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: Freeness and monics

Fabrizio Romano Genovese (Mar 28 2024 at 14:04):

I have a very simple question:
The definition of monic morphism is: $f \circ g = f \circ h \implies g = h$. Now take a free category generated from a graph. The only morphisms we have around are:

• The generators (edges of the graph);
• The identities;
• Compositions of generators and identities (graph paths).

Here equality is almost at the typographical level, the only equations we enforce are associativity and identity laws. So am I correct in thinking that any morphism in a free category is vacuously monic, since basically the only equalities we have are between stuff that is _boringly_ equal?

If that's so, does it hold also for freely generated symmetric/commutative monoidal categories?

Morgan Rogers (he/him) (Mar 28 2024 at 14:14):

Yes, morphisms in free categories are both monic and epic (for instance, the free monoid on n generators, which is the free category on n endomorphisms, is both left and right cancellative). The same is true for the free monoidal category... but with symmetry you have to be more careful: is $f \otimes f ; \tau = f \otimes f$, for $\tau$ a transposing two copies of the same object?

Peva Blanchard (Mar 28 2024 at 14:18):

Regarding composition, I think it comes down to the fact that the free category generated by a graph can be explicitly described as follows:

• objects: vertices of the graph
• arrow $a \xrightarrow{\pi} b$ is a finite path $\pi = (e_1, \dots, e_n)$ of compatible edges of the graph, starting from $a$ and ending with $b$
• identities are the empty paths
• composition is concatenation
  And left and right cancellation applies to path concatenation.

Fabrizio Romano Genovese (Mar 28 2024 at 14:47):

Morgan Rogers (he/him) said:

Yes, morphisms in free categories are both monic and epic (for instance, the free monoid on n generators, which is the free category on n endomorphisms, is both left and right cancellative). The same is true for the free monoidal category... but with symmetry you have to be more careful: is $f \otimes f ; \tau = f \otimes f$, for $\tau$ a transposing two copies of the same object?

Generally, no. The best one can say is $f \otimes f ; \tau = \tau ; f \otimes f$. But it is indeed $f \otimes f; \tau = f \otimes f$ if the category is commutative monoidal. But in that case $\tau = id$ by definition, so I don't think this is very problematic...

Morgan Rogers (he/him) (Mar 28 2024 at 17:25):

That sounds right to me on further thought, at which point everything is both monic and epic in these cases too!