So I think we’re ready, at least for the statement of Yoneda’s lemma. It says that for any locally small category Image may be NSFW.
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, if Image may be NSFW.
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is an object in Image may be NSFW.
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, and Image may be NSFW.
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a functor, then
Image may be NSFW.
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Moreover, the isomorphism is natural in both Image may be NSFW.
Clik here to view. and Image may be NSFW.
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.
Wow that looks complicated. Let’s parse some of the notation that I haven’t even explained yet. So certainly we know what the righthand-side means. It’s Image may be NSFW.
Clik here to view. applied to Image may be NSFW.
Clik here to view.. That’s just a set. As for the lefthand-side, Image may be NSFW.
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and Image may be NSFW.
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I haven’t explained.
So Image may be NSFW.
Clik here to view. is a functor we mentioned briefly, but I used a different notation. It’s the representable functor Image may be NSFW.
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. I write it as Image may be NSFW.
Clik here to view. here to avoid over using parentheses and therefore complicating this business well beyond it’s current level of complication. As a reminder, Image may be NSFW.
Clik here to view. is a functor from Image may be NSFW.
Clik here to view. to Image may be NSFW.
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(the same as Image may be NSFW.
Clik here to view.). It takes objects Image may be NSFW.
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in Image may be NSFW.
Clik here to view. to the set of homomorphisms Image may be NSFW.
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. Remember we’re in a locally small category, so Image may be NSFW.
Clik here to view. really is a set. It takes morphisms Image may be NSFW.
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to a map Image may be NSFW.
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by sending
Image may be NSFW.
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We checked all the necessary details in an earlier post to make sure this really was a functor.
So the last thing is that Image may be NSFW.
Clik here to view.. It’s the collection of all natural transformations between the functors Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view.. So Yoneda’s lemma claims that there is a one to one correspondence between the the natural transformations from Image may be NSFW.
Clik here to view. to Image may be NSFW.
Clik here to view. and the set Image may be NSFW.
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.
In particular, it claims that Image may be NSFW.
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is a set. This is because Image may be NSFW.
Clik here to view. is a locally small category, and so each natural transformation (defined as a collection of morphisms, each of which was a set) is a set. But there are only so many collections of morphisms, not even all of which are natural transformations. The collection is small enough to be a set. If you don’t care about this set theory business. Then disregard the paragraph you probably just read angrily.
It’s worth mentioning now, that Yoneda’s lemma is a generalization of some nice theorems. We can (and will) use it to derive Cayley’s theorem (every group embeds into a symmetric group). We can (and will) use it to derive the important fact that Image may be NSFW.
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in the category of Image may be NSFW.
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-modules. I bet in that one you can already start to see the resemblance.
We’ll prove Yoneda’s lemma over the next few posts.
Image may be NSFW.
Clik here to view.
