Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: The left orthogonal of monos is not always made of epis

fosco (Dec 30 2025 at 10:33):

The usual definition of the class of "strong epimorphisms" in a completely general category is $Epi^s := Epi\cap {}^\perp(Mono)$: a "strong epi" in $C$ is an epi $p : E \to B$ which, in addition, is also left orthogonal to all monos.

The conditions under which asking $p$ to be an epi is redundant are, however, extremely weak: it is enough for $C$ to have equalizers, and in fact it seems to be enough that the diagonal $B\to B\times B$ exists, and that $p$ is orthogonal to it, to ensure $p$ is an epimorphism.

How to build a counterexample, then? Can one find a (possibly finite) category $C$ such that there exists a $p$, which is not an epimorphism, and yet $p\in {}^\perp(Mono)$?

Does this work? Take the free category on the graph [image], subject to the relations $f\circ p=g\circ p$, $p\circ a_p=p\circ b_p$, $f\circ a_f = f\circ b_f$, $g\circ a_g=g\circ b_g$, chosen in order to make $p$ not an epi, and $p,f,g$ not monos; in particular, $g\circ a_f\ne g\circ b_f$, $f\circ a_g\neq f\circ b_g$, etc.: so, $p$ is not an epi, and there are no nontrivial lifting problems to solve against the monos (for example, the arrows $a_x,b_x : Z_x \to ...$ are vacuously mono, but $\hom(E,Z_x)$ is empty).

Is there a simpler counterexample?

image.png

Ivan Di Liberti (Dec 30 2025 at 10:34):

Man if you dont have products, are you even a category?

fosco (Dec 30 2025 at 10:38):

Products, in this economy!

Morgan Rogers (he/him) (Dec 31 2025 at 09:24):

I don't think you need the objects $Z_x$ to be distinct, so you could have fewer arrows, but conceptually this is pretty much the simplest example: you need enough arrows around to stop the relevant ones from being epi/mono, after all.