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: pushout of monos in Cat

Naso (Nov 20 2021 at 15:03):

Is the pushout of monos a mono in the category $\mathsf{Cat}$ of small categories ? Is $\mathsf{Cat}$ adhesive?

Naso (Nov 20 2021 at 15:47):

I think the answer to both is "no": https://www.ioc.ee/~pawel/papers/adhesive.pdf

Naso (Nov 20 2021 at 16:15):

What about for the category of monoids $\mathsf{Mon}$?

Evan Patterson (Nov 20 2021 at 19:19):

That's right: the coprojections of a pushout of monics in Cat are not necessarily monic, and thus Cat is not adhesive.

Some results, counterexamples, and references about the first problem are in this paper: https://doi.org/10.4153/CMB-2009-030-5

John Baez (Nov 20 2021 at 21:52):

A monic in Cat is just a functor whose map on objects and map on morphisms are both injections.

What's a simple example of pushout of monics in Cat whose coprojections are not both monic?

John Baez (Nov 20 2021 at 21:53):

(I'm too lazy to gain access to that paper right now.)

Reid Barton (Nov 20 2021 at 22:20):

You can freely invert a morphism $f : x \to y$ of a category $C$ by forming the pushout of the functor $\{0 \to 1\} \to C$ picking out $f$ with the inclusion $\{0 \to 1\} \to \{0 \cong 1\}$. Those are both monomorphisms if $x \ne y$ (if $x = y$ so $f$ is an endomorphism, then you can push out along functor $B \mathbb{N} \to B \mathbb{Z}$ instead, which is also a monomorphism).

Reid Barton (Nov 20 2021 at 22:21):

Now we could construct an example like: take the category with three objects $A$, $B$, $C$, two distinct parallel maps $f$, $g : A \to B$ and a map $h : B \to C$ with $hf = hg$. Then if we invert $h$, we cause $f$ and $g$ to become equal so the map to the localization is not a mono.

John Baez (Nov 20 2021 at 22:34):

Nice, thanks!

With a suitable pushout along a mono we can invert any morphism $h$ in a category, so if we had $hf = hg$ in that category, we now have $f = g$ in the pushout, even if it wasn't true to start with. So we've collapsed two morphisms down to one! This "collapse" is not a monomorphism in Cat.

(That's my summary of hat you said, to help me remember this.)

John Baez (Nov 20 2021 at 22:39):

So, about this question....

Nasos Evangelou-Oost said:

What about for the category of monoids $\mathsf{Mon}$?

we can now see that also in $\mathsf{Mon}$, the pushout along a mono may not be a mono. The same sort of counterexample works.

Naso (Nov 21 2021 at 12:47):

thanks all :)