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I’ve removed this discussion from direct sum:
OK, so which of these is correct for pointed sets? In particular, given pointed sets $A$ and $B$ (where in each we call the basepoint $0$), do we want the wedge sum $A \vee B$ (the image of the coproduct in the product) or the direct product $A \times B$ (the product, since there are only $2$ sets)? I can turn either into a general definition above, but which is the right one?
Or should we use both? There are, after all, two names: ‘direct sum’ and ‘weak direct product’; I naturally use these for the image-of-coproduct and almost-zero-product versions, respectively. Is there already a convention in universal algebra? If not, do people like this terminology? Or do you know that one is definitely what is wanted while the other is useless?
Mike: I’ve only ever heard “direct sum” used to mean “coproduct,” or sometimes “finite biproduct,” in an additive category.
Toby: Well, it’s definitely also used in the sense of direct sum of groups; the same concept is also called ‘weak direct product’. I thought that I once knew how this worked in general, from a universal-algebra perspective, but when I started writing this, I found that I had forgotten (or never really understood) ….
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