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Stream: learning: questions

Topic: unfactorable morphisms?

Gershom (May 07 2020 at 20:54):

Basic terminology here I'm blanking on. Suppose I have a monomorphism $f : A \to B$ and it is the case that if $f = g \circ h$ with $g$ and $h$ both mono, then one of the two must be $id$. Is there a name for this?

John Baez (May 07 2020 at 20:56):

That's an "evil" concept - not invariant under equivalence of categories. A "non-evil" version would be to say one of the two must be an isomorphism. Your version can only occur if neither $A$ nor $B$ have nontrivial automorphisms.

I don't know a name for it, though.

Gershom (May 07 2020 at 21:02):

Iso works fine too. I just want to capture the operation of "adding a single element to a set". Surprised it doesn't seem more common!

Amar Hadzihasanovic (May 07 2020 at 21:31):

In a poset, that would be the fact that $B$ covers $A$. In fact it's equivalent to: the class of the identity on $B$ covers the class of $f$ in the poset of subobjects of $B$. But I don't know of any slick name for it.

Amar Hadzihasanovic (May 07 2020 at 21:33):

Maybe you could call it a co-covering monomorphism.

Gershom (May 07 2020 at 21:34):

i was thinking "minimal" or "generating"

Nathanael Arkor (May 07 2020 at 21:42):

This paper uses unfactorisable, while this math.stackexchange question suggests atomic, indecomposable or irreducible (not necessarily for monomorphisms).

T Murrills (May 07 2020 at 21:45):

I wonder how characterizing “adding an element to a set” in this way relates to saying that B is a coproduct of A with the terminal object of Set, when generalizing the notion to other categories with terminal objects and coproducts...what “makes” these two properties coincide, so to speak? is mono-irreducibility just somehow always equivalent to being a leg of such a coproduct cone when it’s available...? If not, what else do you need in your category to make that true?

Nathanael Arkor (May 07 2020 at 21:53):

The category of finite sets is the free coproduct completion of a point (i.e. the singleton set), so in a sense this is characteristic of sets.

sarahzrf (May 07 2020 at 21:57):

finite coproduct completion, might be worth noting

sarahzrf (May 07 2020 at 21:58):

Set is the coproduct completion

সায়ন্তন রায় (May 08 2020 at 05:17):

@Gershom Well, I don't think I have seen a standard terminology for this. But maybe this line of thinking gives you some additional food for thought to decide upon one (if you haven't already done so). Let $\bf{A}$ be a category $M$ is a class $\bf{A}$-morphisms. Then we say an $\mathbf{A}$-morphism $f$ to be a "$M$-irreducible morphism" or "$M$-indecomposable morphism" if for every $g,h\in M$ whenever $f=g\circ h$ at least one of $g$ or $h$ is identity. The case under consideration then could be seen by taking $M$ to be the class of all $\mathbf{A}$-monomorphisms.

Gershom (May 08 2020 at 06:08):

One reason this is a concept worth considering is because we actually make use of this fact at times. E.g. it is very handy that every morphism in the simplex category comes from face and degeneracy maps, which is an instance of this. So for "hands on" work, being able to discuss generating morphisms seems quite useful. I would imagine this works out quite nicely in every concrete category, at least.

Pastel Raschke (May 09 2020 at 14:48):

IMG_20200509_154206.jpg

Pastel Raschke (May 09 2020 at 15:02):

algebraic morphisms (of a globular extension) in 'a type-theoretical definition of weak omega-categories', eric finster and samuel mimram

i think this terminology comes from algebraic elements of field extensions? and i think the operation is referred to as adjoining indeterminates? i don't understand group extensions/exact sequences or if/how field extensions relate to any other sort of notion of extension, dual-lift, context, or otherwise (the nlab for field extension isn't very good)