Dualities between cellular sheaves and cosheaves

Abstract

Abstract. This paper affirms a conjecture of MacPherson: that the derived category of cellular sheaves is equivalent to the derived category of cellular cosheaves. We give a self-contained treatment of cellular sheaves and cosheaves and note that certain classical dualities give rise to an exchange of sheaves with cosheaves. Following a result of Pitts that states that cosheaves are cocontinuous functors on the category of sheaves, we use the derived equivalence provided here to gain a novel description of compactly supported sheaf cohomology.

Homology and cohomology of finite spaces

“…This is the global form of duality, which is already done in [9]. We shall give an explicit description of the dualizing complex D X in Theorem 4.3.4 (a related result may be found in [5]). As we mentioned above, we also study the validity of the local form of duality in Theorem 4.3.9.…”

“…We shall give here some basic results about sheaves, cosheaves, direct and inverse images, homology, cohomology and standard resolutions (see also [5,Sections 2., 3. and 6. ]).…”

“…In subsection 4.3 we study Grothendieck-Verdier duality for first cohomology and then homology for a morphism f : X → Y . Some related duality results can also be found in [5]. Verdier duality for cohomology deals with the existence of a right adjoint f !…”

Sheaf Theory Approach to Distributed Applications: Analysing Heterogeneous Data in Air Traffic Monitoring

“…So the next step would be encoding the existing heterogeneous data from the simplicial complex into a sheaf model. To each simplex in the complex there assigns a set which carries all information about the data and enables us to analyze data transmission in distributed network system [11]. Definition 2.…”

Toward a spectral theory of cellular sheaves

“…We will work exclusively with cellular sheaves [Cur14]. Fix a (regular, locally finite) cell complex -a triangulated surface will suffice for purposes of imagination.…”