# Topos

This page is a sub-page of our page on Mathematical Concepts.

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Related KMR-pages:

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Other relevant sources of information:

An Elementary Theory of the Category of Sets (long version) with commentary,
by F. William Lawvere.
An informal introduction to Topos theory, by Tom Leinster, arXiv, 2011.
Ologs – A categorical framework for knowledge representation, by David Spivak and Robert Kent, 2016.

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Groucho Marx is supposed to have said that
I don’t want to be a member of a club that wants me as a member.”

Trans-complex numbers:

Suppose that a complex number discovers that its imaginary part has become zero
and wants to “rebrand” itself as a real number.

Using classical aristotelian logic, this is not possible. “Once a member, always a member” is the credo of ZFC-based set membership. There is no “logical room” for “trans-complex” numbers, i.e., complex numbers that want to (re)present as real numbers, because their values have become real.

Of course, a complex number can never “discover” anything at all, since all such discoveries are made somewhere else within the computational context that makes use of the complex number in question. This is why a complex number is a “data-type” but not an “object” in the OOP programming sense. Moreover, when the discovery involves a change of value, for the “value-object” that handles the change there are always at least two different values involved.

The category $\, \mathbf{S_{et}} \,$ of sets and functions relies on a $\, \{ \, \textnormal{true}, \textnormal{false} \, \}$-type of membership detector. For a topos, this binary $\, \{ \, \textnormal{yes}, \textnormal{no} \, \} \,$ membership detector is generalized into a subobject classifier called $\, \Omega \,$, which helps to select a “member” (or rather a subobject) under more subtle circumstances. This gives a mathematical basis upon which the concept of evolutionary entropy can be modeled.

Definition: A topos is a cartesian closed category that has a subobject classifier.