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chloe!
@schala163
something that has always puzzled me a bit when u first start doing category, it kinda seems like injectivity is a more fundamental concept than surjectivity; "f is a monomorphism" straightforwardly translates to "f is injective on generalized elements" 1/3?
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chloe!
@schala163
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) 1/4?
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chloe!
@schala163
"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. 2/4???
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chloe!
@schala163
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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chloe!
@schala163
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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chloe!
@schala163
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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chloe!
@schala163
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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chloe!
@schala163
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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chloe!
@schala163
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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QC (in Berlin 7/29-??? w/ COVID 😷)
@QiaochuYuan
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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