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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeJul 16th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI'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? >---[[Toby Bartels]] >[[Mike Shulman|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](http://secure.wikimedia.org/wikipedia/en/wiki/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) ....

   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?

   —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) ….