PlanetMath: well-pointed topos

(more info)

Math for the people, by the people.

Main Menu

well-pointed topos

(Definition)

The concept of well-pointed topoi was introduced by Freyd in [1]. A topos is well-pointed if it satisfies either the following equivalent conditions:

The terminal object distinguishes morphisms in the sense that if the diagram
$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ 1\ar[r]^x & A\ar@<1ex>[r]^f\ar@<-1ex>[r]_g & B } } \end{xy}$
commutes for every morphism $x\colon 1\to A$ , then in fact
$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A\ar@<1ex>[r]^f\ar@<-1ex>[r]_g & B } } \end{xy}$
commutes, that is, . Moreover, is not isomorphic to the initial object.
The topos $\mathcal{T}$ is complemented and supports split, and the truth object $\Omega$ of $\mathcal{T}$ has exactly two elements, $\top\colon 1\to\Omega$ , and $\bot\colon 1\to\Omega$ . To say that $\mathcal{T}$ is complemented means that if $m\colon X\to Y$ is a monomorphism, then there exists a monomorphism $m'\colon X'\to Y$ such that $m\sqcup m'\colon X\sqcup X'\to Y$ is an isomorphism. To say that $\mathcal{T}$ supports split means that every subobject of is projective.

Every well-pointed topos is a Boolean topos.

Bibliography

1: P. Freyd. Aspects of topoi. Bull. Austral. Math. Soc. 7 (1972), 1-76.
2: P. T. Johnstone. Topos theory. Academic Press, 1977.
3: S. Mac Lane and I. Moerdijk. Sheaves and Geometry in Logic: A First Introduction to Topos Theory, Springer-Verlag, 1992.

Anyone with an account can edit this entry. Please help improve it!

"well-pointed topos" is owned by mps.

(view preamble)

Other names:

well-pointed topoi, well-pointed

Also defines:

complemented, supports split

This object's parent.

Log in to rate this entry.
(view current ratings)

Cross-references: Boolean topos, subobject, isomorphism, monomorphism, truth object, initial object, isomorphic, diagram, morphisms, terminal object, equivalent, topos
There are 2 references to this entry.

This is version 1 of well-pointed topos, born on 2007-01-23.
Object id is 8812, canonical name is WellPointedTopos.
Accessed 1562 times total.

Classification:

18A15 (Category theory; homological algebra :: General theory of categories and functors :: Foundations, relations to logic and deductive systems)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact