In topology, a branch of mathematics, a cosheaf with values in an
∞-category
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. Th ...
''C'' that admits colimits is a functor ''F'' from the category of open subsets of a topological space ''X'' (more precisely its
nerve) to ''C'' such that
*(1) The ''F'' of the empty set is the initial object.
*(2) For any increasing sequence
of open subsets with union ''U'', the canonical map
is an equivalence.
*(3)
is the pushout of
and
.
The basic example is
where on the right is the
singular chain complex of ''U'' with coefficients in an abelian group ''A''.
Example:
[http://www.math.harvard.edu/~lurie/282ynotes/LectureIX-NPD.pdf ] If ''f'' is a continuous map, then
is a cosheaf.
See also
*
sheaf (mathematics)
Notes
References
*http://www.math.harvard.edu/~lurie/282ynotes/LectureVIII-Poincare.pdf
*http://arxiv.org/pdf/1303.3255v1.pdf , section 3, in particular Thm 3.10 p. 34
Algebraic topology
Category theory
Sheaf theory
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