ind-scheme

In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes.

Examples

$\mathbb{C}P^{\infty} = \varinjlim \mathbb{C}P^N$ is an ind-scheme.
Perhaps the most famous example of an ind-scheme is an infinite grassmannian (which is a quotient of the loop group of an algebraic group G.)

A. Beilinson, Vladimir Drinfel'd, Quantization of Hitchin’s integrable system and Hecke eigensheaves on Hitchin system, preliminary version [http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf] {{Webarchive|url=https://web.archive.org/web/20150105083234/http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf |date=2015-01-05 }}
V.Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, notes of the talk at the `Unity of Mathematics' conference. [http://www.math.uchicago.edu/~mitya/langlands/gelf.pdf Expanded version]
http://ncatlab.org/nlab/show/ind-scheme