Abstract
We examine an inverse semigroup T in terms of the universal locally constant covering of its classifying topos 
. In particular, we prove that the fundamental group of 
coincides with the maximum group image of T. We explain the connection between E-unitary inverse semigroups and locally decidable toposes, characterize E-unitary inverse semigroups in terms of a kind of geometric morphism called a spread, characterize F-inverse semigroups, and interpret McAlister’s “P-theorem” in terms of the universal covering.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barr, M., Diaconescu, R.: On locally simply connected toposes and their fundamental groups. Cahiers de Topologie Géom. Différentiele Catég. 22(3), 301–314 (1981)
Barr, M., Paré, R.: Molecular toposes. J. Pure Appl. Algebra 17, 127–152 (1980)
Borceux, F., Janelidze, G.: Galois Theories. Vol. 72 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2001)
Bunge, M.: Galois groupoids and covering morphisms in topos theory. In: Proceedings of the Fields Institute: Workshop on Descent, Galois Theory and Hopf algebras, vol. 43, pp. 131–162. Fields Institute Communications, American Mathematical Society (2004)
Bunge, M., Funk, J.: Quasicomponents in topos theory: the hyperpure, complete spread factorization. Math. Proc. Cambridge Philos. Soc. 141(3), (2006)
Bunge, M., Funk, J.: Singular Coverings of Toposes, Vol. 1980 of Lecture Notes in Mathematics. Springer-Verlag, Heidelberg (2007)
Fox, R.H.: Covering spaces with singularities. In: Fox, R. H. et al. (eds.) Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz, pp. 243–257. Princeton University Press, Princeton (1957)
Funk, J.: Semigroups and toposes. Semigroup Forum 75(3), 480–519 (2007)
Hines, P.: The category theory of self-similarity. Theory Appl. Categ. 6(3), 33–46 (1999)
James, H., Lawson, M.V.: An application of groupoids of fractions to inverse semigroups. Period. Math. Hungar. 38(1–2), 43–54 (1999)
Johnstone, P.T.: Topos Theory. Academic Press, Inc., London (1977)
Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)
Johnstone, P.T.: Sketches of an Elephant: A Topos Theory Compendium. Clarendon Press, Oxford (2002)
Joyal, A., Tierney, M.: An Extension of the Galois Theory of Grothendieck, vol. Memoirs of the American Mathematical Society 309. American Mathematical Society (1984)
Kock, A., Moerdijk, I.: Presentations of étendues. Cahiers de Topologie Géom. Différentiele Catég. 32(2), 145–164 (1991)
Lausch, H.: Cohomology of inverse semigroups. J. Algebra 35, 273–303 (1974)
Lawson, M., Steinberg, B.: Etendues and ordered groupoids. Cahiers de Topologie Géom. Différentiele Catég. 40(2), 127–140 (2002)
Lawson, M.V.: Inverse Semigroups: The Theory of Partial Symmetries. World Scientific Publishing Co., Singapore (1998)
Loganathan, M.: Cohomology of inverse semigroups. J. Algebra 70, 375–393 (1981)
S. Mac Lane, Moerdijk, I.: Sheaves in Geometry and Logic. Springer-Verlag, Berlin (1992)
Munn, W.D.: A note on E-unitary inverse semigroups. Bull. London Math. Soc. 8, 71–76 (1976)
O’Carroll, L.: Inverse semigroups as extensions of semilattices. Glasgow Math. J. 16(1), 12–21 (1975)
O’Carroll, L.: Idempotent determined congruences on inverse semigroups. Semigroup Forum 12(3), 233–243 (1976)
Wraith, G.: Lectures on elementary topoi. In Model theory and topoi, vol. 445, pp. 114–206. Springer-Verlag, New York (1975)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Funk, J., Steinberg, B. The Universal Covering of an Inverse Semigroup. Appl Categor Struct 18, 135–163 (2010). https://doi.org/10.1007/s10485-008-9147-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-008-9147-2