Category theory in general and topos theory in particular provide a basis for a foundation of mathematics that is not only closer to mathematical practice than the one based on (classical) set theory, but that is also unifying, constructive, and provides new insights. Toposes bounded over a base S (the "set theory") may be thought of as generalized topological spaces. My work, of a foundational nature, centers on investigations in areas lof mathematics, such as functional analysis, algebraic topology, differential topology, where topological spaces are involved. The main topics of my proposed research for the next five years are: to deal with several problems left open in the research monograph (written in collaboration with J. Funk) "Singular Coverings of Toposes", LNM 1890, Springer, 2006; to clarify the nature of higher stacks for the intrinsic topology of a topos (in collaboration with C. Hermida); to pursue investigations into the fundamental groupoid of a general topos and Galolis theory, and to continue the study of smooth mappings using Penon infinitesimals (initiated in joint work with F. Gago-Couso).

  • "Extensive 2-categories and Top". February 5, 2002.
  • "A Van Kampen theorem for toposes and applications" February 12, 2002.
  • "Locally constant objcets in a Grothendieck topos". February 19, 2002.
  • "Stack completions rfevisited" October 29, 2002.
  • "Johnstone's theorem and Fox completions" March 2, 2004
  • "Michael cov erings are comprehensive" April 4, 2006
  • "Locally discrete locales" January 23, 2007