Abstract
A theory of bisheaves has been recently introduced to measure the homological stability of fibers of maps to manifolds. A bisheaf over a topological space is a triple consisting of a sheaf, a cosheaf, and compatible maps from the stalks of the sheaf to the stalks of the cosheaf. In this note we describe how, given a bisheaf constructible (i.e., locally constant) with respect to a triangulation of its underlying space, one can explicitly determine the coarsest stratification of that space for which the bisheaf remains constructible.
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Notes
- 1.
The open star of \(\sigma \in \mathcal {M}\) is given by \({{{\mathbf {st}}}}\, \sigma = \{\tau \in \mathcal {M} \mid \sigma \leq \tau \}\).
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Acknowledgements
This work had its genesis in the Abel Symposium on Topological Data Analysis, held in June 2018 amid the breathtaking fjords of Geiranger, where both authors gave invited lectures. AP spoke about [9] and VN about [11], and it became clear to us almost immediately that there were compelling practical reasons to combine these works. It is a sincere pleasure to thank the Abel Foundation and the Abel Symposium organizers, particularly Nils Baas and Marius Thaule, for giving us the opportunity to work in such an inspiring location. The ideas of Robert MacPherson are densely sprinkled throughout not only this paper, but also across both its progenitors [9] and [11]. We are grateful to the Institute for Advanced Study for hosting many of our discussions with Bob. We also thank the anonymous referee for encouraging us to clarify Definition 2 and the subsequent remark.
VN’s work is supported by The Alan Turing Institute under the EPSRC grant number EP/N510129/1, and by the Friends of the Institute for Advanced Study. AP’s work is supported by the National Science Foundation under agreement number CCF-1717159.
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Nanda, V., Patel, A. (2020). Canonical Stratifications Along Bisheaves. In: Baas, N., Carlsson, G., Quick, G., Szymik, M., Thaule, M. (eds) Topological Data Analysis. Abel Symposia, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-030-43408-3_15
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