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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?

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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
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
@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
@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

1 Answer 1

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).

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