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The standard way to encode a group as a category is as a "category with one object and all arrows invertible". All of the arrows are group elements, and composition of arrows is the group operation.

A loop obeys similar axioms to a group, but does not impose associativity. Inverses need not exist, but a "cancellation property" exists -- given $xy = z$, and any two of $x$, $y$, and $z$, the third is uniquely determined.

Quasigroups need not even have a neutral element.

Given the lack of associativity, arrows under composition do not work to encode loop elements.

Is there a natural way to do this?

Travis Willse
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wnoise
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    Semigroups are "categories with one object"; for groups, you must also require every arrow to be invertible. – Arturo Magidin Apr 18 '11 at 21:11
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    The lack of associativity is precisely what makes loops not at all like groups. As you say, arrows under composition can't model elements of a loop, so I don't see how this is a natural question to ask. The category-theoretic formalism is inherently associative. – Qiaochu Yuan Apr 18 '11 at 21:30
  • @Arturo Magidin: thanks, fixed. – wnoise Apr 18 '11 at 21:38
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    @Qiaochu Yuan: Lots of people claim that category theory is a good way to talk about any math sub-field. I'm taking that claim seriously. Yes, nothing as nice as the group construction seems to work, so just how unnatural do we need to go? We could, for instance, try encoding the multiplication group of a loop, but there are groups that do not arise as the multiplication group of any loop, and there are groups that are the multiplication group of loops of different size. – wnoise Apr 18 '11 at 21:42
  • Have you read Postmodern Algebra by Smith and Romanowska? I'm not saying it will help but it has quasigroups and loops and category theory in it, but I'm not sure if it combines them. – GeoffDS Apr 18 '11 at 22:12
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    @wnoise: When people say category theory is a good way to talk about most fields, they mean that in most fields the objects of interest have some sort of morphisms between them defined and that these form a category. I don't think people mean that the objects themselves are best defined in categorical terms. – Omar Antolín-Camarena Apr 18 '11 at 22:27
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    @wnoise: you can use category theory to talk about loops in the sense that you can define a category of loops. I just don't see a natural way to express individual loops as categories. – Qiaochu Yuan Apr 18 '11 at 22:27
  • I'd be interested in an "external" view as well: what coherence properties on the morphisms are required to get a category where the objects are loops. But I suppose that'd be a separate question. – wnoise Apr 19 '11 at 19:03
  • @t.b.: why did you eliminate the "quasigroups" and "loops" tags? It's true that the site probably doesn't need both -- but they have aspects which do not appear to be covered under the other tags. – wnoise Sep 22 '11 at 04:29
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    I understand that last complaint. In the course of cleaning up rarely used tags I removed the (loops) tag from this site altogether because there was no consistent use of it -- several people started using it for "loops" as in "loop-space" or "fundamental group". Then I decided to remove the (quasi-groups) tag because there is no universally recognized meaning for it (as is the case with so many quasi-things) and since there were only two or three questions tagged such anyway, so I saw little use for it. – t.b. Sep 27 '11 at 03:44

2 Answers2

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As expressed by Qiaochi Yuan in this and this comment, the way that category theory applies to studying loops and quasigroups is in the form of a category whose objects are loops, resp. quasigroups.

Only a few structures can actually be described as categories having certain special properties (among which sets, groups, partially ordered sets). For a structure to have any chance of being a "special type of category", it is of course necessary that the defining properties for a category are somehow satisfied by the structure in question.

For loops and quasigroups, this is not obvious to say the least, due to the lack of associativity (which is all-important in category theory).

Community
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Lord_Farin
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I know this is an older thread but I've been thinking about basically this question for a few months now

Something I've come up with is that a quasi-group requires certain automorphisms to exist on the categorical product of 3 elements.

An example of this, in the Category of sets the products AxB, CxA and BxC are all isomorphic to subsets of AxBxC which is isomorphic to CxAxB and BxCxA.

These isomorphisms are what allow us to distinguish between a binairy function and it's left and right inverses, and are generalizable withing the context of categorical products (I believe) as the proper morphisms should exist between categorical products of pairs and products of triples.

Once these automorphisms are established, I believe quasi-groups arise naturally from the existence of morphisms between categorical products of pairs and objects.

This is all really sketchy right now, I'm honestly just getting into category theory but I believe the above works?

stosava
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