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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeJul 16th 2012

    I’ve removed this discussion from direct sum:

    OK, so which of these is correct for pointed sets? In particular, given pointed sets AA and BB (where in each we call the basepoint 00), do we want the wedge sum ABA \vee B (the image of the coproduct in the product) or the direct product A×BA \times B (the product, since there are only 22 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?

    Toby Bartels

    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) ….