André Joyal (UQAM)

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.