this lets you treat monomorphisms like they are just injective functions in a very robust way, using not that much structure on your category (just need finite limits)
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"f is an epimorphism" otoh...
well for one thing the *definition* of an epimorphism is dual to that of a monomorphism, which is to say, it's defined in terms of the *injectivity* of post-composition.
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you need a lot more to be able to work with epimorphisms like surjective maps. There's a whole slew of variants: regular epi, strict epi, effective epi, so on. And correspondingly a whole zoo of different types of category where you ask these to play nice in various ways.
pi/7??
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But something kinda weird happens when you start thinking about *higher* category theory. Suddenly it becomes most natural to think of *injectivity* as a kind of *surjectivity*.
injectivity means: f(x)=f(y) implies x=y
k/k+1
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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.
SmallNum/BigNum
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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
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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