Cosheaf

In topology, a branch of mathematics, a '''cosheaf''' is a dual notion to that of a sheaf that is useful in studying Borel-Moore homology.{{further explanation needed|date=October 2024}}

==Definition==

We associate to a topological space <math>X</math> its category of open sets <math>\operatorname{Op}(X)</math>, whose objects are the open sets of <math>X</math>, with a (unique) morphism from <math>U</math> to <math>V</math> whenever <math>U \subset V</math>. Fix a category <math>\mathcal{C}</math>. Then a ''precosheaf (with values in <math>\mathcal{C}</math>)'' is a covariant functor <math>F : \operatorname{Op}X \to \mathcal{C}</math>, i.e., <math>F</math> consists of * for each open set <math>U</math> of <math>X</math>, an object <math>F(U)</math> in <math>\mathcal{C}</math>, and * for each inclusion of open sets <math>U \subset V</math>, a morphism <math>\iota_{U,V} : F(U) \to F(V)</math> in <math>\mathcal{C}</math> such that ** <math>\iota_{U,U} = \mathrm{id}_{F(U)}</math> for all <math>U</math> and ** <math>\iota_{U,V} \circ \iota_{V,W} = \iota_{U,W}</math> whenever <math>U \subset V \subset W</math>.

Suppose now that <math>\mathcal{C}</math> is an abelian category that admits small colimits. Then a ''cosheaf'' is a precosheaf <math>F</math> for which the sequence

<math display="block"> \bigoplus_{(\alpha,\beta)}F(U_{\alpha,\beta}) \xrightarrow{\sum_{(\alpha,\beta)} (\iota_{U_{\alpha,\beta},U_\alpha} - \iota_{U_{\alpha,\beta},U_\beta})} \bigoplus_{\alpha} F(U_\alpha) \xrightarrow{\sum_\alpha \iota_{U_\alpha,U}} F(U) \to 0 </math>

is exact for every collection <math>\{U_\alpha\}_\alpha</math> of open sets, where <math>U := \bigcup_\alpha U_\alpha</math> and <math>U_{\alpha,\beta} := U_\alpha \cap U_\beta</math>. (Notice that this is dual to the sheaf condition.) Approximately, exactness at <math>F(U)</math> means that every element over <math>U</math> can be represented as a finite sum of elements that live over the smaller opens <math>U_\alpha</math>, while exactness at <math>\bigoplus_\alpha F(U_\alpha)</math> means that, when we compare two such representations of the same element, their difference must be captured by a finite collection of elements living over the intersections <math>U_{\alpha,\beta}</math>.

Equivalently, <math>F</math> is a ''cosheaf'' if * for all open sets <math>U</math> and <math>V</math>, <math>F(U \cup V)</math> is the pushout of <math>F(U \cap V) \to F(U)</math> and <math>F(U \cap V) \to F(V)</math>, and * for any upward-directed family <math>\{U_\alpha\}_\alpha</math> of open sets, the canonical morphism <math>\varinjlim F(U_\alpha) \to F\left(\bigcup_\alpha U_\alpha\right)</math> is an isomorphism. One can show that this definition agrees with the previous one.<ref>{{cite book | title=Sheaf Theory | isbn=9780387949055 | last1=Bredon | first1=Glen E. | date=24 January 1997 | publisher=Springer |url={{Google books|zGdqWepiT1QC|keywords=cosheaf|plainurl=yes}} }}</ref> This one, however, has the benefit of making sense even when <math>\mathcal{C}</math> is not an abelian category.

==Examples==

A motivating example of a precosheaf of abelian groups is the ''singular precosheaf'', sending an open set <math>U</math> to <math>C_{k}(U; \mathbb{Z})</math>, the free abelian group of singular <math>k</math>-chains on <math>U</math>. In particular, there is a natural inclusion <math>\iota_{U,V} : C_{k}(U; \mathbb{Z}) \to C_{k}(V; \mathbb{Z})</math> whenever <math>U \subset V</math>. However, this fails to be a cosheaf because a singular simplex cannot be broken up into smaller pieces. To fix this, we let <math>s : C_{k}(U; \mathbb{Z}) \to C_{k}(U; \mathbb{Z})</math> be the barycentric subdivision homomorphism and define <math>\overline{C}_{k}(U; \mathbb{Z})</math> to be the colimit of the diagram

<math display="block"> C_{k}(U; \mathbb{Z}) \xrightarrow{s} C_{k}(U; \mathbb{Z}) \xrightarrow{s} C_{k}(U; \mathbb{Z}) \xrightarrow{s} \ldots. </math>

In the colimit, a simplex is identified with all of its barycentric subdivisions. One can show using the Lebesgue number lemma that the precosheaf sending <math>U</math> to <math>\overline{C}_{k}(U; \mathbb{Z})</math> is in fact a cosheaf.

Fix a continuous map <math>f : Y \to X</math> of topological spaces. Then the precosheaf (on <math>X</math>) of topological spaces sending <math>U</math> to <math>f^{-1}(U)</math> is a cosheaf.<ref>{{cite web |last1=Lurie |first1=Jacob |title=Tamagawa Numbers via Nonabelian Poincare Duality, Lecture 9: Nonabelian Poincare Duality in Algebraic Geometry |url=https://www.math.ias.edu/~lurie/282ynotes/LectureIX-NPD.pdf |publisher=School of Mathematics, Institute for Advanced Study.}}</ref>

== Notes == {{reflist}}

== References == {{ref begin}} *{{cite book | title=Sheaf Theory | isbn=9780387949055 | last1=Bredon | first1=Glen E. | date=24 January 1997 | publisher=Springer |url={{Google books|zGdqWepiT1QC|keywords=cosheaf|plainurl=yes}} }} *{{cite journal |doi=10.2140/pjm.1968.25.1 |title=Cosheaves and homology |date=1968 |last1=Bredon |first1=Glen |journal=Pacific Journal of Mathematics |volume=25 |pages=1–32 |doi-access=free }} *{{cite journal |url=http://eudml.org/doc/91560 |title=The display locale of a cosheaf |journal=Cahiers de Topologie et Géométrie Différentielle Catégoriques |date=1995 |volume=36 |issue=1 |pages=53–93 |last1=Funk |first1=J. }} *{{cite journal |doi=10.1007/s13160-015-0173-9 |title=Topological data analysis and cosheaves |date=2015 |last1=Curry |first1=Justin Michael |journal=Japan Journal of Industrial and Applied Mathematics |volume=32 |issue=2 |pages=333–371 |arxiv=1411.0613 |s2cid=256048254 }} *{{cite arXiv |eprint=1209.2995 |last1=Positselski |first1=Leonid |title=Contraherent cosheaves |date=2012 |class=math.CT }} *{{cite book |url={{Google books|HudaEAAAQBAJ|page=306|plainurl=yes}} | title=Sheaf Theory through Examples | isbn=9780262362375 | last1=Rosiak | first1=Daniel | date=25 October 2022 | publisher=MIT Press }} *{{cite web |last1=Lurie |first1=Jacob |author-link1=Jacob Lurie |title=Tamagawa Numbers via Nonabelian Poincare Duality, Lecture 8: Nonabelian Poincare Duality in Topology |url=https://www.math.harvard.edu/~lurie/282ynotes/LectureVIII-Poincare.pdf|publisher=School of Mathematics, Institute for Advanced Study.}} *{{cite book| title=Sheaves, cosheaves and applications|contribution= § 3, in particular Thm 3.10 |page=34 | last1=Curry | first1=Justin | date=2014 | arxiv=1303.3255|id={{ProQuest|1553207954}}|type=Doctoral dissertation|publisher=University of Pennsylvania }} {{ref end}} Category:Algebraic topology Category:Category theory Category:Sheaf theory