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, 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.
^Schreiber, Urs (2011-07-25). "Bohr toposes". The n-Category Café. Retrieved 2018-02-19.
^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.
subject, a Bohr topos is a ringedtopos that plays the role of a quantum phase space. The definition of a topos-version of a "locally ringed space" is not...
small limits. A ringedtopos is a pair (X,R), where X is a topos and R is a commutative ring object in X. Most of the constructions of ringed spaces go through...
_{\infty }} -ring can be associated with a spectrally ringed site. This spectrally ringedtopos can be a spectral scheme if the spectrum of this ring gives an...
topos for objects of a topos is the topos of presheaves over the opposite of the category of finite sets. The classifying topos for rings of a topos is...
framework to Tsallis entropies. The information cohomology is an example of ringedtopos. Multivariate k-Mutual information appear in coboundaries expressions...
generally a non-commutative ring. This can be generalized to the case of sheaves of modules over a ringed space, or ringedtopos. In Grothendieck's duality...
ISBN 978-1119656029. OCLC 1183854105. Hooker, Mark T. (2006). "The Feigned-manuscript Topos". A Tolkienian Mathomium: a collection of articles on J. R. R. Tolkien and...
the American Mathematical Society. Myles Tierney, On the Spectrum of a RingedTopos, Algebra, Topology and Category Theory, (1976) doi:10.1016/B978-0-12-339050-9...
University of Kansas Publications in Humanistic Studies. Online version at the Topos Text Project. Gaius Valerius Flaccus, Argonautica translated by Mozley,...
called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundation of mathematics. A topos can also be considered...
closed point x of X we can consider the local ring Rx at this point, which is a discrete valuation ring whose spectrum has one closed point and one open...
categorified to say that the category of sets (or more generally, any topos) is a 2-rig. Ring of sets – Family closed under unions and relative complements Valuation...
X : Δop → Set) is Cartesian closed. Even more generally, every elementary topos is Cartesian closed. In algebraic topology, Cartesian closed categories...
non-standard NNOs; for example, if z = s z, in which case the category or topos E is trivial.) Freyd showed that z and s form a coproduct diagram for NNOs;...
ones. However, in the case of rings, "epimorphism" is often considered as a synonym of "surjection", although there are ring epimorphisms that are not surjective...
of Oxford TallCat, a research group at Tallinn University of Technology Topos Institute Cybercat Institute Software: DisCoPy, a Python toolkit for computing...
Saunders (1992). Sheaves in geometry and logic : a first introduction to topos theory. New York: Springer-Verlag. p. 13. ISBN 0387977104. nLab, a wiki...
This is nothing but the category of presheaves on Δ. As such, it is a topos. The simplex category Δ is generated by two particularly important families...
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