Joshua Grochow

The notion of epimorphism can be quite different from surjection, e.g. in Rings.

Though I recently learned epimorphisms can be characterized in terms of Isbell's zig-zags: https://en.wikipedia.org/wiki/Isbell%27s_zigzag_theorem.

Whereas monic seems to capture the notion of "injective" quite well in a categorical def. And indeed the two agree on any variety of algebras in the sense of universal algebra.

#algebra #CategoryTheory #UniversalAlgebra #math

Isbell's zigzag theorem - Wikipedia

Oct 14, 2024 at 6:00pmWeb
Show threadJoshua GrochowOct 14, 2024

But! TIL there's a categorical definition that supposedly agrees w/ "surjection" on any variety of algebras:

h is "categorically surjective" (a term I just made up) if for any factorization h=fg with f monic, f must be an iso.

(h/t Knoebel's book https://doi.org/10.1007/978-0-8176-4642-4)

Are there categorical definitions that agree w/ injective (resp. surjective) on all concrete categories?

#algebra #CategoryTheory #math #UniversalAlgebra

Show threadJordiGHOct 14, 2024
@joshuagrochow Don't you need an Abelian category? How do you define monic?
Show threadJoshua GrochowOct 14, 2024
@JordiGH monic=right-cancellative. f is monic if for any two morphisms g, h, if fg=fh, then g=h. Definition makes sense in any category, agrees with injective in most concrete categories that arise naturally.
Show threadJordiGHOct 14, 2024
@joshuagrochow So why not define surjective the same way?
Show threadJoshua GrochowOct 14, 2024

@JordiGH when you do that you precisely get the def of epimorphism.

The issue is that epimorphism and surjective agree a lot less often than monic and injective do. Eg the inclusion of Z into Q is epi in the category of rings, but not surjective.

Show threadJoshua GrochowOct 14, 2024
@JordiGH (in concrete Abelian categories, since you mention those, I think epi and surjective agree)
Show threadJordiGHOct 15, 2024
@joshuagrochow Wait, this doesn't make sense. Why is the inclusion map of ℤ into ℚ as a ring an epimorphism?
Show threadJoshua GrochowOct 15, 2024
@JordiGH Z to Q is epic because any ring homomorphism out of Q is completely determined by what it does to Z. For example, suppose I have a ring hom f:Q->S for another ring S. Once you know f(2), then f(1/2) is uniquely determined: it must be 1/f(2) in S. This example generalizes to show that the inclusion of an integral domain into its field of fractions is epic in the category of rings. And Isbell's zig-zag theorem generalizes this reasoning to characterize epimorphisms in various "algebraic" categories (like rings, semigroups, etc.)
Show threadChristina GrossackOct 14, 2024

@joshuagrochow

Here's something closely related that you might already know:

Any functor that preserves pullbacks will also preserve monomorphisms. Moreover, any faithful functor will reflect monomorphisms.

So if by a "concrete category" you mean a category with a faithful functor U to Set, injections (arrows f with Uf monic in set) will always be monic. The converse is true whenever U preserves pullbacks, for instance whenever U is representable (for instance, whenever U has a left adjoint).

The case for surjections (arrows f with Uf epic in Set) is dual -- functors preserving pushouts preserve epis, and faithful functors reflect epis. So a surjection is always epic in a concrete category, and the converse follows whenever U preserves pushouts (for instance, when U has a right adjoint. Though this situation is much rarer)

You can find this stuff in Chapter II.7 of The Joy of Cats in case you want to read more ^_^

Show threadBoardersOct 14, 2024
@hallasurvivor @joshuagrochow I like this answer. One can get a lot of intuition from meditating on the pullbacks here - pullbacks are given by subsets where constraint equations in the theory (equations using operations from the theory) are satisfied, the subset is automatically a sub-algebra and so one can just compute with the underlying set. Pushouts, on the other hand, typically require us to take a quotient by a congruence relation and so they care about the equations that present the theory - it wouldn’t be reasonable to expect any old equiv relation to be a congruence relation
Show threadtheHigherGeometerOct 15, 2024
@joshuagrochow Just for people reading along, there is a standard term for what you have called "categorically surjective": https://ncatlab.org/nlab/show/extremal+epimorphism
Show threadJoshua GrochowOct 15, 2024
@highergeometer thanks!
Show threadJon SterlingOct 14, 2024
@joshuagrochow Interestingly, even in the world of set-like things, epimorphisms and surjections can disagree: in particular, an epimorphism in an infinity-topos is not the same as a surjection — where epimorphism is defined in terms of a “subterminal extension property”. There is a cool paper by @buchholtz, @de_Jong_Tom, and @egbertrijke exploring this. https://arxiv.org/abs/2401.14106
Epimorphisms and Acyclic Types in Univalent Foundations

We characterize the epimorphisms in homotopy type theory (HoTT) as the fiberwise acyclic maps and develop a type-theoretic treatment of acyclic maps and types in the context of synthetic homotopy theory as developed in univalent foundations. We present examples and applications in group theory, such as the acyclicity of the Higman group, through the identification of groups with 0-connected, pointed 1-types. Many of our results are formalized as part of the agda-unimath library.

arXiv.org