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.[1][2]

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, Exercice 13.9), which is equivalent to saying that all the stalks of the structure ring object are local rings when there are enough points.

Morphisms

A morphism of ringed topoi is a pair consisting of a topos morphism and a ring homomorphism .

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

Notes

  1. ↑ Schreiber, Urs (2011-07-25). "Bohr toposes". The n-Category Café. Retrieved 2018-02-19.
  2. ↑ Heunen, Chris; Landsman, Nicolaas P.; Spitters, Bas (2009-10-01). "A Topos for Algebraic Quantum Theory". Communications in Mathematical Physics. 291 (1): 63–110. arXiv:0709.4364. Bibcode:2009CMaPh.291...63H. doi:10.1007/s00220-009-0865-6. ISSN 0010-3616.

References

Further reading

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