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?



The stable general linear group in algebraic geometry
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.


