. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Modalities
Before embarking on details, here is one general piece of advice.
One often hears that modal (or some other) logic is pointless because it can be translated into some
simpler language in a first-order way. Take no notice of such arguments. There is no weight to the
claim that the original system must therefore be replaced by the new one. What is essential is to
single out important concepts and to investigate their properties. [Scott, 1971]
12 / 30
13. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Modalities as algebra
Basic idea: Modalities are unary operators over a logic basis.
Many logical bases possible: here linear, intuitionistic or classic
logic. Operators can be:
constructive ∧, ∨, →, ¬
or linear ones ( , ` , ⊗
Which modalities are useful? Which basis?
Why? How?
Why so many modalities? How to choose?
Which are the important theorems?
Which are the most useful applications?
Today: using modalities as motivation for comonoids
13 / 30
14. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Modality Rules
Sequent rules for the modality !
!Γ ` B
!Γ `!B
Γ ` B
Γ, !A ` B
Γ, A ` B
Γ, !A ` B
Γ, !A, !A ` B
Γ, !A ` B
Reading them as morphisms in a category:
Γ =!A, B =!A
δ: !A →!!A
Γ = ∅, B = 1
er: !A → 1
Γ = ∅, B = A
der: !A → A
∅, B =!A⊗!A
copy: !A →!A⊗!A
14 / 30
15. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Linear Logic Bang !
Basic idea: each !A needs to be a comonoid with respect to the
tensor product. That means it has natural maps:
!A
copy
- !A⊗!A
!A
counit
- I
satisfying comonoid requirements
Less obviously !A also needs a coalgebra structure:
!A
der
- A
!A
prom
- !!A
These structures need to interact in a consistent and coherent way.
15 / 30
16. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Dialectica construction
Dialectica categories are being discussed, because lenses,
containers and polynomials look like it
Most of the conversation has been about the Dialectica
morphisms and the symmetric monoidal structure
But the hard work on Dialectica was proving that it has
appropriate comonoids/comonads
This story is worth telling as it seems to generalize in
interesting ways
16 / 30
17. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Dialectica Categories
A precise model of Linear Logic
All connectives correspond to distinct structures in the
categories
Two families of models:
for ILL original Dialectica categories Dial2(C)
for CLL/FILL Dialectica-like categories DDial2(C)
Difference is mostly on morphisms, objects are the same, i.e.
triples (U, X, α: A ,→ U × X) where U, X are objects of C
cartesian closed
Different notions of morphism ⇒ different structures in the
cats
Both dialectica cats Dial2(C) and DDial2(C) are symmetric
monoidal closed categories with products.
17 / 30
18. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Original Dialectica Categories
A map in Dial2(C) between objects A = (U, X, α) and
B = (V , Y , β) is a pair of maps in C, (f : U → V , F : U × Y → X)
satisfying the pullback condition below
◦ - A
◦ - U × Y
? hπ1, Fi
- U × X
?
α
B
? β- V × Y
?
f × Y
18 / 30
20. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Back to comonads and comonoids
Surprisingly the ! operator in Dial2(C) is a cofree comonad
Many useful well-known monads like exceptions, powerset,
continuations, etc... Fewer known and loved comonads
In our case a comonoid is simply some object that is a monoid
in C in the second coordinate, this defines a comonad in
Dial(C).
Take !(U, X, α) = (U, X∗, α∗), where (−)∗ is the free
commutative monoid in C.
Theorem: [linear and usual logic together]
The monoidal comonad ! in Dial2(C) above models modalities and
recovers Intuitionistic (and Classical) Logic.
20 / 30
21. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Free comonad in Dial2(C)
First cofree comonad model, not purely syntactic
Tensor product is easy (U, X, α) ⊗ (V , Y , β) is
(U × V , X × Y , α × β)
! is a monoidal comonad. There is a natural transformation
m(−,−) :!A⊗!B →!(A ⊗ B) and a morphism MI : I →!I
satisfying many commutative diagrams
! induces a commutative comonoid structure on !A
!A also has naturally a coalgebra structure induced by the
comonad !
The comonoid and coalgebra structures interact in a nice way.
21 / 30
23. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Comonoids in Dialectica
The approach so far has been:
We wanted to model a logical system, Linear Logic
We know what is necessary for a categorical model ⇒ objects
of a SMCC with both a comonoid and a coalgebra structure
(wrt a tensor product and a comonad) interacting nicely
We prove that we can provide one example of comonoids and
coalgebras interacting nicely in the specific case of the
Dialectica construction Dial(C)
23 / 30
24. Motivation
Dialectica
Dialectica Comonoids
Conclusions
What if we change the approach?
We have a Dialectica construction, applied to a cartesian
closed category C with finite limits and commutative monoids
We know Dial(C) has a notion of tensor product ⊗ which is
easy to calculate
We can describe the category of comonoids for this notion of
tensor product
We know there is at least one object in Dial(C) which is a
comonoid, the object (1, 1, id1).
What are the other comonoids in Dial(C)?
What can we say about the category Comon⊗(C)?
24 / 30
25. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Comonoids and coalgebras in Math
Barr JPAA 1973 ”Coalgebras Over a Commutative Ring”
about K-modules that have a comonoid structure
Hans Porst on local presentability of cats of coalgebras
Agore on limits of algebras, coalgebras and Hopf algebras
a warning: Sometimes “cocommutative coalgebra” in a symmetric
monoidal category is used as a synonym for cocommutative
comonoid object.
https://ncatlab.org/nlab/show/cocommutative+coalgebra
Our terminology comes from MacLane’s CWM, adding cos whenever necessary.
25 / 30
26. Motivation
Dialectica
Dialectica Comonoids
Conclusions
More comonoids and coalgebras in Math
From https:
//ncatlab.org/nlab/show/coalgebra+over+a+comonad
Related concepts:
partial differential equations are the coalgebras of a jet
comonad
well-behaved lenses (in computer science) are the coalgebras
of the costate comonad
model category-structures on coalgebras over a comonad,
Hess and Shipley ”The homotopy theory of coalgebras over a
comonad” arXiv:1205.3979
26 / 30
27. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Two specific examples
Vaughan Pratt, 2003
http://boole.stanford.edu/pub/comonoids.pdf, “Comonoids in
chu: a large cartesian closed sibling of topological spaces”
Niu and Spivak, arXiv:2112.11518v3, “Collectives:
Compositional protocols for contributions and returns”,
comonoids in Poly.
Both Chu and Poly are constructions similar to Dialectica.
Chu has the same objects, different morphisms, equalities instead
of implications.
Poly has objects that are a dependent version of Dialectica objects,
but morphisms as in Chu.
27 / 30
28. Motivation
Dialectica
Dialectica Comonoids
Conclusions
Conclusions
I promised you a story of comonoids in the Dialectica
construction
I could have called these comonoids ‘Linear Modalities’ as the
(co)monad that introduces them behaves like an S4 modality
when considered as logic
I showed you how dialectica cats introduce several different
(co)monads useful to provide models of LL
I have not talked about constructive modal linear logics
Constructive modal logics are interesting for programmers,
logicians and philosophers. Shame they don’t talk much to
each other.
There’s much to say on other applications of comonoids,
especially in programming languages
But also on categorical ways of thinking of differentiation!
28 / 30
29. Appendix Some References
Some References I
V de Paiva
The Dialectica Categories
PhD thesis, University of Cambridge, 1991.
G. Bierman, V de Paiva
On an Intuitionistic Modal Logic
Studia Logica (65):383-416. 2000.
V. Pratt
Comonoids in chu
boole.stanford.edu/pub/comonoids.pdf, 2003.
N. Niu and D. Spivak
Collectives
arxiv 2112.11518v3, 2022.
29 / 30
30. Appendix Some References
Some References II
M. Marvan
A note on the category of partial differential equations
Proceedings of the Conference DIFFERENTIAL GEOMETRY
AND ITS APPLICATIONS, August 24-30, 1986, Brno
30 / 30