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Stream: theory: category theory

Topic: dual of subterminal

Mike Shulman (Apr 06 2021 at 05:05):

What do you call the dual of a subterminal object? That is, an object which admits at most one morphism to any other object? (For instance, the fields $\mathbb{Q}$ and $\mathbb{F}_p$.) The best I can think of is "quot-initial", which is rather unlovely.

Javier Prieto (Apr 06 2021 at 05:46):

I'll bite: why not subinitial?

John Baez (Apr 06 2021 at 05:51):

Such an object is not a subobject of the initial object. It's a quotient object of the initial object (if the category has an initial object).

John Baez (Apr 06 2021 at 05:54):

I can't think of any term better than "quot-initial object" offhand. The word "cosubterminal" leaps to mind, and it rolls off the tongue with a bit less of a clunk than "quot-initial", but I can't mount a serious argument for its virtues.

Morgan Rogers (he/him) (Apr 06 2021 at 08:12):

There are some other latin prefixes meaning "under" that you could employ, such as infra, or a greek one such as hypo. Hypoinitial doesn't sound too bad (although @Fabrizio Genovese won't like it :stuck_out_tongue_wink: )

Chad Nester (Apr 06 2021 at 08:47):

superinitial?

Chad Nester (Apr 06 2021 at 08:48):

subobject $\leftrightsquigarrow$ subset
superobject $\leftrightsquigarrow$ superset

Chad Nester (Apr 06 2021 at 08:55):

I think "cosubterminal" is more precise, although it's a bit arcane.

Kenji Maillard (Apr 06 2021 at 09:08):

(ab)using the terminology of Freyd and Scedrov, you might call it a super-initiator

Fawzi Hreiki (Apr 06 2021 at 10:10):

But super usually means bigger than which is not (usually) the case for quotients

Fabrizio Genovese (Apr 06 2021 at 10:24):

John Baez said:

Such an object is not a subobject of the initial object. It's a quotient object of the initial object (if the category has an initial object).

If it's a quotient it goes out the initial object in some way, right? initial comes from in-eo, "to go in". So you are going out something you are going in (at least etymologically). I'd go for "abinitial object".

Nick Hu (Apr 06 2021 at 10:25):

quasi-*? (q for quotient)

Fabrizio Genovese (Apr 06 2021 at 10:25):

"ab initium", literally "from the beginning", if you wish :smile:

Fabrizio Genovese (Apr 06 2021 at 10:27):

Latin-wise it would mean the same if you said "ainitial", but in English the a- prefix is often considered privative and one could interpret it as "not initial", which may be confusing.

Fabrizio Genovese (Apr 06 2021 at 10:29):

Morgan Rogers (he/him) said:

There are some other latin prefixes meaning "under" that you could employ, such as infra, or a greek one such as hypo. Hypoinitial doesn't sound too bad (although Fabrizio Genovese won't like it :stuck_out_tongue_wink: )

I am not a fan of mixing Greek and Latin, but (surprise surprise!) I'm not a fan of having technical words coming necessarily from Greek/Latin either. I think it would make maths more global if we started to form words/notations based on other languages as well (e.g. I strongly support using the kana よ to denote Yoneda embedding). So if someone has ideas coming from other languages I'm all ears :smile:

Amar Hadzihasanovic (Apr 06 2021 at 10:50):

Subterminal objects seem to be an instance of what Walter Tholen here calls a mono-cone, that is, a cone $\gamma$ over a functor such that $\gamma \circ f = \gamma \circ g$ implies $f = g$ for any parallel pair of morphisms to the tip of $\gamma$. (A monomorphism with target $a$ is in particular a mono-cone over the functor from the one-object discrete category that picks the object $a$). Of course a limit cone is always a mono-cone.

The dual notion should be an epi-(co)cone.

Amar Hadzihasanovic (Apr 06 2021 at 11:03):

So a potential uniform nomenclature would be to substitute for cone or cocone the special name of a cone or cocone of that shape...

A subterminal object is a mono-cone over the only functor from the empty category; such a cone is just an object; so “subterminal object” -> “mono-object” and, for the dual, “epi-object”.

Amar Hadzihasanovic (Apr 06 2021 at 11:04):

The similarly weakened notion of 'product' could be called a mono-span and its dual an epi-cospan... and monomorphisms and epimorphisms fit the naming scheme.

John Baez (Apr 06 2021 at 14:51):

Chad Nester said:

subobject $\leftrightsquigarrow$ subset
superobject $\leftrightsquigarrow$ superset

Unfortunately "superset" is usually used in a different way: a superset of $X$ is a set $Y$ with $X \subseteq Y$. Category theorists would say it's a set $Y$ with a mono $i: X \to Y$.

John Baez (Apr 06 2021 at 14:53):

James Dolan used to like the words "cobigger" and "cosmaller". If we have a mono $i: X \to Y$ we can say $X$ is smaller than $Y$ and $Y$ is bigger than $X$. If we have an epi $p: Y \to X$ we can say $X$ is cosmaller than $Y$ and $Y$ is cobigger than $X$.

John Baez (Apr 06 2021 at 14:54):

So, a subterminal object is smaller than $1$, while a quot-initial object is cosmaller than $0$.

Spencer Breiner (Apr 06 2021 at 15:56):

Fawzi Hreiki said:

But super usually means bigger than which is not (usually) the case for quotients

Maybe "sup-initial" to temper that intuition?

Or how about "epinitial"? I also like "ab-initial".

Nathanael Arkor (Apr 06 2021 at 18:50):

It's worth noting that there's a strong connection to multi-colimits. For instance, the prime fields also form a multi-initial family in the category of fields. In particular, a multi-initial family is always "cosubterminal". It's a slightly stronger property, because it assumes the cosubterminal objects can be collected together in such a way that they do ensure existence of morphisms to every object of the category. However, one can always a new object to the category with unique morphisms to objects not covered by the cosubterminal objects, in such a way that the addition of this object to the family of cosubterminal objects forms a multi-initial family. So being cosubterminal is not very far from being multi-initial at all; one may find that one is really concerned about multi-initial objects rather than cosubterminal objects.

Mike Shulman (Apr 06 2021 at 19:03):

Nathanael Arkor said:

a multi-initial family is always "cosubterminal". It's a slightly stronger property, because it assumes the cosubterminal objects can be collected together in such a way that they do ensure existence of morphisms to every object of the category. However, one can always a new object to the category with unique morphisms to objects not covered by the cosubterminal objects, in such a way that the addition of this object to the family of cosubterminal objects forms a multi-initial family.

I don't think this is true. For a multi-initial family, any object admits a unique morphism from a unique object in the family. The cosubterminal objects may not have this property even if you add a new one: a given object might admit morphisms from more than one of them. Such as, for instance, if there is also an initial object!

Nathanael Arkor (Apr 06 2021 at 19:14):

Ah, that's a very good point! They must instead be considered in isolation. But presumably one may still consider the construction of adding a new "almost initial" object such that the pair of the new object, and a given cosubterminal object, form a multi-initial family (of cardinality 2). Then, in this modified sense, the two concepts are tightly connected.

Nathanael Arkor (Apr 06 2021 at 19:17):

Maybe a simpler way to state this is that cosubterminal objects are initial in their connected components.

Nathanael Arkor (Apr 06 2021 at 19:24):

These are called "partial initial objects" in Makkai's The word problem for computads.

Nathanael Arkor (Apr 06 2021 at 19:25):

And "locally initial objects" in Palm's Categories with slicing.

Mike Shulman (Apr 06 2021 at 20:15):

Nathanael Arkor said:

Maybe a simpler way to state this is that cosubterminal objects are initial in their connected components.

No, not that either. If there's an actual initial object, the whole category is one connected component, so the cosubterminals aren't initial there.

Mike Shulman (Apr 06 2021 at 20:19):

I think I like "epi-object" and "ab-initial" best of the suggestions made so far.

Nathanael Arkor (Apr 06 2021 at 21:10):

Mike Shulman said:

Nathanael Arkor said:

Maybe a simpler way to state this is that cosubterminal objects are initial in their connected components.

No, not that either. If there's an actual initial object, the whole category is one connected component, so the cosubterminals aren't initial there.

Sorry, when I said connected component, I was really thinking of the coslice over that object (a "directed connected component"). (Though I think Makkai and Palm's notion is really the connected component.)

Mike Shulman (Apr 06 2021 at 21:26):

But any object is initial in its coslice... (-:

Nathanael Arkor (Apr 06 2021 at 21:31):

I wondered whether that might still be too ambiguous :flushed: I mean the essential image of the forgetful functor from the coslice into the base category, so whose objects are those for which there exists a morphism from the cosubterminal object, and whose morphisms are the morphisms from the cosubterminal object, or those forming the commuting triangles of such.