Schreiber The (co-)reflective categories of supergravity

A talk that I gave:

• Urs Schreiber,

  The (co-)reflective categories of supergravity

  Category Theory and Algebra Seminar at Brno (org. Jiří Rosický)

  March 12, 2015

  (talk notes at super-Cartan geometry)

Abstract. In string theory, the global formulation of the dynamics of super p-branes requires target super spacetimes to be refined to super-stacks (FSS 13). This raises the mathematical problem of classifying the admissible (“anomaly free”) Cartan geometries for such higher supergeometry. In this talk I present a category theoretic analysis which greatly facilitates solving this problem. I discuss how the (higher) topos of super-stacks is stratified by a system of opposing (co-)reflective subcategories that serve to characterize its geometric content. In closing I will give a very brief outlook on how this serves to solve the classification problem.

$$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \e &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \R & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{contractible}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{differential}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$$

This diagram shows the system of reflectors and coreflectors on the supergeometric topos. The symbol “$\dashv\,\,$” denotes adjunctions of (co-)reflectors and the symbol “$\vee\,\,$” denotes inclusion of (co-)reflective subcategories. More details are at differential cohomology in a cohesive topos.

The discussion that applies this system of operations to the above problem is at Obstruction theory for parameterized higher WZW terms.

Related talks include

• Synthetic prequantum field theory in a cohesive homotopy topos

Related lecture notes include

• Higher Cartan Geometry

• Structure Theory for Higher WZW Terms

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