Oxford

The use of tensed language and the metaphor of set "formation" found in informal descriptions of the iterative conception of set are seldom taken at all seriously. This talk offers an axiomatisation of the iterative conception in a bimodal language and presents some reasons to thus take the tense more seriously than usual (although not literally).

In {\it Was sind und was sollen die Zahlen?} (1888), Dedekind proves the Recursion Theorem (Theorem 126), and applies it to establish the categoricity of his axioms for arithmetic (Theorem 132). It is essential to these results that mathematical induction is formulated using second-order quantification, and if the second-order quantifier ranges over all subsets of the first-order domain (full second-order quantification), the categoricity result shows that, to within isomorphism, only one structure satisfies these axioms. However, the proof of categoricity is correct for a wide class of non-full Henkin models of second-order quantification. In light of this fact, can the proof of second-order categoricity be taken to establish that the second-order axioms of arithmetic characterize a unique structure?

Given a triangulated category A, equipped with a differential

Z/2-graded enhancement, and a triangulated oriented marked surface S, we

explain how to define a space X(S,A) which classifies systems of exact

triangles in A parametrized by the triangles of S. The space X(S,A) is

independent, up to essentially unique Morita equivalence, of the choice of

triangulation and is therefore acted upon by the mapping class group of the

surface. We can describe the space X(S,A) as a mapping space Map(F(S),A),

where F(S) is the universal differential Z/2-graded category of exact

triangles parametrized by S. It turns out that F(S) is a purely topological

variant of the Fukaya category of S. Our construction of F(S) can then be

regarded as implementing a 2-dimensional instance of Kontsevich's proposal

on localizing the Fukaya category along a singular Lagrangian spine. As we

will see, these results arise as applications of a general theory of cyclic

2-Segal spaces.

This talk is based on joint work with Mikhail Kapranov.