does anyone can explain what the heck a Kan extension is
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They kinda act like an inverse to functor composition, like for a right kan extension you get a map Ran K D ∘ K → D, and for a left one you get D → Lan K D ∘ K
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:waitwhat:
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Fair enough lmao
So like there 20000 ways to approach Kan Extensions, but the one I like the best is "They are the natural way of taking a functor with a smaller domain and turning it into one with a larger domain"
PS: pls let me know if you aren't up for an infodump!
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So like, you are already very familiar with the general problem here because this is what the kan operations do!
Like, we have a map from an open box into something, and we want to extend that to a map from a filled cube, like so:
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is it important for it to be a square or a cube here? because this looks like a filled square…? (just checking)
i think it was just an example of a lifting problem, extending something defined on an open box along an inclusion into the boundary of a cube
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Exactly, it’s something that fits the same shape of “extending a map to a bigger domain in a coherent way”
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