We make a first suggestion for an axiomatic approach to string field theory in the form of what we will call a string field functor. We give a plausibility argument for the existence of a string field functor and show that it has to be unique up to a natural gauge freedom. We prove some elementary properties of string field functors and argue that such a functorial formulation is only possible ...

For a small category B and a double category D, let LaxN (B,D) denote the category whose objects are vertical normal lax functors B //D and morphisms are horizontal lax transformations. It is well known that LaxN (B,Cat) ≃ Cat/B, where Cat is the double category of small categories, functors, and profunctors. In [19], we generalized this equivalence to certain double categories, in the case whe...

We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the no...

à 4.0 Short Comment on Sheaves, Bundles, and Representable Functors. The notion of a sheaf over a topological space X generalize that of a bundle over X . We're more-or-less aware of this. I want to spend some time looking the specifics of this generalization. I'll focus on schemes. Fix a ring A and let 1 :=A 1 be the affine line. The first thing I want to point out is that the global section...

1. Even periodic ring theories and formal group laws 1 2. Formal groups 5 3. Geometry of formal groups 8 4. Heights of formal groups 12 5. Homology of Z× BU and related spaces 15 6. The Thom isomorphism 19 7. Elliptic spectra 24 8. The generalized Weierstrass equation 25 9. Modular forms 29 10. Elliptic curves and the group structure 30 11. Weierstrass parameters and Thom spectrum of ΩU(4) 33 1...

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f : X → BF , where BF denotes the classifying space for stable spherical fibrations. To do this, we consider symmetric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identifies the topological Hochschild h...

The goal of this paper is to explain the analogy between certain results in algebraic geometry, namely the Riemann-Roch theorems due to Baum,Fulton, and MacPherson ([BFM2],and [FMac]); and recent results in geometric topology due to Dwyer, Weiss and myself [DWW]. One reason for doing this is that the bivariant viewpoint introduced by Fulton-MacPherson in their memoir [FMac], becomes particularl...

This paper presents a novel treatment of the canonical extension of a bounded lattice, in the spirit of the theory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M. Ploščica, and the canonical extension can be obtained in the same manner as can be done in the distributive case by exploiting Priestley duality. To encompass ...

Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jabłoński, and Mrozek suggests an equivalent description of barcodes as functors R → Mch, where R is the poset category of real numbers and Mch is the category whose obje...

The geometric and algebraic properties of Gray categories with duals are investigated by means of a diagrammatic calculus. The diagrams are three-dimensional stratifications of a cube, with regions, surfaces, lines and vertices labelled by Gray category data. These can be viewed as a generalisation of ribbon diagrams. The Gray categories present two types of duals, which are extended to functor...