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Category theory is about categories,. One notion which makes sense in the context of a category is the categorical product (http://en.wikipedia.org/wiki/Product

by Chinjut 12y ago

Category theory is about categories,. One notion which makes sense in the context of a category is the categorical product (http://en.wikipedia.org/wiki/Product_%28category_theory%29 http://en.wikipedia.org/wiki/Product_%28category_theory%29). All the examples given are examples of categorical products within suitable categories [e.g., Cartesian product within the category of sets and functions between them (amounting to multiplication of cardinals, if one just cares about the action on objects), least common multiple within the category of positive integers ordered by divisibility (a partial ordering being just a special kind of category), logical conjunction within the category of truth values (which can be thought of as sets with at most one element), and structs or record types in the category whose objects are the types of your favorite programming language and morphisms are the programs between them].

JadeNB 12y ago
Certainly, I didn't mean to imply that you (EDIT: I mean chas) were wrong, just that it didn't feel like the examples that you brought up were in the 'spirit' of category theory. As I mentioned in a sister comment (https://news.ycombinator.com/item?id=8780829 https://news.ycombinator.com/item?id=8780829), any sufficiently powerful formalisation can encode any other (proof: definition of 'sufficiently powerful'), so that there is no mathematical structure of which it can be said definitively that it is not an instance of category theory—but that doesn't mean that every structure should be so viewed! For example, to pick on the lcm example (just because it's the one that caught my attention): as you mention, posets are automatically categories, but I don't think that the theory of posets is best viewed as a part of category theory; and, similarly, the product is just a special case of the limit over a discrete category, but I don't think that describing the least common multiple, say, as a limit will be educational to anyone! By contrast, viewing the lcm as part of an unusual algebraic structure on the natural numbers I think can be instructive.
- throwaway_99837 12y ago
  I think you should not attach yourself emotionally to whether a certain perspective is "educational" or "instructive" to anyone. Rather, look at both perspectives in a [detached] clinical manner as just things that are.
- Chinjut 12y ago
  I absolutely think it's instructive to view the LCM as a specific case of meets in a semilattice, and in turn to view these as specialization of a general theory of categorical products. (Why wouldn't it be instructive? If you like, the "unusual algebraic structure" on the natural numbers which might be worth considering is that of a category with products). And I also think that recognition of such analogies and general patterns ubiquitous across mathematics is a large part of the spirit of category theory.

by Chinjut 12y ago

JadeNB 12y ago

Certainly, I didn't mean to imply that you (EDIT: I mean chas) were wrong, just that it didn't feel like the examples that you brought up were in the 'spirit' of category theory. As I mentioned in a sister comment (https://news.ycombinator.com/item?id=8780829 https://news.ycombinator.com/item?id=8780829), any sufficiently powerful formalisation can encode any other (proof: definition of 'sufficiently powerful'), so that there is no mathematical structure of which it can be said definitively that it is not an instance of category theory—but that doesn't mean that every structure should be so viewed! For example, to pick on the lcm example (just because it's the one that caught my attention): as you mention, posets are automatically categories, but I don't think that the theory of posets is best viewed as a part of category theory; and, similarly, the product is just a special case of the limit over a discrete category, but I don't think that describing the least common multiple, say, as a limit will be educational to anyone! By contrast, viewing the lcm as part of an unusual algebraic structure on the natural numbers I think can be instructive.

throwaway_99837 12y ago
I think you should not attach yourself emotionally to whether a certain perspective is "educational" or "instructive" to anyone. Rather, look at both perspectives in a [detached] clinical manner as just things that are.
Chinjut 12y ago
I absolutely think it's instructive to view the LCM as a specific case of meets in a semilattice, and in turn to view these as specialization of a general theory of categorical products. (Why wouldn't it be instructive? If you like, the "unusual algebraic structure" on the natural numbers which might be worth considering is that of a category with products). And I also think that recognition of such analogies and general patterns ubiquitous across mathematics is a large part of the spirit of category theory.