Let $S$ be a set and $\cdot$ a partial binary operation on $S$. A subset $F\subseteq S$ is $\cdot$-closed if the following condition holds:

  • for all $f,g\in F$, if $(f,g)\in\mathrm{dom}(\cdot)$, then $f\cdot g\in F$.

Fact: the intersection of two closed subsets of $S$ is closed.

A coordinate map for $S$ is a pair $(U,\varphi,G)$ such that $U\subseteq S$ is a closed non-empty subset, $G=(G,\circ)$ is a semicategory, and $\varphi\colon U\rightarrow \mathrm{mor}(G)$ is a bijection such that $(u,v)\in \mathrm{dom}(\cdot)$ iff $(\varphi(u),\varphi(v))\in\circ$ and $(u,v)\in\mathrm{dom}(\cdot)$ implies $\varphi(u\cdot v)=\varphi(u)\circ\varphi(v)$ for all $u,v\in U$.

An atlas for $(S,\cdot)$ is a collection of coordinate maps $(U_\alpha,\varphi_\alpha,G_\alpha)$ such that $S = \cup_\alpha U_\alpha$, $\alpha\in I$ an index set.

Fact: the transition maps $\tau_{\alpha,\beta}\colon \varphi_\alpha(U_\alpha \cap U_\beta)\rightarrow \varphi_\beta(U_\alpha \cap U_\beta)$, $\tau_{\alpha,\beta}=\varphi_\beta\circ\varphi_\alpha^{-1}$ are semigroupoid $\circ$-isomorphisms, i.e. a composability- and composition-preserving correspondence of the morphisms.

The pair $(S,\cdot)$ together with an atlas is an algebraic manifold.

Has such an "algebraic manifold" been defined before? Are there any non-trivial examples?

  • 5
    $\begingroup$ Perhaps a better term is "locally a semicategory", as "algebraic manifold" already refers to a smooth variety... $\endgroup$ Jan 4, 2021 at 0:51
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    $\begingroup$ Would you first list what examples you have in mind and which ones you consider as "trivial"? $\endgroup$
    – YCor
    Jan 4, 2021 at 1:26
  • $\begingroup$ Hi Kevin, I'll consider "locally a semicategory" for future questions. @YCor I will leave "trivial" up to the reader to define. I mean a non-semicategory. For example the object set and binary relation $(\{a,b,c,d,e\},\{((a,b),c),((d,e),c)\})$ has an atlas with two charts as each of the closed subsets $\{a,b,c\}$ and$\{c,d,e\}$ is isomorphic to the semicategory $(\{a,b,c\},\{((a,b),c)\})$. However it's also a semicategory in its own right, hence trivial. It is possible that with these conditions, $S$ is always a semicategory, i.e. the morphisms of some semicategory. $\endgroup$ Jan 4, 2021 at 2:14
  • $\begingroup$ cf ncatlab.org/nlab/show/pseudogroup#definition $\endgroup$
    – David Roberts
    Jan 4, 2021 at 3:18
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    $\begingroup$ You can edit the question rather than adding info in comments. $\endgroup$
    – YCor
    Jan 4, 2021 at 8:36


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