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Stream: learning: questions

Topic: I forgot a bunch of stuff

Daniel (Mar 01 2021 at 19:10):

If C is a category, what was the C(a, -) again?

Nathanael Arkor (Mar 01 2021 at 19:12):

It's a functor $\mathbf C(a, {-}) : \mathbf C \to \mathrm{Set}$ that takes an object $b \in \mathbf C$ and produces the hom-set $\mathbf C(a, b)$.

James Wood (Mar 01 2021 at 19:12):

Covariant hom-functor

Joshua Meyers (Mar 01 2021 at 19:40):

Also consider its action on morphisms: if $f:b\to b'$ is a morphism in $\mathbf{C}$ then $\mathbf{C}(a,-)$ sends $f$ to the morphism $\mathbf{C}(a,f):\mathbf{C}(a,b)\to\mathbf{C}(a,b')$ in $\mathbf{Set}$ defined by $\mathbf{C}(a,f)(g)=f\circ g$.

Daniel (Mar 01 2021 at 21:53):

what does [C, Set](C(a, -), F) then mean? Here F is supposed to be a functor

Ralph Sarkis (Mar 01 2021 at 21:54):

This is the set of natural transformations from the Hom functor $\mathbf{C}(a,-)$ to $F$.

Ralph Sarkis (Mar 01 2021 at 21:56):

Note that the fact that this is a set is not benign (the collection of natural transformations between two arbitrary functors may be a proper class --- or whatever is bigger than a set in your foundations), it follows from the Yoneda lemma.

Joe Moeller (Mar 01 2021 at 21:56):

This is a particular instance of the notation C(x,y) for the set of maps from x to y in C, but where the category is [C,Set], and you're looking at maps from the representable C(a,-) to some other object of [C, Set] (aka functor) F.

Daniel (Mar 01 2021 at 22:07):

So, just to make sure I'm getting to getting this: [C, Set] is the category of functors from C to Set, C(a, -) is a covariant which maps every element to its hom-set with a and C, Set is a hom-set in the category [C, Set]` containing natural transformations from C to Set

Ralph Sarkis (Mar 01 2021 at 22:23):

Your notation broke (you can use double dollar signs for writing math), but anyway only the last three words are incorrect. Natural transformations are morphisms between functors, so the hom-set $\mathbf{C},\mathbf{Set}$ contains natural transformations between $\mathbf{C}(a,-)$ and $F$.