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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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    @Martin: thank you, I didn't know.2012-10-07

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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.