What is a topos?

**Abstract**

The first mutation was provoked by the introduction of the notion of
elementary topos by Lawvere and Tierney in the late 60; it is
connecting topos theory with intuitionistic logic.

The second mutation was provoked by the introduction of the notion of
model topos by Charles Rezk in the late 90; it was extensively
developed by Jacob Lurie in his book "Higher topos theory"; it is
connecting topos theory with homotopy theory and higher category
theory.

A third mutation is presently emerging under the impulse of homotopy
type theory, the new field of mathematics initiated by the homotopy
interpretation of Martin-Lof type theory by Awodey, Warren and
Voevodski.

There is a profound unity between these developements.

We believe that topos theory is best understood from a dual algebraic
point of view.

We propose calling the dual notion an "arena" (tentative name).

The theory of arenas has many things in common with the theory of
commutative rings.