(no subject)

[timelets]

I can't come up with a relevant and/or real-life example of two different functors mapping one category to another. How would I apply a natural transformation so that it makes a difference?

Comments

[ecreet]

I know lots of examples, but they of course involve functors in the mathematical sense, as in a bunch of data related to categories, which are also understood as a bunch of data. Not "words with arrows between them". Are you sure your understanding of "relevant and/or real-life" does not contradict your request of there being not just one obvious way to organize real-life notions into diagrams with arrows?

[timelets]

I'm not sure, although after sleeping on it I managed to find a possible application - need to work through it to verify. It would be great if you could share your examples, whether purely mathematical or computational.

[ecreet]

A huge class of examples is representable functors from any category to sets (or abelian groups, or vector spaces, or whatever the initial category may be enriched over). For any object X in any category Hom(X,-) is a functor from this category to sets. For non-isomorphic X and Y these functors would not be isomorphic (follows from Yoneda's Lemma).

[timelets]

In general, I'm familiar with these examples, but I don't have a good feel for them yet. I should probably dig into representable functors before taking on another topic. Thanks.