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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.)

formal scheme

References

A. Beilinson, Vladimir Drinfel'd, Quantization of Hitchin’s integrable system and Hecke eigensheaves on Hitchin system, preliminary version [1]
V.Drinfeld, Infinite-dimensional vector bundles in algebraic geometry, notes of the talk at the `Unity of Mathematics' conference. Expanded version
http://ncatlab.org/nlab/show/ind-scheme