Manifold calculus and homotopy sheaves

By Pedro Boavida de Brito, Michael Weiss. 2012

73 citations recorded by the source corpus.

Manifold calculus is a form of functor calculus concerned with functors from some category of manifolds to spaces. A weakness in the original formulation is that it is not continuous in the sense that it does not handle well the natural enrichments. In this paper, we correct this by defining an enriched version of manifold calculus which essentially extends the discrete setting. Along the way, we recast the Taylor tower as a tower of homotopy sheafifications. As a spin-off we obtain a natural connection to operads: the limit of the Taylor tower is a certain (derived) space of right module maps over the framed little discs operad.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/37b06fea7107e18dd3b8a1b7fa858928a07e370d

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the rational homology of high dimensional analogues of spaces of long knots (2011, background): “10 that deals with right modules was proved by de Brito and Weiss in [6].”
The rational homotopy of mapping spaces of E${}_n$ operads (2017, background): “In another domain, the embedding calculus developed by Goodwillie and Weiss [26, 53] allowed to express the homotopy of spaces of smooth embeddings of manifolds in terms of mapping spaces associated to modules over the little discs operads (see [1, 5, 48]).”
Spaces of smooth embeddings and configuration categories (2015, methodology): “As in [7], we fix a smooth (m − 1)-manifold L without boundary and re-define Man so that the objects are smooth m-manifoldsM with boundary, together with a diffeomorphism L→ ∂M .”