Well, take the positive integers () with addition as your semigroup. Then integer linear combinations of these form a ring:

Hm, the latex rendering there doesn’t seem to make a clear distinction between bold (ring generator) and non-bold (coefficient) numbers. I trust you can figure it out from context.

]]>Sorry, you’re correct here. I’m sort of busy on other things these days and not able to give it the attention it deserves.

]]>I’m not sure that I understand, forgive me for asking again. A coproduct in \mathbf{Grp} is the same as a pushout in \mathbf{Grp} over the trivial group \mathbf{0}, and a pushout in \mathbf{Grp} over the group G is the same as a coproduct in the comma category (G\downarrow\mathbf{Grp}).

In this sense they are indeed the same thing. But as I understood it the amalgamated free group of A and B over C is still only a coproduct of A and B *in the comma category*, but it is not a coproduct of A and B in the category \mathbf{Grp}, since the universal property of the coproduct in \mathbf{Grp} does not account for the identification of the two copies of C.

Is it a terminological convention to use pushout and coproduct interchangeably in categories with an initial object, and elide the switch to/from the comma category? I’m probably missing something again, so thank you for your patience.

]]>They’re the same thing.

]]>Thanks for writing this blog, I enjoy reading it very much. Sadly it’ll be a while before I catch up with the present.

]]>I missed that part. I’ll be over there in the corner, looking ashamed ðŸ™‚

]]>Simone did say at the end that “Clearly you have to suppose “.

]]>It’s been 11 years, but Is it too late to be pedantic and point out that the ONLY IF part fails for q = 0?

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