Definition 0.1. The two axioms that define an elementary topos, or a standard topos, as a special
category τ are:
- i. τ has finite limits
- ii. τ has power objects Ω(A) for objects A in τ.
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 τ subject to the following axioms:
- 𝕋1. τ is cartesian closed
- 𝕋2. τ has a subobject classifier.
One can show that axioms i. and ii. also imply axioms 𝕋1 and 𝕋2; one notes that property 𝕋2 can
also be expressed as the existence of a representable subobject functor.
References
[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.
