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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*".)
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.
(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.)