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The book by Adámek–Rosický–Vitale uses a different definition of an algebraic category: an algebraic category is a category equivalent to the category of algebras over an algebraic theory, i.e., the category of functors T→Set that preserve finite products, where T is a small category with finite products.
This definition seems to be much more widely used these days than the older definition of Adámek–Herrlich–Strecker.
Should we adjust the article accordingly? Is the older definition of Adámek–Herrlich–Strecker actually used anywhere other than in their book?
I wonder if it is possible to generalize from algebraic categories to algebraic (infinity,1)-categories and thus whether it is possible to talk about trivial algebras in algebraic (infinity,1)-categories, such as trivial H-spaces and trivial A-infinity spaces.
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