topos axioms | Physics Library

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

i. $\tau$ has finite limits
ii. $\tau$ has power objects $\Omega(A)$ for objects in $\tau$ .

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:

$\mathbb{T}_1$ . $\tau$ is cartesian closed
$\mathbb{T}_2$ . $\tau$ has a subobject classifier.

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

1: R.J. Wood. 2004. Ordered Sets via Adjunctions, in Categorical Foundations.,
2: M. C. Pedicchio and W. Tholen, Eds. 2000. Cambridge, UK: Cambridge University Press.
3: 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.
5: J. Lambek and P. J. Scott. Introduction to higher order categorical logic. Cambridge University Press.
6: S. Mac Lane. 1997. Categories for the Working Mathematician, 2nd ed. Springer-Verlag.
7: 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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