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EDIT: Are there references to the literature that works out dervied functors for categories enriched over abelian monoids? (I narrowed down the question in the hope for an answer.)

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In my book, semigroups have associative multiplication (or let us say addition in the abelian case). Since he assumes a neutral element, I assume he means commutative monoid, in which case I find this question to be a very natural one. If Colin doesn't mean to include associtativity as an axiom, I find the question far less natural. – Todd Trimble Sep 18 '11 at 15:07
    
(Semigroups -abelian or not- are defined to be associative!) – Qfwfq Sep 18 '11 at 15:10
    
Yeah, I was confused because Colin forgot to list associativity as one of the axioms (unless he really meant the non-associative analogue!). – Harry Gindi Sep 18 '11 at 15:20
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(In case my initial comment seems confusing, it was in response to a comment of Harry which he later deleted.) – Todd Trimble Sep 18 '11 at 20:23
    
yes. i do mean associative. usually i find the nomenclature "monoid" confusing. i find the analogy between ring and semiring, and group and semigroup. – user2529 Sep 19 '11 at 2:47

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