Categories as Contexts

From Of All Their Possible Variations, by Agnes Cameron (2022-08-29), insightful observations about Category Theory:

... Emily Riehl describes categories as 'contexts' in which to think about a class of mathematical objects, with category theory itself providing a language to move between these vantage points. Continuing with the language metaphor, Barry Mazur describes objects and morphisms (two fundamental components of any category) as the 'nouns and verbs' of a mathematical sentence.

F William Lawvere and Stephen Schanuel's book Conceptual Mathematics uses diagrams as a primary representation from which to explore category theory. To demonstrate this, they frame a thought experiment of Galileo's–on understanding the flight path of a bird through space as a function of time–as an example of what category theorists call 'composition,' where simple representations (or 'maps') of the links between different mathematical spaces may be combined to reason about more complex objects. In this instance, they show how the flight path of a bird can be understood by composing the passage of time with the location of the bird's shadow on the ground, and its corresponding level in the air.

(cf Greatest Inventions (2011-06-09), Cakes, Custard, and Category Theory (2016-02-14), Category Theory Concepts (2016-04-25), Ultimate Abstraction (2017-08-24), Put the Vast Storehouse in Order (2017-10-04), Category Theory is like a Lighthouse (2018-12-24), Why Care about Category Theory (2019-03-03), ...) - ^z - 2022-11-16