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Fatima Afzali (Nov 04 2021 at 11:14):It's been a pet project of mine to systematize the use of intuition in mathematics, and one of the things I've noticed is that intuitive explanations tend to cluster around some 'canonical' cases which directly link to everyday experience, objects that have already been internalized, and/or visuospatial intuition.
I'm calling these clusters "conceptual perspectives". For instance, one conceptual perspective on category theory sees the objects of a category as states of the system and the morphisms as processes that go between states. If we choose to intuit categories via this perspective, we can then intuit functors as measurement processes (namely, if , then is the concealed system, is the system we directly observe; if is in state , then we observe to be in state , a concealed process causes an observed process — note that we're assured of the functor laws entirely by virtue of our intuiting these processes as occurring through time — and so on). Obviously, we can intuit functors in other ways compatible with this conceptual perspective, and other concepts — adjoints, natural transformations, etc. — can each adopt a variety of different interpretations in the context of this perspective. So, following this train of thought, I've been trying to build a sort of encyclopedia of conceptual perspectives in category theory.
So far, I've isolated a couple of conceptual perspectives on categories themselves: table.png
I know I've missed many important/prominent ones, and I don't have the mathematical experience to know what functors, natural transformations, adjoints, and so on could/should naturally correspond to in each of these perspectives, so I'd like to get some feedback on this project and other perspectives I should include in it.
(Alternatively, to exploit Cunningham's law: this pitiful table is complete, prove me wrong!)
Martti Karvonen (Nov 04 2021 at 13:02):I'm not adding a new perspective to the list, but the resource theories (as understood in physics) can be understood categorically (see e.g. https://arxiv.org/abs/1409.5531 ) and many of these would fit the intuition of "objects are states, morphisms transform states to others", so picking some of these would let you fill the example column for that row.
Robin Kaarsgaard (Nov 04 2021 at 19:45):What you're describing as the computation perspective describes one particular paradigm, namely typed functional programming. The imperative paradigm is essentially what you describe as the system perspective, where we think of objects as describing the current state of the machine, and morphisms as state transformations.
Nathaniel Virgo (Nov 05 2021 at 01:00):I like to take the string diagram perspective, where an object is a type of wire and a morphism is a box between two types of wire, like this:
Morgan Rogers (he/him) (Nov 05 2021 at 10:42):There's a perspective of objects as vector spaces and morphisms as matrices.
To contrast the spatial examples, you could think of objects of any category as algebraic objects (groups, rings...)
Morgan Rogers (he/him) (Nov 05 2021 at 10:44):A lot of category theory amounts to determining the extent to which these analogies are precise: to what extent is my category comparable in structure to a category of [spaces, algebras, processes...]? Is that enough to lift constructions (and hence intuition) from those cases?
Fatima Afzali (Nov 07 2021 at 00:20):Martti Karvonen said:
I'm not adding a new perspective to the list, but the resource theories (as understood in physics) can be understood categorically (see e.g. https://arxiv.org/abs/1409.5531 ) and many of these would fit the intuition of "objects are states, morphisms transform states to others", so picking some of these would let you fill the example column for that row.
It seems that a significant portion of Coecke's oeuvre is about showing not only how things can be modeled by monoidal categories with various additional structures, but explaining intuitively why they should be. String diagrams also seem to fit into the state-transformation perspective, as in e.g. categorical quantum mechanics, where they form a visual language for manipulating objects and morphisms which conceives of them as states and transformations... Thanks for this paper, it seems very useful.
Nathaniel Virgo (Nov 07 2021 at 01:44):String diagrams are often associated with the state transformation perspective (with good reason), but I don't think they have to be. For example, here's a string diagram equation in the category consisting of the natural numbers as a monoid under addition:
You can probably choose to see those morphisms as transforming a state in some way, but you don't have to - they're just boxes that can move on a wire and combine in a certain way. YMMV but in my mind the string diagram perspective is distinct from the others and useful in its own right.