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

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?

• Perhaps a better term is "locally a semicategory", as "algebraic manifold" already refers to a smooth variety... Jan 4, 2021 at 0:51
• Would you first list what examples you have in mind and which ones you consider as "trivial"?
– YCor
Jan 4, 2021 at 1:26
• 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. Jan 4, 2021 at 2:14
• Jan 4, 2021 at 3:18
• You can edit the question rather than adding info in comments.
– YCor
Jan 4, 2021 at 8:36