5.6 KiB
- a class
- on 2021-05-30
Idea of what category theory is. Basic definition of mathematical concepts.
Notions that are important in category theory. Commutative diagrams; categories as kinds of context. At the end, how this fits in a bigger picture.
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Aside: I'm reading Feynman and this fits in amazingly well with the introduction of that book, six easy pieces.
An elegant framework for reasoning about situations where we are composing stuff together. modeling, computation.
Category theory is about composing stuff together.
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Example: composing journeys.
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Example: composing processes. Mathematical functions, algorithms, physical processes.
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Notation: boxes instead of arrows. Directionality is from left to right.
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The output of the first box is the same type as the input of the second.
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You can also compose physical components, of course.
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And physical processes.
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The fundamental example is mathematical; composing functions as sets.
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Aside: we should put this on youtube.
We'll be used 'then' notations for compositions, not traditional.
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We'll relate objects. The things doing the relating are morphisms.
(here there should be a better screenshot, I couldn't get it on time)
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Each morphism has a source and a target.
We can compose morphisms when the target of the first matches the source of the second.
Common feature of all examples: they are associative. The order of composition doesn't matter; the order of application may. (check)
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This was true for all the examples we looked at so far.
Associativity means that brackets are not needed.
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Aside: going from a class in real time to screenshots is one morphism; going from screenshots to notes is another. Going from notes to a chapter or blog post is another.
Associativiy in plugs:
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Identity morphisms: morphisms that do nothing. Like a zero in addition.
Identity morphism for functions: the identity function.
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A converter that converts to the same electrical standard is just an extension that can become a converter of any type with one additional composition.
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identity journeys (aside: I call these excursions)
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Mathematical notion of a category
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Aside: I think most concepts had been introduced before except one?
"F is a morphism from X to Y":
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Identity morphisms:
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Composition:
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Unitality: identity "works" (I think this was the concept I thought hadn't been defined previously):
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The category of sets and functions plays a central role in category:
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The category of plugs and chords:
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The category of journeys:
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(Aside: some screenshots might be duplicate, they are likely from two moments that felt significant)
Any directed graph generates a category.
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(Aside: a directed graph generates dependency trees.)
But categories are richer than directed graphs, so they aren't used that much as such.
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Category theory is interested in relations between morphisms.
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if f;g is equal to h, this diagram commutes.
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Aside: when you say 'if you compose' something, it'd be nice to have the right notation for the composition on screen.
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Sameness: isomorphism. Identity is equal to roundtrip.
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(in the category of sets/functions, an isomorphism is equivalent to a bijection. in the category of topological spaces and continuous maps, it is a homeomorphism)
A context.
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(Aside: I needed more time in this slide).
Contexts leads to universal constructions and universal properties.
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Example: categorical product.
You start with objects, no morphisms. From the left diagram, you construct one with morphisms. In this one, we generalize cartesian product.
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pushouts Using the information given by f and g, you integrate two objects (check).
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Additional structures: monoidal products. A way to compose objects and morphisms "in parallel".
(Aside: it would be nice to have original and post application of composition side by side or top to bottom).
(Aside: missing screenshot here, find in Pictures)
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