when you're used to thinking of equality as a *property*, it's easy to not think much of this. But once you start thinking of equality as a *structure*, the picture starts to change: This is saying that f is *surjective* on equations.
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if X is a set, you can treat it as a boring category (as "setoid") where objects are elements of X and morphisms x->y are equations x=y
then a function f:X->Y between sets easily upgrades to a functor between setoids.
almost done/not quite done
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a function f:X->Y is surjective iff it is essentially surjective, or "surjective on 0-cells" when viewed as a functor between setoids
a function f:X->Y is injective iff it is full, or "surjective on 1-cells", when viewed as a functor between setoids.
wake me up/can't wake up
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tldr:
when ur a little bit category-brained, injectivity looks like a more fundamental concept than surjectivity
but if ur a bit *more* category-brained, then it looks like injectivity is a form of surjectivity, just shifted one dimension up
so umm, what's the deal with that??
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hmm so part of the deal: the categories we like in practice differ systematically from their opposites, so category theory in practice isn’t self-dual; e.g. we like locally presentable categories and the opposite of a lp category isn’t lp (unless it’s a preorder or sth)
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we like the yoneda embedding and not its opposite, and the yoneda embedding reflects and preserves limits, hence reflects and preserves kernel pairs, hence reflects and preserves monomorphisms; but it ignores colimits and substitutes new colimits in practice
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moving up a categorical level, if f : X -> Y is a map of spaces / oo-groupoids, we like thinking of it as an object in the slice category over Y, not as an object in the coslice category under X, because spaces are locally cartesian closed so this is a family of spaces over Y
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once we’ve made that choice it becomes natural to write down the relative postnikov tower of f, which gives us two infinite families of conditions generalizing injectivity and surjectivity: “f is n-connected” and “f is n-truncated”
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where (-1)-connected reduces to surjections for sets and (-1)-truncated reduces to injections for sets. this is quite funny though because it’s a definition of surjections based on limits, not colimits: a map of sets is a surjection iff its fibers are non-empty
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(the relevant nlab page: ncatlab.org/nlab/show/%28n )
i didn’t know where i was going with this in advance but i get a sense from all this that it’s sort of an accident that epimorphisms are surjections for sets. there’s a similar flavor of thing going on with the disjoint union
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where i think it’s sort of an accident that coproducts are disjoint unions for sets. one manifestation of this is that sum types in type theory are supposed to be coproducts but their defining feature is that a term of type A+B is either a term of type A or a term of type B
which is not the universal property of the coproduct, since it’s about maps in and not maps out! i still don’t quite get what’s going on here but it seems that the sum type is really supposed to be an “extensive coproduct”
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