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I want to say that a group object in a category (e.g. a discrete group, topological group, algebraic group...) is the image under a product-preserving functor of the "group object diagram", $D$. One problem with this idea is that this diagram $D$ as a category on its own doesn't have enough structure to make the object labelled $``G\times G"$ really the product of $G$ with itself in $D$.

Is there a category $U$ with a group object $G$ in it such that every group object in every other category $C$ is the image of $G$ under a product-preserving functor $F:U\rightarrow C$, unique up to natural isomorphism?

(It's okay with me if "product-preserving" or "up to natural isomorphism" are replaced by some other appropriate qualifiers, like "limit preserving"...)

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@Sergiy: please stop making minor edits to old posts. It bumps them back up to the front page. –  Qiaochu Yuan Nov 22 '13 at 18:45
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4 Answers

up vote 31 down vote accepted

Yes, the category U is the opposite of the full subcategory of Grp on the free groups on 0, 1, 2, ... generators. This is an instance of Lawvere's theory of "theories". See this nLab entry for a discussion (of this example in fact).

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Awesome, thank-you! –  Andrew Critch Oct 20 '09 at 4:38
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Perhaps this is a good spot to put my standard propaganda that the definition of "group object" is wrong. Recall that having inverses is typically a property not a structure. The problem is that the inverse is typically not a morphism, instead it's an "anti-morphism." This is true anywhere that the notion of anti-morphism makes sense (in particular: noncommutative rings, Poisson manifolds).

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Very interesting. How then does one express "anti-morphism" in a category theoretic way? –  Kevin H. Lin Oct 28 '09 at 13:55
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I disagree with you about the inverse morphism. 1) It's not meant to be an automorphism of the group object, but of the underlying C-object; 2) including it as part of the structure give us "good" working definitions of algebraic groups, topological groups, etc; 3) as defined, an abelian group is precisely a group object in the category of groups, which is cool. I think the lesson here is that a group object is more than a monoid object "with inverses", but one such that inversion is a suitably well-behaved operation. –  Andrew Critch Nov 1 '09 at 8:10
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Except that it doesn't give good working defintions of Poisson-Lie groups or quantum groups. And those are exactly the cases where anti-morphism makes sense. –  Noah Snyder Nov 2 '09 at 0:07
    
Dear Noah Snyder, I agree with you that having inverses is a property. Nevertheless, I think it makes sense to ask that $\text{Hom}_{C}(S, G)$ is a group, if $G$ is a group object in $C$, and $S$ any other object in $C$. Then Yoneda gives a morphism $i \colon G \to G$ in $C$, that is the 'inversion' morphism, right? (I know nothing of Poisson-Lie groups or quantum groups, but I would hope that they satisfy the "$\text{Hom}(S,G)$ is a group"-approach.) More general: how do you formulate the property of having inverses? –  jmc Nov 22 '13 at 10:52
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I would really suggest looking at Steve Awodey's lecture notes on Categorical Logic., found here http://www.andrew.cmu.edu/user/awodey/catlog/notes/ The category you are looking for is called "the theory of groups". I find these notes much more digestible than Lawvere's original papers on the subject.

Essentially, you can form a category in which all objects are products of a single object G, and the only morphisms between them are those morphisms you get out of the basic definition of a group object. Then a group in any category is a product preserving functor from this category. Actually, in this case, it is easy to see that the appropriate category is just the opposite of the full subcategory 1, F(1), F(2), ... , F(n), ... where F(k) is the free group on k generators.

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To elaborate on Reid's answer, there is a general notion of "operad" which provides universal "algebraic" objects in the sense that you want.

One interesting feature of groups: a group object is a product-preserving functor from a certain category. A Hopf algebra object is a monoidal functor from a related category, but where the monoidal structure need not be product. See, if your monoidal structure is Product, then every object of your category is a coassociative counital coalgebra in a unique way, and so half of the Hopf axioms are trivial.

Why does this matter? There are no interesting group objects in Vect. Indeed, there are no interesting (unital) monoid objects, because the product in Vect is also a coproduct. A monoid object is an object V and maps e : {pt} → V and m : V x VV, unital and associative. Well, in Vect x is ⊕, and {pt} is 0. There is only one linear map from 0 to V, so we know what e is. If the monoid is unital, then m(v \oplus 0) = v = m(0 \oplus v), and linearity of m takes care of the rest.

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I don't think operads are quite relevant here, as there's no operad for the theory of groups. (Or to put it slightly less precisely, there's no operad whose algebras are groups.) Operads are only capable of encoding a rather restricted class of algebraic theories. –  Tom Leinster Oct 20 '09 at 17:46
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