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## What is an algebraic object? [closed]

The question is in the title. In order to give a bit more backround about the question, one knows that their are several different notions of an algebraic object. One approach is that of Lavere and his algebraic theories. One may also talk about monads. Another approach is an operadic/prop type approach. Yet another way to define an algebraic object is to use sketches of various types. Their may be other approaches that have not been enumerated. My understanding is that all of the things we would like to consider algebraic are not covered by one of these approaches. So this is where the question in the title comes from. Is their a context where all of the things we would like to consider algebraic fit into the aforementioned context? (of course I may be mistaken)

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 I think you are mistaken, primarily because you intend to incorporate a vast array of perspectives on something and call the result an algebraic object. The closest I come to such a thing (in my limited experience) is an algebraic structure or universal algebra. While that is general enough to satisfy me, there is putting it in several different contexts: as a building block, a member of a class of other objects, a ground for interpretation of or into other objects, and so on. I haven't mentioned category theory yet. Gerhard "Slices, Dices, Makes Julienne Fries" Paseman, 2011.06.20 – Gerhard Paseman Jun 21 2011 at 4:45 I imagined the possibility that I might be trying to put too many concepts in a single context. Still it is a bit unsatisfying. For example Hopf algebras and certain generalizations of Drinfeld are considered to be "algebraic" (I'm not sure they form a universal algebra??). – no-1 Jun 21 2011 at 4:54 It seems like you haven't really motivated your question. What's the purpose of having a definition of algebraic object -- does it help accomplish something? – Ryan Budney Jun 21 2011 at 6:27 In my opinion, there is nothing wrong with this question, but it is unlikely that there is such a context. The word 'algebra' generally refers to finite syntactic manipulation of symbols, but it is used formally in some contexts but is used informally in many others, and often with different meanings. In "algebra vs analysis", it suggests one thing, but in "algebra vs coalgebra", as you point out, it suggests something much more specific. It is the same with "geometry", which can mean something as vague as "visualizable" or as precise as "equipped with a Riemannian metric". – James Borger Jun 21 2011 at 7:44