# Gelfand spectra and Wallman compactifications

@article{Caramello2012GelfandSA, title={Gelfand spectra and Wallman compactifications}, author={Olivia Caramello}, journal={arXiv: Category Theory}, year={2012} }

We carry out a systematic, topos-theoretically inspired, investigation of Wallman compactifications with a particular emphasis on their relations with Gelfand spectra and Stone-Cech compactifications. In addition to proving several specific results about Wallman bases and maximal spectra of distributive lattices, we establish a general framework for functorializing the representation of a topological space as the maximal spectrum of a Wallman base for it, which allows to generate different…

## 2 Citations

### Topos-theoretic background

- Mathematics
- 2018

ing these two fundamental ingredients in the construction of categories of sheaves is precisely what led Grothendieck to introduce the notion of site reviewed in the last section. Sheaves on a…

## References

SHOWING 1-10 OF 27 REFERENCES

### The spectral theory of commutative C*-algebras: The constructive spectrum

- Mathematics
- 2000

This paper introduces the notion of a commutative C*-algebra in a Grothendieck topos E and subsequently that of the spectrum MFn A of A, presented as the locale determined by an appropriate…

### The spectral theory of commutative C*-algebras: The constructive Gelfand-Mazur theorem

- Mathematics
- 2000

It is shown, for a commutative C*-algebra in any Grothendieck topos E, that the locale MFn A of multiplicative linear functionals on A is isomorphic to the locale Max A of maximal ideals of A,…

### A topos-theoretic approach to Stone-type dualities

- Mathematics
- 2011

We present an abstract unifying framework for interpreting Stone-type dualities; several known dualities are seen to be instances of just one topos-theoretic phenomenon, and new dualities are…

### Alexandroff algebras and complete regularity

- Mathematics
- 1979

We characterize lattice theoretically the topological notion of complete regularity and study the implications of this characterization in the setting of local lattices (complete distributive…

### Sheaves in geometry and logic: a first introduction to topos theory

- Mathematics
- 1992

This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various…

### On Rings of Continuous Functions on Topological Spaces

- Mathematics
- 1991

This paper is related to studies byM.H.Stone [2] and to the above paper by G.E. Shilov. In contrast to the latter, we consider the ring of continuous functions on a topological space as a purely…

### A General Theory of Spectra. I: I.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1940

The mathematical theory of spectra deals with the characteristic value Vroblem (Eigefnwertproblem) for linear operators, and provides a general unifying treatment for typical instances of the problem…

### Rings of continuous functions

- Mathematics
- 1960

Contents: Functions of a Topological Space.- Ideals and Z-Filters.- Completely Regular Spaces.- Fixed Ideals. Compact Spaces.- Ordered Residue Class Rings.- The Stone-Czech Compactification.-…