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Lawvere–Tierney sheaves in Algebraic Set Theory

Published online by Cambridge University Press:  12 March 2014

S. Awodey
Affiliation:
Department of Philosophy, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pa 15213, USA, E-mail: awodey@cmu.edu
N. Gambino
Affiliation:
Department of Computer Science, University of Leicester, University Road, Leicester Lei 7Rh, UK, E-mail: nicola.gambino@gmail.com
P. L. Lumsdaine
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pa 15213, USA, E-mail: plumsdai@andrew.cmu.edu
M. A. Warren
Affiliation:
Department of Philosophy, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pa 15213, USA, E-mail: mwarren@andrew.cmu.edu

Abstract

We present a solution to the problem of denning a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2009

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