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Fix a typo (#12)

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Patrick Stevens
2024-05-12 10:25:47 +01:00
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hugo/content/posts/2024-04-13-yoneda.md
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@@ -79,7 +79,7 @@ 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) : \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\\). 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.)