Hi Simon,

>> The implementation of FreeSmallCategory in #12630 is somewhat related
>> to what I need. I do not understand why the FreeSmallCategory can be a
>> Magma ? The composition is not defined everywhere...
>
> Multiplication of paths *is* defined everywhere. That's why the 
> FreeSmallCategory
> of #12630 contains an element called "invalid path".

I see. Thanks. Did you discuss the pros and cons of
 - adding an extra element NotAPath (your "invalid path") but having a
multiplication defined everywhere
 - having a partial multiplication
I do not like the extra element in my case because in all algorithm I
will need to check that (recall that I want morphisms from the set of
paths to other set of paths).

>> In my case, I
>> want it to be a groupoid because I do have inversion, but in your case
>> you don't and it looks like a monoidoid (?)
>
> There has been a decision on the name on a different thread on
> sage-combinat-devel, and I wouldn't like to change it again.

Sure. I was not discussing the name but the category here. I guess
that your FreeSmallCategory has some special features dedicated to
Quiver and it has no meaning to change it. Nevertheless, it is still
possible to factorize code.

Best,
Vincent

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