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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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closed as not a real question by Franz Lemmermeyer, Ryan Budney, Dan Petersen, David Roberts, Harry Gindi Jun 21 '11 at 8:17

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

    
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 '11 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 '11 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 '11 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". –  JBorger Jun 21 '11 at 7:44
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