Conversation

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?
2
6
26
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?
1
8
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???
2
11
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??
2
8
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
1
11
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
2
10
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
1
7
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
2
15
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??
3
15
QC (in Berlin 7/29-??? w/ COVID 😷)
@QiaochuYuan
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)
1
3
QC (in Berlin 7/29-??? w/ COVID 😷)
@QiaochuYuan
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”
1
Show replies