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Jun 16, 2014 at 19:16 comment added Todd Trimble Thanks; this is close to the answer I would have given myself. A really good situation to be in is where the underlying category of the monoidal category is regular, and the tensor product preserves colimits in each argument (or countable coproducts and coequalizers if you prefer). Then the category of monoids is also regular, and various notions of epimorphism of monoids (regular, strong, extremal) coincide.
Jun 16, 2014 at 17:51 history edited Qiaochu Yuan CC BY-SA 3.0
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Jun 16, 2014 at 17:45 history edited Qiaochu Yuan CC BY-SA 3.0
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Jun 16, 2014 at 12:38 vote accept Christian Fischmann
Jun 16, 2014 at 6:05 history edited Qiaochu Yuan CC BY-SA 3.0
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Jun 16, 2014 at 5:59 history answered Qiaochu Yuan CC BY-SA 3.0