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Notions of topos

Published online by Cambridge University Press:  17 April 2009

Ross Street
School of Mathematics and Physics, Macquarie University, North Ryde, New South Wales 2113, Australia.
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A Grothendieck topos has the property that its Yoneda embedding has a left-exact left adjoint. A category with the latter property is called lex-total. It is proved here that every lex-total category is equivalent to its category of canonical sheaves. An unpublished proof due to Peter Freyd is extended slightly to yield that a lex-total category, which has a set of objects of cardinality at most that of the universe such that each object in the category is a quotient of an object from that set, is necessarily a Grothendieck topos.

Research Article
Copyright © Australian Mathematical Society 1981


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