Yorgos Lanthimos and Efthymis Filippou developed the premise for the film out of the idea of people who allege something which is fabricated, for example via prank calls or by announcing their own deaths. The story took form as they needed a setting which could work well cinematically. Lanthimos considers it the complete opposite of his previous film, Dogtooth, which he says "is the story of a person who tries to escape a fictitious world. Alps is about a person who tries to enter a fabricated world."
The film was produced by the Greek company Chaos(Χάος) Film, which previously had produced Lanthimos' 2005 film Kinetta. The budget included funding from the Greek Film Center. Filming started in October 2010. Some scenes were added on the set and parts of the dialogue were improvised by the actors.
Image is an original novel based on the U.S. television series Angel.
Plot summary
Cordelia Chase has a vision of a child being attacked by a squidlike demon. Meanwhile, Gunn is trying to rescue a young artist; the artist's studio is being attacked by vampires. Cordelia goes to investigate the mansion from her vision. She soon finds herself surrounded by baby products, portraits, and chased by a tentacled monster.
When Angel arrives on the scene, he is surprised to discover that he recognizes some of the portraits. He holds distant memories of him and Darla spending a night with storytellers and artists. Angel reveals that he and Darla were present at the party where Mary Shelley was inspired to write Frankenstein; indeed, they witnessed the event that gave Mary the initial idea.
An old evil is trying to use a painting to preserve the life of its body, which, in the terms of the story, inspired the novel The Picture of Dorian Gray. In their efforts to save a child the villain is focused on, Team Angel will learn not to judge everything by its image.
For any object Z with a morphism and a monomorphism such that , there exists a unique morphism such that .
Remarks:
such a factorization does not necessarily exist
g is unique by definition of monic (= left invertible, abstraction of injectivity)
m is monic.
h=lm already implies that m is unique.
k=mg
The image of f is often denoted by im f or Im(f).
One can show that a morphism f is epic if and only if f = im f.
Examples
In the category of sets the image of a morphism is the inclusion from the ordinary image to . In many concrete categories such as groups, abelian groups and (left- or right) modules, the image of a morphism is the image of the correspondent morphism in the category of sets.
Image is a board game developed by 3M released in 1971. The object of the game is to put together cards that represent an image, a description of a famous person.
Speech is the vocalized form of humancommunication. It is based upon the syntactic combination of lexicals and names that are drawn from very large (usually about 1,000 different words) vocabularies. Each spoken word is created out of the phonetic combination of a limited set of vowel and consonant speech sound units. These vocabularies, the syntax which structures them, and their set of speech sound units differ, creating the existence of many thousands of different types of mutually unintelligible human languages. Most human speakers are able to communicate in two or more of them, hence being polyglots. The vocal abilities that enable humans to produce speech also provide humans with the ability to sing.
A gestural form of human communication exists for the deaf in the form of sign language. Speech in some cultures has become the basis of a written language, often one that differs in its vocabulary, syntax and phonetics from its associated spoken one, a situation called diglossia. Speech in addition to its use in communication, it is suggested by some psychologists such as Vygotsky is internally used by mental processes to enhance and organize cognition in the form of an interior monologue.
Todd Thomas (born October 25, 1968), better known by the stage name Speech, is an American rapper and musician. He is a member of the progressive hip hop group Arrested Development and has released a number of solo albums.
In 1987, Speech joined with fellow DJ Headliner to form the group Arrested Development. After over three years together, the group released their inaugural album, 3 Years, 5 Months & 2 Days in the Life Of..., which produced several hits and sold very well. Speech performed lead vocals, and produced the group's tracks. The group's follow-up album, Zingalamaduni, fared poorly by comparison, but was critically acclaimed. Speech would later go on to pursue his solo career. Speech also started Vagabond Productions since 1994 – Vagabond started as a vehicle for Grammy winners Arrested Development and their business dealings and became a promoter of neo soul, hip-hop, and rock concerts in Atlanta. The company then switched to representing artists in U.S. and Japan. Since then, Vagabond has become an all around production house that presents music to various labels and on-line outlets across the globe.
The First Real Application of Category Theory #SoME3
this is a video about category theory... for #SoME3
haha jk this is just algebraic topology
btw i forgot the ray is supposed to go the other way (f(x) to x) im just dumb af pls dont crucify me
published: 17 Aug 2023
Category Theory is Impossible Without These 6 Things
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---
🎬 Welcome to our Channel! Today, we’re diving into the fascinating world of Category Theory!📚✨
---
🧑🏫 So, what exactly is Category Theory?🤔
- At its core, Category Theory is all about understanding mathematical structures and the relationships between them. ...
published: 29 Jun 2024
Glossary of category theory
If you find our videos helpful you can support us by buying something from amazon.
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Glossary of category theory
This is a glossary of properties and concepts in category theory in mathematics.Especially for higher categories, the concepts from algebraic topology are also used in the category theory.
=======Image-Copyright-Info========
License: Creative Commons Attribution-Share Alike 4.0 (CC BY-SA 4.0)
LicenseLink: http://creativecommons.org/licenses/by-sa/4.0
Author-Info: IkamusumeFan
Image Source: https://en.wikipedia.org/wiki/File:Section_retract.svg
=======Image-Copyright-Info========
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Attribution:
Article text available under CC-BY-SA
image source in video
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published: 22 Jan 2016
WE MUST ADD STRUCTURE TO DEEP LEARNING BECAUSE...
Dr. Paul Lessard and his collaborators have written a paper on "Categorical Deep Learning and Algebraic Theory of Architectures". They aim to make neural networks more interpretable, composable and amenable to formal reasoning. The key is mathematical abstraction, as exemplified by category theory - using monads to develop a more principled, algebraic approach to structuring neural networks.
We also discussed the limitations of current neural network architectures in terms of their ability to generalise and reason in a human-like way. In particular, the inability of neural networks to do unbounded computation equivalent to a Turing machine. Paul expressed optimism that this is not a fundamental limitation, but an artefact of current architectures and training procedures.
The power of ab...
published: 01 Apr 2024
Category Theory 2.1: Functions, epimorphisms
Functions, epimorphisms
published: 01 Sep 2016
Category Theory 4.1: Terminal and initial objects
Terminal and initial objects
published: 15 Sep 2016
2024-09-03 - NITheCS & SU Category Theory Research Seminar: ‘Workshop on pointfree ... part 4
2024-09-03 - NITheCS & SU Category Theory Research Seminar
Workshop on pointfree topology and constructive mathematics – part 4
Dr Graham Manuell
(Stellenbosch University)
ABSTRACT
Constructive logic is a generalisation of classical logic that applies in more situations, including mathematical universes (toposes) where propositions can take different values at different locations in some space or where every function from ℕ to ℕ is computable. Unfortunately, in this general setting many classical results of point-set topology fail. However, almost all of these results can be recovered if we reformulate topology in terms of lattices of opens without predefined underlying sets of points. Moreover, this perspective also sheds light on many not-obviously-topological aspects of constructive ma...
published: 04 Sep 2024
Category Theory in Life - Eugenia Cheng
This presentation was the opening keynote of Lambda World 2017 by Dr. Eugenia Cheng.
Follow:
-https://www.twitter.com/47deg
-https://www.twitter.com/lambda_world
Visit:
-https://www.47deg.com/events for more details.
___
Category theory can be thought of as being 'very abstract algebra'. It is thought of as 'too abstract' by some people, and as 'abstract nonsense' by some others. In this talk, I will show that while it is abstract, it is far from being nonsense. I will argue that the abstraction has a purpose and that broad applicability is one of the powerful consequences. To demonstrate this, I will show how I apply concepts of category theory to important questions of life such as prejudice, privilege, blame and responsibility. I will introduce the category theory concepts from s...
published: 03 Nov 2017
How to Create a Digital Catalogue for Your Business - 5 November 2024
With more businesses moving online each day, creating a standout Digital Catalogue is essential for showcasing your products & services and reaching potential customers effectively. A well-designed catalogue simplifies the buying process, making it easy for customers to find & purchase from you.
Join us on 5 November (Tuesday) for a session where we’ll guide you through the steps of building a digital catalogue that can attract, engage, and convert customers.
Key Topics
- Need for Digital Catalogue
- Types of Digital Catalogues
- Guide to Create a Digital Catalogue
- Ensuring Completeness & Quality
- Q&A Session
#globallinker #digitalcatlogue #onlinesales #ecommerce
this is a video about category theory... for #SoME3
haha jk this is just algebraic topology
btw i forgot the ray is supposed to go the other way (f(x) to...
this is a video about category theory... for #SoME3
haha jk this is just algebraic topology
btw i forgot the ray is supposed to go the other way (f(x) to x) im just dumb af pls dont crucify me
this is a video about category theory... for #SoME3
haha jk this is just algebraic topology
btw i forgot the ray is supposed to go the other way (f(x) to x) im just dumb af pls dont crucify me
😎 Become a member to have exclusive access:
https://www.youtube.com/channel/UC3Z1rXCFFadHw69-PZpQRYQ/join
📈 Check out my Udemy courses (you may find something ...
😎 Become a member to have exclusive access:
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📈 Check out my Udemy courses (you may find something that interests you 😉): https://www.udemy.com/user/luca-di-beo/
📊 Do you need PRIVATE CLASSES on Math & Physics, or do you know somebody who does? I might be helpful! Our email: [email protected]
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🥹 Consider supporting us on Patreon:
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---
🎬 Welcome to our Channel! Today, we’re diving into the fascinating world of Category Theory!📚✨
---
🧑🏫 So, what exactly is Category Theory?🤔
- At its core, Category Theory is all about understanding mathematical structures and the relationships between them. Imagine having a universal language that can describe patterns and connections across various fields – that’s the power of Category Theory! 🌐🔗
---
🌟 Why is Category Theory so important?🧠
- Category Theory provides a unifying framework that simplifies complex concepts. It allows mathematicians to see connections between seemingly unrelated areas, leading to groundbreaking discoveries and innovations. 🧩✨
- Think of it as the ultimate toolbox for mathematicians. By focusing on the relationships and structures, rather than individual elements, Category Theory offers a deeper, more holistic understanding of mathematics. 🔧🔍
---
📚 Real-World Applications🌍
- Beyond pure mathematics, Category Theory has applications in computer science, physics, and even linguistics! Whether it’s improving algorithms, understanding quantum mechanics, or analyzing languages, Category Theory is a versatile tool that’s making waves in various disciplines. 🌊🔬
---
🔭 Why should you watch this video?📺
- If you’re curious about how mathematics can bridge gaps between different fields, this video is for you! 🌉
- Whether you’re a student, a professional, or just a math enthusiast, understanding Category Theory will open your eyes to new ways of thinking and problem-solving. 💡
- Join us as we break down complex concepts into easy-to-understand explanations, with real-world examples that highlight the beauty and power of Category Theory. 🎓📈
---
- Drop a comment below if you have any questions or if there’s a topic you’d like us to cover next! 👇
- Hit the like button if you enjoyed this video and subscribe to our channel for more exciting content! 🎥💡
---
Thanks for watching, and we’ll see you in the next video! 🌟👋
—-
#CategoryTheory #Mathematics #MathExplained #MathTheory #MathematicalStructures #AbstractMath #MathEducation #MathLovers #STEM #Science #Technology #Engineering #Math #PureMath #AppliedMath #CategoryTheoryBasics #MathConcepts #MathFramework #MathConnections #MathInnovations #MathCommunity #MathResearch #MathDiscovery #MathFundamentals #MathApplications #ComputerScience #Physics #Linguistics #QuantumMechanics #Algorithms #ToposTheory #AbstractAlgebra #GroupTheory #Topology #Functor #Morphism #HigherCategoryTheory #MathVideo #EducationalVideo #LearningMath #MathStudents #MathTeachers #MathEnthusiasts #STEMEducation #STEMCommunity #MathPassion #MathGeek #MathNerd #MathChannel #MathTutorial #ExploringMath
—-
Image credits:
Keys
https://commons.wikimedia.org/wiki/File:Vachette_Radial_NT_key.jpg
https://commons.wikimedia.org/wiki/File:Taylor_539_Bit_Key_Blank.jpg
Eilenberg and Mac Lane
https://commons.wikimedia.org/wiki/File:1-image0.jpg
https://commons.wikimedia.org/wiki/File:Samuel_Eilenberg_MFO.jpeg
William Lawvere
https://commons.wikimedia.org/wiki/File:William_Lawvere.jpg
Columbia University
https://commons.wikimedia.org/wiki/File:Columbia_University_-_Low_Memorial_Library_(48170370506).jpg
Alexander Grothendieck
https://commons.wikimedia.org/wiki/File:Alexander_Grothendieck.jpg
😎 Become a member to have exclusive access:
https://www.youtube.com/channel/UC3Z1rXCFFadHw69-PZpQRYQ/join
📈 Check out my Udemy courses (you may find something that interests you 😉): https://www.udemy.com/user/luca-di-beo/
📊 Do you need PRIVATE CLASSES on Math & Physics, or do you know somebody who does? I might be helpful! Our email: [email protected]
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🥹 Consider supporting us on Patreon:
https://www.patreon.com/user?u=86646021
---
🎬 Welcome to our Channel! Today, we’re diving into the fascinating world of Category Theory!📚✨
---
🧑🏫 So, what exactly is Category Theory?🤔
- At its core, Category Theory is all about understanding mathematical structures and the relationships between them. Imagine having a universal language that can describe patterns and connections across various fields – that’s the power of Category Theory! 🌐🔗
---
🌟 Why is Category Theory so important?🧠
- Category Theory provides a unifying framework that simplifies complex concepts. It allows mathematicians to see connections between seemingly unrelated areas, leading to groundbreaking discoveries and innovations. 🧩✨
- Think of it as the ultimate toolbox for mathematicians. By focusing on the relationships and structures, rather than individual elements, Category Theory offers a deeper, more holistic understanding of mathematics. 🔧🔍
---
📚 Real-World Applications🌍
- Beyond pure mathematics, Category Theory has applications in computer science, physics, and even linguistics! Whether it’s improving algorithms, understanding quantum mechanics, or analyzing languages, Category Theory is a versatile tool that’s making waves in various disciplines. 🌊🔬
---
🔭 Why should you watch this video?📺
- If you’re curious about how mathematics can bridge gaps between different fields, this video is for you! 🌉
- Whether you’re a student, a professional, or just a math enthusiast, understanding Category Theory will open your eyes to new ways of thinking and problem-solving. 💡
- Join us as we break down complex concepts into easy-to-understand explanations, with real-world examples that highlight the beauty and power of Category Theory. 🎓📈
---
- Drop a comment below if you have any questions or if there’s a topic you’d like us to cover next! 👇
- Hit the like button if you enjoyed this video and subscribe to our channel for more exciting content! 🎥💡
---
Thanks for watching, and we’ll see you in the next video! 🌟👋
—-
#CategoryTheory #Mathematics #MathExplained #MathTheory #MathematicalStructures #AbstractMath #MathEducation #MathLovers #STEM #Science #Technology #Engineering #Math #PureMath #AppliedMath #CategoryTheoryBasics #MathConcepts #MathFramework #MathConnections #MathInnovations #MathCommunity #MathResearch #MathDiscovery #MathFundamentals #MathApplications #ComputerScience #Physics #Linguistics #QuantumMechanics #Algorithms #ToposTheory #AbstractAlgebra #GroupTheory #Topology #Functor #Morphism #HigherCategoryTheory #MathVideo #EducationalVideo #LearningMath #MathStudents #MathTeachers #MathEnthusiasts #STEMEducation #STEMCommunity #MathPassion #MathGeek #MathNerd #MathChannel #MathTutorial #ExploringMath
—-
Image credits:
Keys
https://commons.wikimedia.org/wiki/File:Vachette_Radial_NT_key.jpg
https://commons.wikimedia.org/wiki/File:Taylor_539_Bit_Key_Blank.jpg
Eilenberg and Mac Lane
https://commons.wikimedia.org/wiki/File:1-image0.jpg
https://commons.wikimedia.org/wiki/File:Samuel_Eilenberg_MFO.jpeg
William Lawvere
https://commons.wikimedia.org/wiki/File:William_Lawvere.jpg
Columbia University
https://commons.wikimedia.org/wiki/File:Columbia_University_-_Low_Memorial_Library_(48170370506).jpg
Alexander Grothendieck
https://commons.wikimedia.org/wiki/File:Alexander_Grothendieck.jpg
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Glossary of category theory
This...
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Glossary of category theory
This is a glossary of properties and concepts in category theory in mathematics.Especially for higher categories, the concepts from algebraic topology are also used in the category theory.
=======Image-Copyright-Info========
License: Creative Commons Attribution-Share Alike 4.0 (CC BY-SA 4.0)
LicenseLink: http://creativecommons.org/licenses/by-sa/4.0
Author-Info: IkamusumeFan
Image Source: https://en.wikipedia.org/wiki/File:Section_retract.svg
=======Image-Copyright-Info========
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
https://www.youtube.com/watch?v=MJBLsvLGXHQ
If you find our videos helpful you can support us by buying something from amazon.
https://www.amazon.com/?tag=wiki-audio-20
Glossary of category theory
This is a glossary of properties and concepts in category theory in mathematics.Especially for higher categories, the concepts from algebraic topology are also used in the category theory.
=======Image-Copyright-Info========
License: Creative Commons Attribution-Share Alike 4.0 (CC BY-SA 4.0)
LicenseLink: http://creativecommons.org/licenses/by-sa/4.0
Author-Info: IkamusumeFan
Image Source: https://en.wikipedia.org/wiki/File:Section_retract.svg
=======Image-Copyright-Info========
-Video is targeted to blind users
Attribution:
Article text available under CC-BY-SA
image source in video
https://www.youtube.com/watch?v=MJBLsvLGXHQ
Dr. Paul Lessard and his collaborators have written a paper on "Categorical Deep Learning and Algebraic Theory of Architectures". They aim to make neural netwo...
Dr. Paul Lessard and his collaborators have written a paper on "Categorical Deep Learning and Algebraic Theory of Architectures". They aim to make neural networks more interpretable, composable and amenable to formal reasoning. The key is mathematical abstraction, as exemplified by category theory - using monads to develop a more principled, algebraic approach to structuring neural networks.
We also discussed the limitations of current neural network architectures in terms of their ability to generalise and reason in a human-like way. In particular, the inability of neural networks to do unbounded computation equivalent to a Turing machine. Paul expressed optimism that this is not a fundamental limitation, but an artefact of current architectures and training procedures.
The power of abstraction - allowing us to focus on the essential structure while ignoring extraneous details. This can make certain problems more tractable to reason about. Paul sees category theory as providing a powerful "Lego set" for productively thinking about many practical problems.
Towards the end, Paul gave an accessible introduction to some core concepts in category theory like categories, morphisms, functors, monads etc. We explained how these abstract constructs can capture essential patterns that arise across different domains of mathematics.
Paul is optimistic about the potential of category theory and related mathematical abstractions to put AI and neural networks on a more robust conceptual foundation to enable interpretability and reasoning. However, significant theoretical and engineering challenges remain in realising this vision.
Please support us on Patreon. We are entirely funded from Patreon donations right now.
https://patreon.com/mlst
If you would like to sponsor us, so we can tell your story - reach out on mlstreettalk at gmail
Links:
Categorical Deep Learning: An Algebraic Theory of Architectures
Bruno Gavranović, Paul Lessard, Andrew Dudzik,
Tamara von Glehn, João G. M. Araújo, Petar Veličković
Paper: https://categoricaldeeplearning.com/
Symbolica:
https://twitter.com/symbolica
https://www.symbolica.ai/
Dr. Paul Lessard (Principal Scientist - Symbolica)
https://www.linkedin.com/in/paul-roy-lessard/
Neural Networks and the Chomsky Hierarchy (Grégoire Delétang et al)
https://arxiv.org/abs/2207.02098
Interviewer: Dr. Tim Scarfe
Pod: https://podcasters.spotify.com/pod/show/machinelearningstreettalk/episodes/Dr--Paul-Lessard---CategoricalStructured-Deep-Learning-e2hqqlq
Transcript:
https://docs.google.com/document/d/1NiHJKTkeqYdpcgr6lGCTwqMKl6YA9tS5R1jCgi987gA/edit?usp=sharing
More info about NNs not being recursive/TMs:
https://www.youtube.com/watch?v=4KIQH1VEwBI
Geometric Deep Learning blueprint:
https://www.youtube.com/watch?v=bIZB1hIJ4u8
TOC:
00:00:00 - Intro
00:05:07 - What is the category paper all about
00:07:19 - Composition
00:10:42 - Abstract Algebra
00:23:01 - DSLs for machine learning
00:24:10 - Inscrutability
00:29:04 - Limitations with current NNs
00:30:41 - Generative code / NNs don't recurse
00:34:34 - NNs are not Turing machines (special edition)
00:53:09 - Abstraction
00:55:11 - Category theory objects
00:58:06 - Cat theory vs number theory
00:59:43 - Data and Code are one and the same
01:08:05 - Syntax and semantics
01:14:32 - Category DL elevator pitch
01:17:05 - Abstraction again
01:20:25 - Lego set for the universe
01:23:04 - Reasoning
01:28:05 - Category theory 101
01:37:42 - Monads
01:45:59 - Where to learn more cat theory
Dr. Paul Lessard and his collaborators have written a paper on "Categorical Deep Learning and Algebraic Theory of Architectures". They aim to make neural networks more interpretable, composable and amenable to formal reasoning. The key is mathematical abstraction, as exemplified by category theory - using monads to develop a more principled, algebraic approach to structuring neural networks.
We also discussed the limitations of current neural network architectures in terms of their ability to generalise and reason in a human-like way. In particular, the inability of neural networks to do unbounded computation equivalent to a Turing machine. Paul expressed optimism that this is not a fundamental limitation, but an artefact of current architectures and training procedures.
The power of abstraction - allowing us to focus on the essential structure while ignoring extraneous details. This can make certain problems more tractable to reason about. Paul sees category theory as providing a powerful "Lego set" for productively thinking about many practical problems.
Towards the end, Paul gave an accessible introduction to some core concepts in category theory like categories, morphisms, functors, monads etc. We explained how these abstract constructs can capture essential patterns that arise across different domains of mathematics.
Paul is optimistic about the potential of category theory and related mathematical abstractions to put AI and neural networks on a more robust conceptual foundation to enable interpretability and reasoning. However, significant theoretical and engineering challenges remain in realising this vision.
Please support us on Patreon. We are entirely funded from Patreon donations right now.
https://patreon.com/mlst
If you would like to sponsor us, so we can tell your story - reach out on mlstreettalk at gmail
Links:
Categorical Deep Learning: An Algebraic Theory of Architectures
Bruno Gavranović, Paul Lessard, Andrew Dudzik,
Tamara von Glehn, João G. M. Araújo, Petar Veličković
Paper: https://categoricaldeeplearning.com/
Symbolica:
https://twitter.com/symbolica
https://www.symbolica.ai/
Dr. Paul Lessard (Principal Scientist - Symbolica)
https://www.linkedin.com/in/paul-roy-lessard/
Neural Networks and the Chomsky Hierarchy (Grégoire Delétang et al)
https://arxiv.org/abs/2207.02098
Interviewer: Dr. Tim Scarfe
Pod: https://podcasters.spotify.com/pod/show/machinelearningstreettalk/episodes/Dr--Paul-Lessard---CategoricalStructured-Deep-Learning-e2hqqlq
Transcript:
https://docs.google.com/document/d/1NiHJKTkeqYdpcgr6lGCTwqMKl6YA9tS5R1jCgi987gA/edit?usp=sharing
More info about NNs not being recursive/TMs:
https://www.youtube.com/watch?v=4KIQH1VEwBI
Geometric Deep Learning blueprint:
https://www.youtube.com/watch?v=bIZB1hIJ4u8
TOC:
00:00:00 - Intro
00:05:07 - What is the category paper all about
00:07:19 - Composition
00:10:42 - Abstract Algebra
00:23:01 - DSLs for machine learning
00:24:10 - Inscrutability
00:29:04 - Limitations with current NNs
00:30:41 - Generative code / NNs don't recurse
00:34:34 - NNs are not Turing machines (special edition)
00:53:09 - Abstraction
00:55:11 - Category theory objects
00:58:06 - Cat theory vs number theory
00:59:43 - Data and Code are one and the same
01:08:05 - Syntax and semantics
01:14:32 - Category DL elevator pitch
01:17:05 - Abstraction again
01:20:25 - Lego set for the universe
01:23:04 - Reasoning
01:28:05 - Category theory 101
01:37:42 - Monads
01:45:59 - Where to learn more cat theory
2024-09-03 - NITheCS & SU Category Theory Research Seminar
Workshop on pointfree topology and constructive mathematics – part 4
Dr Graham Manuell
(Stellenbosch ...
2024-09-03 - NITheCS & SU Category Theory Research Seminar
Workshop on pointfree topology and constructive mathematics – part 4
Dr Graham Manuell
(Stellenbosch University)
ABSTRACT
Constructive logic is a generalisation of classical logic that applies in more situations, including mathematical universes (toposes) where propositions can take different values at different locations in some space or where every function from ℕ to ℕ is computable. Unfortunately, in this general setting many classical results of point-set topology fail. However, almost all of these results can be recovered if we reformulate topology in terms of lattices of opens without predefined underlying sets of points. Moreover, this perspective also sheds light on many not-obviously-topological aspects of constructive mathematics itself.
In the penultimate lecture of the series, we will cover Hausdorffness, discreteness, compactness, overtness. These have elegant logical formulations in terms of equality in, and quantification over, locales.
2024-09-03 - NITheCS & SU Category Theory Research Seminar
Workshop on pointfree topology and constructive mathematics – part 4
Dr Graham Manuell
(Stellenbosch University)
ABSTRACT
Constructive logic is a generalisation of classical logic that applies in more situations, including mathematical universes (toposes) where propositions can take different values at different locations in some space or where every function from ℕ to ℕ is computable. Unfortunately, in this general setting many classical results of point-set topology fail. However, almost all of these results can be recovered if we reformulate topology in terms of lattices of opens without predefined underlying sets of points. Moreover, this perspective also sheds light on many not-obviously-topological aspects of constructive mathematics itself.
In the penultimate lecture of the series, we will cover Hausdorffness, discreteness, compactness, overtness. These have elegant logical formulations in terms of equality in, and quantification over, locales.
This presentation was the opening keynote of Lambda World 2017 by Dr. Eugenia Cheng.
Follow:
-https://www.twitter.com/47deg
-https://www.twitter.com/lambda_wo...
This presentation was the opening keynote of Lambda World 2017 by Dr. Eugenia Cheng.
Follow:
-https://www.twitter.com/47deg
-https://www.twitter.com/lambda_world
Visit:
-https://www.47deg.com/events for more details.
___
Category theory can be thought of as being 'very abstract algebra'. It is thought of as 'too abstract' by some people, and as 'abstract nonsense' by some others. In this talk, I will show that while it is abstract, it is far from being nonsense. I will argue that the abstraction has a purpose and that broad applicability is one of the powerful consequences. To demonstrate this, I will show how I apply concepts of category theory to important questions of life such as prejudice, privilege, blame and responsibility. I will introduce the category theory concepts from scratch so no prior knowledge is needed. These concepts will include objects and morphisms, isomorphisms and universal properties.
This presentation was the opening keynote of Lambda World 2017 by Dr. Eugenia Cheng.
Follow:
-https://www.twitter.com/47deg
-https://www.twitter.com/lambda_world
Visit:
-https://www.47deg.com/events for more details.
___
Category theory can be thought of as being 'very abstract algebra'. It is thought of as 'too abstract' by some people, and as 'abstract nonsense' by some others. In this talk, I will show that while it is abstract, it is far from being nonsense. I will argue that the abstraction has a purpose and that broad applicability is one of the powerful consequences. To demonstrate this, I will show how I apply concepts of category theory to important questions of life such as prejudice, privilege, blame and responsibility. I will introduce the category theory concepts from scratch so no prior knowledge is needed. These concepts will include objects and morphisms, isomorphisms and universal properties.
With more businesses moving online each day, creating a standout Digital Catalogue is essential for showcasing your products & services and reaching potential c...
With more businesses moving online each day, creating a standout Digital Catalogue is essential for showcasing your products & services and reaching potential customers effectively. A well-designed catalogue simplifies the buying process, making it easy for customers to find & purchase from you.
Join us on 5 November (Tuesday) for a session where we’ll guide you through the steps of building a digital catalogue that can attract, engage, and convert customers.
Key Topics
- Need for Digital Catalogue
- Types of Digital Catalogues
- Guide to Create a Digital Catalogue
- Ensuring Completeness & Quality
- Q&A Session
#globallinker #digitalcatlogue #onlinesales #ecommerce
With more businesses moving online each day, creating a standout Digital Catalogue is essential for showcasing your products & services and reaching potential customers effectively. A well-designed catalogue simplifies the buying process, making it easy for customers to find & purchase from you.
Join us on 5 November (Tuesday) for a session where we’ll guide you through the steps of building a digital catalogue that can attract, engage, and convert customers.
Key Topics
- Need for Digital Catalogue
- Types of Digital Catalogues
- Guide to Create a Digital Catalogue
- Ensuring Completeness & Quality
- Q&A Session
#globallinker #digitalcatlogue #onlinesales #ecommerce
this is a video about category theory... for #SoME3
haha jk this is just algebraic topology
btw i forgot the ray is supposed to go the other way (f(x) to x) im just dumb af pls dont crucify me
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🎬 Welcome to our Channel! Today, we’re diving into the fascinating world of Category Theory!📚✨
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🧑🏫 So, what exactly is Category Theory?🤔
- At its core, Category Theory is all about understanding mathematical structures and the relationships between them. Imagine having a universal language that can describe patterns and connections across various fields – that’s the power of Category Theory! 🌐🔗
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🌟 Why is Category Theory so important?🧠
- Category Theory provides a unifying framework that simplifies complex concepts. It allows mathematicians to see connections between seemingly unrelated areas, leading to groundbreaking discoveries and innovations. 🧩✨
- Think of it as the ultimate toolbox for mathematicians. By focusing on the relationships and structures, rather than individual elements, Category Theory offers a deeper, more holistic understanding of mathematics. 🔧🔍
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📚 Real-World Applications🌍
- Beyond pure mathematics, Category Theory has applications in computer science, physics, and even linguistics! Whether it’s improving algorithms, understanding quantum mechanics, or analyzing languages, Category Theory is a versatile tool that’s making waves in various disciplines. 🌊🔬
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🔭 Why should you watch this video?📺
- If you’re curious about how mathematics can bridge gaps between different fields, this video is for you! 🌉
- Whether you’re a student, a professional, or just a math enthusiast, understanding Category Theory will open your eyes to new ways of thinking and problem-solving. 💡
- Join us as we break down complex concepts into easy-to-understand explanations, with real-world examples that highlight the beauty and power of Category Theory. 🎓📈
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- Drop a comment below if you have any questions or if there’s a topic you’d like us to cover next! 👇
- Hit the like button if you enjoyed this video and subscribe to our channel for more exciting content! 🎥💡
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Thanks for watching, and we’ll see you in the next video! 🌟👋
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Image credits:
Keys
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Eilenberg and Mac Lane
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William Lawvere
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Columbia University
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Alexander Grothendieck
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Glossary of category theory
This is a glossary of properties and concepts in category theory in mathematics.Especially for higher categories, the concepts from algebraic topology are also used in the category theory.
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https://www.youtube.com/watch?v=MJBLsvLGXHQ
Dr. Paul Lessard and his collaborators have written a paper on "Categorical Deep Learning and Algebraic Theory of Architectures". They aim to make neural networks more interpretable, composable and amenable to formal reasoning. The key is mathematical abstraction, as exemplified by category theory - using monads to develop a more principled, algebraic approach to structuring neural networks.
We also discussed the limitations of current neural network architectures in terms of their ability to generalise and reason in a human-like way. In particular, the inability of neural networks to do unbounded computation equivalent to a Turing machine. Paul expressed optimism that this is not a fundamental limitation, but an artefact of current architectures and training procedures.
The power of abstraction - allowing us to focus on the essential structure while ignoring extraneous details. This can make certain problems more tractable to reason about. Paul sees category theory as providing a powerful "Lego set" for productively thinking about many practical problems.
Towards the end, Paul gave an accessible introduction to some core concepts in category theory like categories, morphisms, functors, monads etc. We explained how these abstract constructs can capture essential patterns that arise across different domains of mathematics.
Paul is optimistic about the potential of category theory and related mathematical abstractions to put AI and neural networks on a more robust conceptual foundation to enable interpretability and reasoning. However, significant theoretical and engineering challenges remain in realising this vision.
Please support us on Patreon. We are entirely funded from Patreon donations right now.
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Links:
Categorical Deep Learning: An Algebraic Theory of Architectures
Bruno Gavranović, Paul Lessard, Andrew Dudzik,
Tamara von Glehn, João G. M. Araújo, Petar Veličković
Paper: https://categoricaldeeplearning.com/
Symbolica:
https://twitter.com/symbolica
https://www.symbolica.ai/
Dr. Paul Lessard (Principal Scientist - Symbolica)
https://www.linkedin.com/in/paul-roy-lessard/
Neural Networks and the Chomsky Hierarchy (Grégoire Delétang et al)
https://arxiv.org/abs/2207.02098
Interviewer: Dr. Tim Scarfe
Pod: https://podcasters.spotify.com/pod/show/machinelearningstreettalk/episodes/Dr--Paul-Lessard---CategoricalStructured-Deep-Learning-e2hqqlq
Transcript:
https://docs.google.com/document/d/1NiHJKTkeqYdpcgr6lGCTwqMKl6YA9tS5R1jCgi987gA/edit?usp=sharing
More info about NNs not being recursive/TMs:
https://www.youtube.com/watch?v=4KIQH1VEwBI
Geometric Deep Learning blueprint:
https://www.youtube.com/watch?v=bIZB1hIJ4u8
TOC:
00:00:00 - Intro
00:05:07 - What is the category paper all about
00:07:19 - Composition
00:10:42 - Abstract Algebra
00:23:01 - DSLs for machine learning
00:24:10 - Inscrutability
00:29:04 - Limitations with current NNs
00:30:41 - Generative code / NNs don't recurse
00:34:34 - NNs are not Turing machines (special edition)
00:53:09 - Abstraction
00:55:11 - Category theory objects
00:58:06 - Cat theory vs number theory
00:59:43 - Data and Code are one and the same
01:08:05 - Syntax and semantics
01:14:32 - Category DL elevator pitch
01:17:05 - Abstraction again
01:20:25 - Lego set for the universe
01:23:04 - Reasoning
01:28:05 - Category theory 101
01:37:42 - Monads
01:45:59 - Where to learn more cat theory
2024-09-03 - NITheCS & SU Category Theory Research Seminar
Workshop on pointfree topology and constructive mathematics – part 4
Dr Graham Manuell
(Stellenbosch University)
ABSTRACT
Constructive logic is a generalisation of classical logic that applies in more situations, including mathematical universes (toposes) where propositions can take different values at different locations in some space or where every function from ℕ to ℕ is computable. Unfortunately, in this general setting many classical results of point-set topology fail. However, almost all of these results can be recovered if we reformulate topology in terms of lattices of opens without predefined underlying sets of points. Moreover, this perspective also sheds light on many not-obviously-topological aspects of constructive mathematics itself.
In the penultimate lecture of the series, we will cover Hausdorffness, discreteness, compactness, overtness. These have elegant logical formulations in terms of equality in, and quantification over, locales.
This presentation was the opening keynote of Lambda World 2017 by Dr. Eugenia Cheng.
Follow:
-https://www.twitter.com/47deg
-https://www.twitter.com/lambda_world
Visit:
-https://www.47deg.com/events for more details.
___
Category theory can be thought of as being 'very abstract algebra'. It is thought of as 'too abstract' by some people, and as 'abstract nonsense' by some others. In this talk, I will show that while it is abstract, it is far from being nonsense. I will argue that the abstraction has a purpose and that broad applicability is one of the powerful consequences. To demonstrate this, I will show how I apply concepts of category theory to important questions of life such as prejudice, privilege, blame and responsibility. I will introduce the category theory concepts from scratch so no prior knowledge is needed. These concepts will include objects and morphisms, isomorphisms and universal properties.
With more businesses moving online each day, creating a standout Digital Catalogue is essential for showcasing your products & services and reaching potential customers effectively. A well-designed catalogue simplifies the buying process, making it easy for customers to find & purchase from you.
Join us on 5 November (Tuesday) for a session where we’ll guide you through the steps of building a digital catalogue that can attract, engage, and convert customers.
Key Topics
- Need for Digital Catalogue
- Types of Digital Catalogues
- Guide to Create a Digital Catalogue
- Ensuring Completeness & Quality
- Q&A Session
#globallinker #digitalcatlogue #onlinesales #ecommerce
Yorgos Lanthimos and Efthymis Filippou developed the premise for the film out of the idea of people who allege something which is fabricated, for example via prank calls or by announcing their own deaths. The story took form as they needed a setting which could work well cinematically. Lanthimos considers it the complete opposite of his previous film, Dogtooth, which he says "is the story of a person who tries to escape a fictitious world. Alps is about a person who tries to enter a fabricated world."
The film was produced by the Greek company Chaos(Χάος) Film, which previously had produced Lanthimos' 2005 film Kinetta. The budget included funding from the Greek Film Center. Filming started in October 2010. Some scenes were added on the set and parts of the dialogue were improvised by the actors.