Re: What is the right abstract definition of "connected"?

["Marta Bunge" <martabunge@hotmail.com>]

Dear Vaughan

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An object of a category is *connected* when its every morphism to a
nonempty coproduct factors through an inclusion thereof.
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Your proposed definition above is precisely the notion of *abstractly unary*
from my J.Algebra '69 paper. It was so termed (instead of *connected*) since
it does not need a terminal object to state it (precisely your motivation)
and since one does not want to restrict to binary coproducts.

When there is a terminal object, and when the coproducts considered are just
the binary ones, it is enough to consider morphisms into the coproducts 1+1
(as I show in my thesis) and, in that case, it should be simply called
*connected*. In another guise, this is the definition of *connected* given
in Cats and Alligators, and it is the one directly inspired by topology. I
see no reason to change the terminology.

In short, your connected objects I have called abstractly unary. They came
about in connection with atoms. An object A in a cocomplete (concrete)
category E is an *atom* if HOM(A,-):E--->Set preserves colimits. More
objectively, if E has exponentiation, Lawvere uses the notion of an *A.T.O.*
instead, meaning that the functor (-)^A : E---> E has a right adjoint (the
"amazing right adjoint").

I hope this helps,
Cordially,
Marta