topos axioms

Definition 0.1   The two axioms that define an elementary topos, or a standard topos, as a special category $ \tau$ are:

To complete the axiomatic definition of topoi, one needs to add the ETAC axioms which allow one to define a category as an interpretation of ETAC. The above axioms imply that any topos has finite colimits, a subobject classifier (such as a Heyting logic algebra), as well as several other properties.

Alternative definitions of topoi have also been proposed, such as:

Definition 0.2   A topos is a category $ \tau$ subject to the following axioms:

One can show that axioms i. and ii. also imply axioms $ \mathbb{T}_1$ and $ \mathbb{T}_2$ ; one notes that property $ \mathbb{T}_2$ can also be expressed as the existence of a representable subobject functor.

Bibliography

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R.J. Wood. 2004. Ordered Sets via Adjunctions, in Categorical Foundations.,

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M. C. Pedicchio and W. Tholen, Eds. 2000. Cambridge, UK: Cambridge University Press.

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W.F. Lawvere. 1963. Functorial Semantics of Algebraic Theories. Proc. Natl. Acad. Sci. USA, 50: 869-872

4
W. F. Lawvere. 1966. The Category of Categories as a Foundation for Mathematics. , In Proc. Conf. Categorical Algebra-La Jolla, 1965, Eilenberg, S et al., eds. Springer-Verlag: Berlin, Heidelberg and New York, pp. 1-20.

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J. Lambek and P. J. Scott. Introduction to higher order categorical logic. Cambridge University Press.

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S. Mac Lane. 1997. Categories for the Working Mathematician, 2nd ed. Springer-Verlag.

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S. Mac Lane and I. Moerdijk. 1992. Sheaves and Geometry in Logic: A First Introduction to Topos Theory, Springer-Verlag: Berlin.



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