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Stream: learning: questions

Topic: Is there a name for an "empty object"?

Andre Knispel (Apr 04 2020 at 16:32):

Is there a name for objects that have no morphisms into it (other than id of course)? I'm tempted to call this an "empty object" because its functor of points is as empty as possible...

Amar Hadzihasanovic (Apr 04 2020 at 16:42):

I've seen people use “source” (only morphisms out of it) and “sink” (only morphisms into it) by analogy with graph theory.

James Wood (Apr 04 2020 at 21:04):

It probably isn't named because it doesn't respect isomorphism. If you have two empty objects, it's natural to imagine that they'd be isomorphic. But if you make them isomorphic, they're no longer empty, because they have maps coming from each other.

James Wood (Apr 04 2020 at 21:05):

The nearest well behaved notion probably ends up being an initial object.

Vinay Madhusudanan (Apr 05 2020 at 04:22):

Related to this, I was thinking the other day that in a [non-singleton] discrete category, no two distinct objects are isomorphic, but in some sense they are all identical! I mean if this was a graph, it would be a transitive one, which would mean that all its vertices are identical for all purposes — there would be no way to differentiate between any of them (without labelling them).

Clearly this is the case even if it is a category. There is no category theoretical property that lets us distinguish between the objects, except the *statement *that "these objects are not isomorphic to one another" [I don't mean this statement is made by a person — it's made by the category itself, in the form of lack of isomorphisms]. That simultaneously makes sense and seems to go against the idea of category theory (that the difference between objects comes from their relations to one another — all objects are identical here in that respect)

Andre Knispel (Apr 05 2020 at 10:31):

James Wood said:

It probably isn't named because it doesn't respect isomorphism. If you have two empty objects, it's natural to imagine that they'd be isomorphic. But if you make them isomorphic, they're no longer empty, because they have maps coming from each other.

In the case I was dealing with, that was kind of the point. I cared about morphisms out of these objects and this was really their only common feature. But I've now realized that I have a much better way of doing the same thing, by slightly tweaking my category. Now I can just use the monoidal unit instead :)

Andre Knispel (Apr 05 2020 at 10:33):

But yeah, this notion is evil

sarahzrf (Apr 06 2020 at 02:19):

it is not

sarahzrf (Apr 06 2020 at 02:19):

it does respect isomorphism

sarahzrf (Apr 06 2020 at 02:20):

rather, it's more like... if you yoneda embed both objects, then restrict those presheaves to the subcategory that contains neither of the objects, then they will be isomorphic there

sarahzrf (Apr 06 2020 at 02:20):

but that's a bit subtler

sarahzrf (Apr 06 2020 at 04:11):

actually wait maybe it is evil, even if it does respect isomophism :thinking:

sarahzrf (Apr 06 2020 at 04:11):

i think the way that argument was phrased was a little confusing tho >.>

sarahzrf (Apr 06 2020 at 04:11):

if you tweak the definition to "any morphism in is an isomorphism", it shouldn't be evil

Vinay Madhusudanan (Apr 06 2020 at 05:15):

sarahzrf said:

if you tweak the definition to "any morphism in is an isomorphism", it shouldn't be evil

I thought that'd solve it too. But this doesn't prevent the existence of other such objects with no morphisms at all between them. So you can still have two such objects that are not isomorphic. Or do you mean something else entirely?

Vinay Madhusudanan (Apr 06 2020 at 05:16):

Again, take the example of a discrete category. For every object: Is it true that every in-morphism is an isomorphism? Yes (the only one is id). Are distinct objects isomorphic? No

sarahzrf (Apr 06 2020 at 05:28):

sure, but there's still nothing evil going on

sarahzrf (Apr 06 2020 at 05:30):

not every property needs to imply uniqueness

Vinay Madhusudanan (Apr 06 2020 at 05:32):

@sarahzrf Lol okay, even I didn't find it "evil", but I'm really confused by the example of the discrete category. The existence of that category itself seems to go against the philosophy of the category. If we are to understand (and distinguish between) objects purely by their relations to all other objects in the category — well then, in the discrete category all objects are "alike", in the sense that none has any non-identity morphisms into or out of it. But this (necessarily) doesn't make them mutually isomorphic

sarahzrf (Apr 06 2020 at 05:32):

ive honestly grappled with this issue myself

sarahzrf (Apr 06 2020 at 05:32):

but. thats just not how it works

sarahzrf (Apr 06 2020 at 05:33):

i think i may have tried to ask before about a notion of equivalence between objects that would identify things like this

sarahzrf (Apr 06 2020 at 05:33):

but i never got much of an answer

Vinay Madhusudanan (Apr 06 2020 at 05:36):

I see. I guess the Yoneda viewpoint (if I can call it that) doesn't really claim that you can distinguish between objects by looking at the structure of their hom-sets — it might be my graph theory bias that led me to understand it that way at first. But the actual hom functors act like Kronecker deltas or characteristic functions, so…

Vinay Madhusudanan (Apr 06 2020 at 05:37):

I suppose I have to make peace with that

Thibaut Benjamin (Apr 06 2020 at 09:45):

I don't know any name for it, but I would be tempted to call this a "left discrete object". Also, as pointed out by @sarahzrf, this notion is not invariant under equivalence of categories. You can have to equivalent categories, with one having a left discrete object and not the other one : just consider the category with one single object, and the category with only two objects that are isomorphic. You can correct this and make it invariant, by stating that every morphism into this object is an isomorphism. If you were to consider the simpler notion of a "discrete object", ie every morphism in or out of it is an isomorphism, then you can describe is equivalently as a "full subcategory that is a connected groupoid", which might sound a lot more scary, but makes it immediately invariant. I am not sure if there is such a way of describing more globally for the left discrete objects

Soichiro Fujii (Apr 06 2020 at 10:17):

I don’t know the name, but here is a characterization.

Suppose $\mathcal{C}$ is a locally small category, and let $P\colon \mathcal{C}\longrightarrow\mathbf{Set}$ be a covariant presheaf on it. Then we can take the cograph/collage of $P$ and obtain the category $\tilde{P}$ with

• the set of objects is $\mathrm{ob}(\tilde{P})=\mathrm{ob}(\mathcal{C})+\{\ast\}$,
• the hom sets are ($A,B\in\mathrm{ob}(\mathcal{C})$)
  • $\tilde{P}(A,B)=\mathcal{C}(A,B)$,
  • $\tilde{P}(A,\ast)=\emptyset$,
  • $\tilde{P}(\ast,B)=PB$,
  • $\tilde{P}(\ast,\ast)=\{id\}$,

with the obvious composition. Then the object $\ast$ is an “empty object” in $\tilde{P}$.

And every “empty object” in a locally small category arises in this way.

John Baez (Apr 06 2020 at 17:58):

Vinay Madhusudanan said:

I'm really confused by the example of the discrete category. The existence of that category itself seems to go against the philosophy of the category. If we are to understand (and distinguish between) objects purely by their relations to all other objects in the category — well then, in the discrete category all objects are "alike", in the sense that none has any non-identity morphisms into or out of it. But this (necessarily) doesn't make them mutually isomorphic.

Right. If you want to make precise the sense in which all objects of a discrete category are "alike" - and when, in general, two objects of a category count as "alike" in this way - you can use the fact that any category $C$ has a group of autoequivalences, i.e. equivalences $F: C \to C$. If this group acts transitively on the objects of $C$ then all objects are "alike" in the sense you're talking about. If for two objects $x, x' \in C$ there's an autoequivalence of $C$ mapping $x$ to $x'$ then we can say these two objects are "alike".

(I don't really recommend the term "alike" for this, but it's not terrible.)

By the way, there's really a 2-group $AUT(C)$ of autoequivalences of $C$ and natural isomorphisms between these. We can think of this as the 2-group of symmetries of the category $C$.

nadia esquivel márquez (Apr 06 2020 at 18:06):

Seems like one can think of isomorphism as an "inner" notion of equivalence and the alike-ness relationship @John Baez is talking about as an "outer" notion of equivalence

John Baez (Apr 06 2020 at 18:08):

Right.

Sam Tenka (Apr 06 2020 at 18:25):

sarahzrf said:

i think i may have tried to ask before about a notion of equivalence between objects that would identify things like this

Partially a special case and partially a generalization of @John Baez's notion:
the objects in the discrete category C, considered as functors from 1 to C, are all related by postcomposition with some automorphism of C. For k-transitivity, one can consider the discrete source with k objects, and for "structured" transitivity, one can consider general source categories.

Morgan Rogers (he/him) (Apr 06 2020 at 21:51):

I'll mention the idea of a strict initial object in this discussion, since it's precisely an initial object satisfying the "every morphism into it is an isomorphism" that you all arrived at.

Sam Tenka (Apr 06 2020 at 21:55):

Morgan Rogers said:

I'll mention the idea of a strict initial object in this discussion, since it's precisely an initial object satisfying the "every morphism into it is an isomorphism" that you all arrived at.

Ooh could you say more? What's a way of viewing these notions of equivalence in terms of an initial object? (For example, the discrete category with object set equal to a set S is initial among all such categories (when we allow only functors that are the identity on S), but that seems to be a bit different than the notions above of transitivity)

Morgan Rogers (he/him) (Apr 07 2020 at 09:37):

My comment was in response to earlier stuff, since no one had mentioned it yet. In a topos, the initial object is always strict (eg the empty set has this property in Set).
As for the automorphisms, the existence of "outer automorphisms" (symmetries of a structure that aren't constructible within the structure) creates problems in all sorts of areas. For groups, the inner automorphisms are those described by conjugation, which amount to the natural transformations on the identity functor when viewing these as categories. The alternating group $A_5$ has a non-trivial outer automorphism (ie a symmetry which is not induced by conjugation). The automorphisms of Dynkin diagrams give outer automorphisms of Lie algebras too, if you've met them before.

Morgan Rogers (he/him) (Apr 07 2020 at 10:10):

As for the Yoneda paradigm... if you want to recover it in your discrete category example, perhaps you could think of it in a more relative way. Each object of a category can only "see" or "interact" with another through morphisms. Each object in a discrete category is aware of itself and none of the others; we can express the situation in a more complicated category in similar terms, all the way up to modelling interactions and relationships in the world in this categorical way (I'm aware of some philosophical attempts to do things like this). If someone applied an outer automorphism to the world, replacing each sentient thing in it with something which, according to all of its own senses, was indistinguishable from its previous state, then by definition no one in the world would be able to tell that anything had changed! Comparisons can be drawn with Star Trek's transporters, or even just the act of sleeping; when you wake up, how do you know you're the "same" person as when you went to bed last night?
What I'm trying to get at is that I (a member of a some category, say) can distinguish "me" from "not me" inside the category where I live (up to isomorphism), and that is the non-trivial distinction identified by the Yoneda paradigm in the discrete case. Sure, everyone says the same thing about everyone else, "I am me, they're not", so they look identical from outside the category, but everyone also agrees that they're not the others!

Morgan Rogers (he/him) (Apr 07 2020 at 10:12):

(What I like the most about making a personifying analogy like this is that the word for the morphism that allows me to distinguish myself from others is my identity morphism :heart_eyes: )

Gershom (Apr 07 2020 at 10:17):

I'm now imagining a chorus of objects, singing in unison, the Kinks' "I'm not like everybody else" :-)

Sam Tenka (Apr 07 2020 at 16:27):

@Morgan Rogers thanks for these explanations and intuitions! Alas, I don't know about dynkin diagrams or the classification of lie groups. But the inner vs outer automorphisms with groups makes sense to me! Yeah, I like that personified perspective: It reminds me of the quip, "a recession is when my neighbor loses their job; a depression is when /I/ lose my job"

John Baez (Apr 07 2020 at 17:26):

Morgan Rogers said:

The alternating group $A_5$ has a non-trivial outer automorphism (ie a symmetry which is not induced by conjugation).

I think you mean $A_6$. $A_5$ doesn't have any nontrivial outer automorphisms. $A_6$ has three.

Morgan Rogers (he/him) (Apr 08 2020 at 09:14):

Finally a case where I don't have to completely concede to you correcting me, John! That Wikipedia page points out that for every $n$ larger than 2, $A_n$ has an outer automorphism given by conjugation in $S_n$ by an odd permutation, so I didn't have to go as high as $n=5$. The case $n = 6$ is more interesting because of the existence of an "exceptional outer automorphism", which is probably the thing I was misremembering to motivate the example in the first place, though :rolling_on_the_floor_laughing:

John Baez (Apr 08 2020 at 15:40):

You're right. I don't think about the alternating groups enough. I mainly remember that the symmetric group $S_n$ has a nontrivial outer automorphism iff $n = 6$. I was so surprised by this when I learned it that I wrote this:

• Some thoughts on the number 6.

It's amazing that the groupoid of 6-element sets behaves in a fundamentally different way than all the rest: the identity functor on this groupoid has a nontrivial natural isomorphism from it to itself!