Terminal Objects: Defining & Identifying

[icantadd]

This is building off a prior post, but I feel I have a problem is of the nature: how can one tell, given an arbitrary category

1) whether it has terminal objects
and
2) how can a limit be created to define them

?

 

[matt grime]

1) You think about it, I'm afraid. There isn't a way that works to identify all objects with some property in an arbitrary category. Categories are just too disparate for that to ever be the case (Waits for someone to provide some counter examples to that blanket assertion).

2) Why would you want to?

Perhaps if you gave some examples of categories you're interested in, then you'll be able to get examples, and spot the 'kind of thing' that is terminal. E.G. in sets it's anyone point set. In abelian category (I think) there is a unique terminal object (and it's the unique initial object too), 0. All abelian categories are, essentially, modules for some ring.

Very loosely speaking: a/the terminal object is something that is a 'quotient' of every object in you category, and an initial object is something that is a subobject of everything. That might help you decide what they are.

 

[icantadd]

I think I may be in a situation where I know what my question is, but I can't get words on it. My understanding is that terminal objects are unique, in any category that contains them, with the condition that any member maps to them. My actual problem is related to an artificial intelligence idea called conceptual blending. Suppose I have a categorical specification of 'concepts'. Then I have categories of concepts, with functors from limits in one to not necessarily limits in another. But those functions mapped to will all map in some way to a terminal object, if it exists. Terminal objects in this sense are of utmost importance because they 'define the concept'. I tend to agree that all objects would be indescernible, and in that I have no interest. My interest is in: is there any algorithm, by which I can look at a category, and infer that it has a terminal object based on some clues (it can be a heuristic) without actually checking for the confluence to the formal definition.

 

[matt grime]

The categorical definition of unique is somewhat strict. Take SET: *any* one element set is terminal. Now that's a proper class of clearly 'unequal' objects. But categorically they are all isomorphic by a unique isomorphism - this is stronger even than merely being in the same isomorphism class. Some times it is nice to remember that an isomorphism class is really a class in many cases.

 

[icantadd]

So it is not that a terminal object is unique in the sense that there exists only one, or that there are many that are all equivalent, but that any terminal object has an isomorphism to any other, by virtue that it is terminal. The isomorphism allows them to be treated as the same, even though they may be separable if one were to peer at the objects. ??

I have not seen the word class or isomorphism class, what does that mean?

 

[matt grime]

An isomorphism class is a collection of (all) isomorphic objects. It is called a class, because it is in general too large to be a set.

What do you mean by 'the same'?

You should understand the notion of unique in a category:

Defn: Given some set of properties, we say X is the unique object in a category with those properties if for any Y with the same properties there is a _unique_ (in the normal sense, i.e. exactly one) isomorphism X->Y.

There will in general be many objects satisfying some set of properties. Often these will not be uniquely determined.

But terminal objects are unique. Why? Suppose T is terminal, then there is a unique map from T to T, which is then necessarily the identity. Now suppose S is another terminal object. Then there are unique morphisms S->T and T->S, all that is required is to show that they are isomorphisms. But this is trivial since the composition S->T->S must be the identity (the unique map from S to S), and similarly T->S->T is the identity on T.

 

[icantadd]

Okay, so

?

Let C and 1 be categories. C is any category, 1 is the category with exactly one object. If there is an isomorphic functor F from C to 1, then there should be an adjoint with unit natural transformation between F and F^(-1) where N : ID_1 -> F^(-1) . F and a co-unit natural transformation where E : F.F^(-1) -> ID_C. In this diagram, the image of 1 would be projected into ID_C by the co-unit acting as an initial object. Thus if I took the dual by reversing the arrows, I would get a terminal object (if the reversed arrows still maintain isomorphism)

?

 

[matt grime]

What is an 'isomorphic functor'?

 

[icantadd]

Ah, perhaps I am wrong. I meant a functor that had an inverse that followed the isomorphic rule, so that F^(-1).F = ID_C and F.F^(-1) = ID_1. Then I could have natural transformations from ID_1 and ID_C to the functors between the two categories. Now I am confused. What I was trying to do is figure out a way to turn identifying an initial object and by dual a terminal object into a problem stated in terms of using adjoint functors. And to do so I would need two natural transformations a unit and co-unit, and two functors one from Category C to Category 1 and one from Category 1 to Category C. Thus I thought of the two categories being isomorphic about a functor, which would give me a functor from 1 -> C and from C -> 1, so that the translation was simple, and the image of 1 in C was guaranteed to be initial if there was an isomorphic functor between 1 and C.