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from Part Two - Doing Category Theory
Published online by Cambridge University Press: 13 October 2022
This chapter brings together all that we’ve done in one of the pinnacles of abstraction in category theory. First, we revisit the sense in which a category sees isomorphic objects as the same, and show that our argument from Chapter 14 is in fact an isomorphism in Set between some particular sets of morphisms. We then show how this arises from some particular types of functor called representable functors. We then go up another level and introduce the Yoneda embedding as a functor from our base category to the category of presheaves on it, and we show that it is full and faithful. We describe the principle behind the Yoneda Lemma, and then state the Yoneda Lemma. Although we have all the technology required for the proof, we stop just short of giving it. We end the chapter with a brief discussion of Mac Lane’s comment that all concepts are Kan extensions.
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