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A category with just one object is a monoid. A category with two objects (which are distinguished) can be described by the following data (imagine the picture $\stackrel{M}{\curvearrowright} \bullet {\stackrel{X}{\longrightarrow} \atop \stackrel{Y}{\longleftarrow}} \bullet \stackrel{N}{\curvearrowleft}$):

• a monoid $M$ and a monoid $N$
• a $(N,M)$-set $X$ and a $(M,N)$-set $Y$
• a homomorphism $\alpha : X \otimes_M Y \to N$ of $(N,N)$-sets and a homomorphism $\beta : Y \otimes_N X \to M$ of $(M,M)$-sets
• such that under the obvious identifications $\alpha \otimes X = X \otimes \beta$ and $\beta \otimes Y = Y \otimes \alpha$

The same description works for enriched categories, where you would call $X$ and $Y$ bimodules. And by some coincidence, the above description reminds heavily of the definition of dualizable bimodules; but there $\beta$ (or $\alpha$) goes in the other direction. One can easily write down the $1$-morphisms as well as the $2$-morphisms in the category above.

Question. Does this category of "mixed bimodules" have been studied anywhere?

More specifically: How can we use this algebraic category to understand the category of all categories with two objects? What are some specific examples defined by generators and relations? For example, one could write down the free category on two objects $x,y$ such that $x$ is a retract of $y$. Background is Tilman's $\mathbb{Z}[C]$-conjecture, where $C$ may be assumed to be a category with just two objects. (EDIT: Oh, meanwhile Winfried Dreckmann has solved it!?)it! So let's call it the Tilman-Dreckmann-Theorem :-))

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# Categories with two objects, or mixed bimodules

A category with just one object is a monoid. A category with two objects (which are distinguished) can be described by the following data (imagine the picture $\stackrel{M}{\curvearrowright} \bullet {\stackrel{X}{\longrightarrow} \atop \stackrel{Y}{\longleftarrow}} \bullet \stackrel{N}{\curvearrowleft}$):

• a monoid $M$ and a monoid $N$
• a $(N,M)$-set $X$ and a $(M,N)$-set $Y$
• a homomorphism $\alpha : X \otimes_M Y \to N$ of $(N,N)$-sets and a homomorphism $\beta : Y \otimes_N X \to M$ of $(M,M)$-sets
• such that under the obvious identifications $\alpha \otimes X = X \otimes \beta$ and $\beta \otimes Y = Y \otimes \alpha$

The same description works for enriched categories, where you would call $X$ and $Y$ bimodules. And by some coincidence, the above description reminds heavily of the definition of dualizable bimodules; but there $\beta$ (or $\alpha$) goes in the other direction. One can easily write down the $1$-morphisms as well as the $2$-morphisms in the category above.

Question. Does this category of "mixed bimodules" have been studied anywhere?

More specifically: How can we use this algebraic category to understand the category of all categories with two objects? What are some specific examples defined by generators and relations? For example, one could write down the free category on two objects $x,y$ such that $x$ is a retract of $y$. Background is Tilman's $\mathbb{Z}[C]$-conjecture, where $C$ may be assumed to be a category with just two objects. (EDIT: Oh, Winfried Dreckmann has solved it!?)