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2024-04-13 18:16:29 +01:00
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@@ -27,7 +27,7 @@ Recall what the functor laws in this context are:
Note what we're *not* requiring of our instantiations: that they're somehow "fully preserving all the structure". Note what we're *not* requiring of our instantiations: that they're somehow "fully preserving all the structure".
If \\(\mathcal{C}\\) has two objects \\(A\\) and \\(B\\), we're perfectly happy to instantiate both of them to the same type, as long as all the arrows keep composing correctly. If \\(\mathcal{C}\\) has two objects \\(A\\) and \\(B\\), we're perfectly happy to instantiate both of them to the same type, as long as all the arrows keep composing correctly.
In particular, every category has a trivial instantiation to the universe where there's only one set \\(\emptyset\\), and only one arrow \\(\mathrm{id} : \emptyset \to \emptyset\\). In particular, for example, our instantiation might squash away arbitrarily much of the category's structure: every category has a trivial instantiation to the universe where there's only one set \\(\emptyset\\), and only one arrow \\(\mathrm{id} : \emptyset \to \emptyset\\).
# Homomorphisms between diagrams # Homomorphisms between diagrams