29,442 research outputs found
The tropicalization of the moduli space of curves
We show that the skeleton of the Deligne-Mumford-Knudsen moduli stack of
stable curves is naturally identified with the moduli space of extended
tropical curves, and that this is compatible with the "naive" set-theoretic
tropicalization map. The proof passes through general structure results on the
skeleton of a toroidal Deligne-Mumford stack. Furthermore, we construct
tautological forgetful, clutching, and gluing maps between moduli spaces of
extended tropical curves and show that they are compatible with the analogous
tautological maps in the algebraic setting.Comment: v2: 55 pages. Expanded Section 2 with improved treatment of the
category of generalized cone complexes. Clarified the role of the coarse
moduli space and its analytification in the construction of the skeleton for
a toroidal DM stac
Iterated spans and classical topological field theories
We construct higher categories of iterated spans, possibly equipped with
extra structure in the form of "local systems", and classify their fully
dualizable objects. By the Cobordism Hypothesis, these give rise to framed
topological quantum field theories, which are the framed versions of the
"classical" TQFTs considered in the quantization programme of
Freed-Hopkins-Lurie-Teleman.
Using this machinery, we also construct an infinity-category of Lagrangian
correspondences between symplectic derived algebraic stacks and show that all
its objects are fully dualizable.Comment: Accepted version, plus corrections to Remarks 10.5 and 10.7. 64 page
A categorical approach to the maximum theorem
Berge's maximum theorem gives conditions ensuring the continuity of an
optimised function as a parameter changes. In this paper we state and prove the
maximum theorem in terms of the theory of monoidal topology and the theory of
double categories.
This approach allows us to generalise (the main assertion of) the maximum
theorem, which is classically stated for topological spaces, to
pseudotopological spaces and pretopological spaces, as well as to closure
spaces, approach spaces and probabilistic approach spaces, amongst others. As a
part of this we prove a generalisation of the extreme value theorem.Comment: 45 pages. Minor changes in v2: this is the final preprint for
publication in JPA
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