nLab
structure sheaf

A Grothendieck topos or more generally a Grothendieck-Rezk-Lurie (∞,1)-topos is a collection of sheaves or more generally of (∞,1)-sheaves on a topological space or, more generally, on some site.

Singling out one of these sheaves as that sheaf which deserves to be regarded as a sheaf of functions on the underlying space is a method of equipping that underlying space with extra structure: for instance for sheaves on a topological space, the chosen sheaf may be smaller than the sheaf of all continuous functions $U\to ℝ$. For instance if $X$ has the extra structure of a manifold, the chosen sheaf may be the sheaf of all smooth functions.

This way singling out a certainm sheaf from all sheaves is a way to encode extra structure. Accordingly such chosen sheafs are called structure sheaf .

A grand formalization of what a structure sheaf is is given by the notion of a structured (∞,1)-topos, which subsumes the notion of generalized scheme:

there, for $T$ an (∞,1)-topos a structure sheaf on $T$ is a

• a geometry (for structured (∞,1)-toposes) $G$;

• and an (∞,1)-functor $G\to T$

  • that preserves the singled out morphisms in $G$ (as described at structured (∞,1)-topos).

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