Ringed topos

In mathematics, a ringed topos is a generalization of a ringed space; that is, the notion is obtained by replacing a "topological space" by a "topos". The notion of a ringed topos has applications to deformation theory in algebraic geometry (cf. cotangent complex) and the mathematical foundation of quantum mechanics. In the latter subject, a Bohr topos is a ringed topos that plays the role of a quantum phase space.

The definition of a topos-version of a "locally ringed space" is not straightforward, as the meaning of "local" in this context is not obvious. One can introduce the notion of a locally ringed topos by introducing a sort of geometric conditions of local rings (see SGA4, Exposé IV, Exercise 13.9), which is equivalent to saying that all the stalks of the structure ring object are local rings when there are enough points.

A morphism ${\displaystyle (T,{\mathcal {O}}_{T})\to (T',{\mathcal {O}}_{T'})}$ of ringed topoi is a pair consisting of a topos morphism ${\displaystyle f:T\to T'}$ and a ring homomorphism ${\displaystyle {\mathcal {O}}_{T'}\to f_{*}{\mathcal {O}}_{T}}$.

If one replaces a "topos" by an ∞-topos, then one gets the notion of a ringed ∞-topos.

One of the key motivating examples of a ringed topos comes from topology. Consider the site ${\displaystyle {\text{Open}}(X)}$ of a topological space ${\displaystyle X}$, and the sheaf of continuous functions

${\displaystyle C_{X}^{0}:{\text{Open}}(X)^{op}\to {\text{CRing}}}$

sending an object ${\displaystyle U\in {\text{Open}}(X)}$, an open subset of ${\displaystyle X}$, to the ring of continuous functions ${\displaystyle C_{X}^{0}(U)}$ on ${\displaystyle U}$. Then, the pair ${\displaystyle ({\text{Sh}}({\text{Open}}(X)),C_{X}^{0})}$ forms a ringed topos. Note this can be generalized to any ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$ where

${\displaystyle {\mathcal {O}}_{X}:{\text{Open}}(X)^{op}\to {\text{Rings}}}$