Let $A$ be a $ \mathbb{C}$-algebra. Then we have the following increasing union: $$ {\rm GL}_1(A) \subseteq {\rm GL}_2(A) \subseteq {\rm GL}_3(A) \subseteq \dots \subset {\rm GL}(A) $$ We call $ {\rm GL}(A)$ the stable general linear group over $A$. The stable general linear groups together form a functor $$ {\rm GL} : \text{\( \mathbb{C} \)-algebras} \to \text{Groups} $$ What if anything can be said about $ {\rm GL}$? I am pretty sure it is not representable. Is it even a sheaf for any reasonable topology?
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4$\begingroup$ I think it is an ind-group, i.e., a group object in the category of ind-schemes, although I might be overlooking some subtlety. You can read about such things in Kumar's book Kac-Moody groups, their flag varieties and representations, or depending on your taste, in SGA 4. $\endgroup$– Dave AndersonSep 23, 2014 at 1:22
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It's an ind-affine ind-algebraic group - the usual choice of increasing union gives a diagram of closed embeddings of affine algebraic groups. It is a sheaf in any topology where $GL_n$ is a sheaf, e.g., fpqc, fppf, \'etale, Zariski. You can define a topological coordinate ring as an inverse limit of the coordinate rings of the groups in the diagram.