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Categories with exactly one object are in 1:1 correspondence with the well-known algebraic structures called monoids.

Is there a similar correspondence for categories with exactly two objects? Are there genuinely algebraic structures they are in 1:1 correspondence with?

What about categories with exactly $n$ objects?

What about categories with countably many ($\omega$) objects?

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You are asking for other, algebraic names of structures which are defined and recognized most easily by viewing them as categories. I believe there are none such names, but I don't know much. The thing with monoids is that –- since they consist of only one object -- the binary operation isn't partial. This makes them easy to describe as common algebraic structures. You can of course go on to view groups as categories with exactly one object and only isos etc. – k.stm Oct 3 '12 at 14:13
I sometimes like to call categories with two objects "bioids." – Noah Snyder Oct 3 '12 at 14:49
@Noah: Honestly? – Hans Stricker Oct 3 '12 at 16:55
Semi-honestly. Mostly I do that up a categorical dimension. A monoidal category is a 2-category with one object, but I do a lot of work on 2-categories with 2-objects, and often in talks or informal settings I'll call them bioidal categories. – Noah Snyder Oct 3 '12 at 17:05
I have asked the same question here:… – Martin Brandenburg Oct 6 '12 at 13:32
up vote 7 down vote accepted

Categories with 2 objects: So called Morita contexts in the bicategory of monoids and biacts. If we have 2 objects, say $X$ and $Y$, then we get 2 (endomorphism-)monoids, say $M:=End(X)$ and $N:=End(Y)$, and they act on $\hom(X,Y)$ and $\hom(Y,X)$ on the proper sides (left or right), so that $\hom(X,Y)$ becomes an $M-N$ bimodule and $\hom(Y,X)$ an $N-M$ bimodule, and the associativity is giving an extra connection between them.

These are analogous to the generalized matrix rings $\begin{pmatrix} R&M\\N&S\end{pmatrix}$ where $R$ and $S$ are rings and ${}_RM_S$ and ${}_SN_R$ bimodules equipped with 'products' $M\otimes N\to R$ and $N\otimes M\to S$..

It is also possible to generalize Morita contexts from $2$ objects to $n$ (generalized $n\times n$ matrix rings), but I'm not sure if that has a (different) name.

For more details, see for example my paper.

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