topos axioms
Definition 0.1
The two axioms that define an
elementary topos, or a
standard topos, as a special category

are:
- i.
has finite limits
- ii.
has power
objects
for objects
in
.
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

subject to the following axioms:
-
.
is cartesian closed
-
.
has a subobject classifier.
One can show that axioms i. and ii. also imply axioms
and
; one notes that property
can also be expressed as the existence of a representable subobject functor.
- 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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