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1) whether it has terminal objects

and

2) how can a limit be created to define them

?

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- Thread starter icantadd
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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, whether it has terminal objects or not? I'm afraid there isn't a way to identify all objects with some property in an arbitrary category. Categories are just too disparate for that to ever be the case.f

- #1

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1) whether it has terminal objects

and

2) how can a limit be created to define them

?

- #2

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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.

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

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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.

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?

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)

?

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What is an 'isomorphic functor'?

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