WebAt an informal level: sheaves can be thought of as functions and cosheaves as measures (you can integrate a sheaf against a cosheaf using a coend, multiply a cosheaf by a sheaf to get a cosheaf etc...). Functions tend to be determined by values at points (i.e. stalks) but for measure the "value at a point" is something like μ ( { x }) and it ... WebMay 8, 2024 · In topology, a branch of mathematics, a cosheaf with values in an ∞-category C that admits colimits is a functor F from the category of open subsets of a topological …
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WebSeries: Graduate Texts in Mathematics, 0072-5285 ; 170; Contents: I Sheaves and Presheaves -- Definitions -- 2 Homomorphisms, subsheaves, and quotient sheaves -- 3 … WebJun 26, 2024 · from the (m-shifted) Borel-Moore to the singular homology over \(\mathcal {U}\).These maps commute with restriction maps of the sheaf and extension maps of the cosheaf by naturality of the cap product. This data, consisting of a sheaf plus a cosheaf along with such maps is the prototypical and motivating example of a bisheaf. cite strong and thorough textual evidence
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WebMar 6, 2024 · In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be … WebDec 3, 2024 · The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes. The Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to pe WebApr 5, 2024 · The cosheaf \(\mathcal {A}\) will assign the category of chain complexes at smooth points, and the category of exact triangles (aka \(\mathrm {Perf}(\bullet \rightarrow \bullet )\)) at the trivalent vertices, with the obvious cogenerization morphisms. This case was studied in . diane oleary