A tutorial on category theory
for consciousness researchers
By Nao Tsuchiya & Hayato Saigo
2022 Mar 9 13:30-15:00 (JST), 15:30-17:00 (AEST)
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Format
5 min break after ~30 min
We strongly encourage any question.
We will try to answer all questions both in Zoom (raised hand, chat box) as well as Youtube comments.
Without questions, this tutorial is meaningless!
Don’t worry about the time. We will launch a follow-up if we cannot get to the end of the contents
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Links
We follow the structure of our paper
Tsuchiya & Saigo 2021 Neuroscience of Consciousness ,
土谷＆西郷 2019 認知科学
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Preprint (Free)
Neurosci Conscious
Contents
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The goal of this tutorial is to understand what the Yoneda lemma is and why we we think it can be used for characterizing qualia
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What does it mean for qualia to constitute a category?
What does it mean for the structure to be related / similar / same with the structure of information?
How can we characterize and quantify “structure” using category theory?
What do we mean by “different levels of sameness” in category theory?
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References
Consciousness researchers
Spivak DI. Category Theory for the Sciences. Massachusetts: MIT Press, 2014.
Lawvere FW, Schanuel SH. Conceptual mathematics. Cambridge: CUP, 2009.
Computer scientists
Fong B, Spivak DI. An Invitation to Applied Category Theory. Cambridge: CUP, 2019.
Mathematicians (mathematically oriented people)
Awodey S. Category Theory. Oxford: OUP, 2010.
Leinster T. Basic Category Theory. Vol. 143 Cambridge: Cambridge University Press, 2014.
Harold Simmons: An Introduction to the category theory
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What is a category?
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Free category of a directed graph
There may be a lot of arrows between any two objects.
If there is only one arrow, it’s called pre-order.
Many arrows: Category of states and state transitions
Q. Does f;g count as indirect connection or as direct as f and g?
What is a category?
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isomorphism
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Q. Is f;g the same as g;f ?
For example, f is the function 2x and g is the function 1+, then (2) f ; g = 5, but (2) g ; f = 6.
Q. is similarity arrow then invertible?
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Different levels of “sameness” in category theory
Q. Is this similarity hierarchy properly contained? (every isomorphism between categories is also a functor)
What is a functor?
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What is a functor?
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What is a functor?
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What is a natural transformation?
A key concept in category theory!
We’ll discuss this in more detail.
This is a cornerstone of understanding category theory.
This concept allows us to characterize a structure in a way that other theoretical tools cannot.
Essence: thinking about a lot of arrows at the same time.
Natural transformations are the basis for other important concepts that we plan to introduce into consciousness research in the future (e.g., adjunction)
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What is a natural transformation?
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Natural transformation is very unique to CT.
Very frequent
Coordinate transformation
Fourier transform
Psychology, analogy
A:B :: C: D
2-map
2nd order function
Q. Could we get some intuition about difference between isomorphism and Natural Transformation?
Q. Is it that isomorphism is within-category, and nat. transformation is between-categories ?)
What is a natural transformation?
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What is a natural transformation?
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What is a natural transformation?
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What is a functor category?
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What is categorical equivalence?
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Category C and E are categorically equivalent if there are invertible natural transformations from 1C to F;G and from 1E to G;F.