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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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    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.2011-09-27

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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).