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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    $\begingroup$ Semigroups are "categories with one object"; for groups, you must also require every arrow to be invertible. $\endgroup$ – Arturo Magidin Apr 18 '11 at 21:11
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    $\begingroup$ 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. $\endgroup$ – Qiaochu Yuan Apr 18 '11 at 21:30
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    $\begingroup$ @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. $\endgroup$ – wnoise Apr 18 '11 at 21:42
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    $\begingroup$ @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. $\endgroup$ – Omar Antolín-Camarena Apr 18 '11 at 22:27
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    $\begingroup$ @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. $\endgroup$ – Qiaochu Yuan Apr 18 '11 at 22:27

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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