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@@ -62,7 +62,7 @@ For example, there are \\(\mathbb{R}\\)-many functions \\(\mathbb{N} \to \mathbb
This diagram wants to have way more structure than there was in \\(\mathcal{C}\\), and it's only by shutting our eyes and ignoring the set-structure we don't care about that we recover anything that looks remotely like \\(\mathcal{C}\\)! This diagram wants to have way more structure than there was in \\(\mathcal{C}\\), and it's only by shutting our eyes and ignoring the set-structure we don't care about that we recover anything that looks remotely like \\(\mathcal{C}\\)!
It turns out there's a sweet spot of diagrams with "exactly the amount of structure \\(\mathcal{C}\\) specified, and no more". It turns out there's a sweet spot of diagrams with "exactly the amount of structure \\(\mathcal{C}\\) specified, and no more".
The construction is: assume there's only one element in \\(FA\\) (for some \\(A\\)), and then chase through everything else that \\(\mathcal{C}\\) says should exist. The construction is: assume there's only one element in \\(FA\\) (for some \\(A\\)), and then chase through everything else that \\(\mathcal{C}\\) says should exist (which may involve adding in more elements of \((FA\)) if there are morphisms telling us there should be more).
What does \\(\mathcal{C}\\) say should exist? What does \\(\mathcal{C}\\) say should exist?
If \\(f : A \to B\\) is a morphism of \\(\mathcal{C}\\), and if \\(a \in FA\\) is an element of our concrete instantiation of the object \\(A\\), we really want there to be a distinct object \\((Ff)(a) \in FB\\) so that we've preserved the information "\\(f\\) was a morphism \\(A \to B\\)". If \\(f : A \to B\\) is a morphism of \\(\mathcal{C}\\), and if \\(a \in FA\\) is an element of our concrete instantiation of the object \\(A\\), we really want there to be a distinct object \\((Ff)(a) \in FB\\) so that we've preserved the information "\\(f\\) was a morphism \\(A \to B\\)".
@@ -75,7 +75,7 @@ This somehow "captures exactly all the structure that \\(\mathcal{C}\\) said \\(
(I've used the symbol \\(\mathrm{Rep}\\) to denote these diagrams, because the category-theoretic term for a diagram isomorphic to one of these is "*representable functor*".) (I've used the symbol \\(\mathrm{Rep}\\) to denote these diagrams, because the category-theoretic term for a diagram isomorphic to one of these is "*representable functor*".)
Note that I haven't yet written down the functions in these concrete instantiations of \\(\mathcal{C}\\); there's only one thing it could plausibly be. Note that I haven't yet written down the functions in these concrete instantiations of \\(\mathcal{C}\\); there's only one thing it could plausibly be.
Given \\(f : B \to C\\) a morphism of \\(\mathcal{C}\\), the corresponding function \\(\mathrm{Rep}_A(f) : \mathrm{Rep}_A(B) \to \mathrm{Rep}_A(C)\\) (that is, the function \\(\mathrm{Rep}_A(f) : \{\text{morphisms $A \to B$ in $\mathcal{C}$}\} \to \{\text{morphisms $A \to C$ in $\mathcal{C}$}\}\\) is defined to be given by composing with \\(f\\): we send \\(g : A \to B\\) to \\(f \circ g : A \to C\\). Given \\(f : B \to C\\) a morphism of \\(\mathcal{C}\\), the corresponding function \\(\mathrm{Rep}_A(f) : \mathrm{Rep}_A(B) \to \mathrm{Rep}_A(C)\\) (that is, the function \\(\mathrm{Rep}_A(f) : \langle\text{morphisms $A \to B$ in $\mathcal{C}$}\rangle \to \langle\text{morphisms $A \to C$ in $\mathcal{C}$}\rangle\\) is defined to be given by composing with \\(f\\): we send \\(g : A \to B\\) to \\(f \circ g : A \to C\\).
These particular diagrams, the *representable functors* (one for every object in \\(\mathcal{C}\\)), together tell you everything there is to know about the category. These particular diagrams, the *representable functors* (one for every object in \\(\mathcal{C}\\)), together tell you everything there is to know about the category.
(That is kind of intuitive, by their definition as "the sets of morphisms": we can list out every morphism in the category, just by writing down every element of every object in each of these concrete instantiations.) (That is kind of intuitive, by their definition as "the sets of morphisms": we can list out every morphism in the category, just by writing down every element of every object in each of these concrete instantiations.)
@@ -86,7 +86,7 @@ What might be less intuitive and requires a bit more thinking is their character
The representable functors contain so little structure that their homomorphisms are forced to be quite simple. The representable functors contain so little structure that their homomorphisms are forced to be quite simple.
Given any diagram \\(G : \mathcal{C} \to \mathrm{\mathbf{Set}}\\), and given any \\(A \in |\mathcal{C}|\\), we can precisely characterise the homomorphisms \\(\mathrm{Rep}_A \to G\\). Given any diagram \\(G : \mathcal{C} \to \mathrm{\mathbf{Set}}\\), and given any \\(A \in |\mathcal{C}|\\), we can precisely characterise the homomorphisms \\(\mathrm{Rep}_A \to G\\).
Remember, \\(\mathrm{Rep}_A\\) was defined by supposing there's just one element \\(a \in \mathrm{Rep}_A\\), and then throwing in all the other elements that are forced to exist by morphisms of \\(\mathcal{C}\\). Remember, \\(\mathrm{Rep}_A\\) was defined by supposing there's just one "free generator" element \\(a \in \mathrm{Rep}_A(A)\\), and then throwing into every object all the other elements that are forced to exist by morphisms of \\(\mathcal{C}\\).
So intuitively speaking it should be the case that a homomorphism \\(\mathrm{Rep}_A \to G\\) is precisely defined by where that one element \\(a\\) is sent; once we've defined that, *everything* else in our instantiation \\(\mathrm{Rep}_A\\) should be forced by the "preserves morphism structure" property of homomorphisms. So intuitively speaking it should be the case that a homomorphism \\(\mathrm{Rep}_A \to G\\) is precisely defined by where that one element \\(a\\) is sent; once we've defined that, *everything* else in our instantiation \\(\mathrm{Rep}_A\\) should be forced by the "preserves morphism structure" property of homomorphisms.
This reasoning is made formal in the *Yoneda lemma*: This reasoning is made formal in the *Yoneda lemma*:
@@ -104,3 +104,5 @@ The precise construction is as follows:
* Given a homomorphism \\(\alpha : \mathrm{Rep}_A\\) to \\(G\\), define an element of \\(G(A)\\) by \\(\alpha(\mathrm{id}_A : A \to A)\\). * Given a homomorphism \\(\alpha : \mathrm{Rep}_A\\) to \\(G\\), define an element of \\(G(A)\\) by \\(\alpha(\mathrm{id}_A : A \to A)\\).
* Show that these are inverse to each other. * Show that these are inverse to each other.
* Prove the naturality conditions. * Prove the naturality conditions.
Exercise: do those formally!