Homotopy colimit and limit functors and homotopical ones

2005 ◽  
pp. 147-170
Author(s):  
William Dwyer ◽  
Philip Hirschhorn ◽  
Daniel Kan ◽  
Jeffrey Smith
Keyword(s):  
Homotopy Colimit

scholarly journals ON THE VOEVODSKY MOTIVE OF THE MODULI STACK OF VECTOR BUNDLES ON A CURVE

2020 ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Vector Bundles ◽  
Intersection Theory ◽  
Line Bundles ◽  
Projective Curve ◽  
Classifying Space ◽  
Smooth Projective Curve ◽  
Quot Schemes ◽  
Homotopy Colimit ◽  
Moduli Stack ◽  
Fixed Rank

Abstract We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.

HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF FINITE GROUPS ASSOCIATED TO MODULAR REPRESENTATIONS

2001 ◽  
Vol 64 (2) ◽  
pp. 472-488 ◽  
Author(s):  
D. NOTBOHM
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Homotopy Theory ◽  
Modular Representation ◽  
Modular Representations ◽  
Classifying Spaces ◽  
Modular Representation Theory ◽  
Classifying Space ◽  
Homotopy Colimit ◽  
A Finite Group

For a prime p, a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod-p homology. Associated to a modular representation G → Gl(n; [ ]p), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so-called subgroup, centralizer and normalizer decompositions. For an action of G on an [ ]p-vector space V, this collection consists of all subgroups of G with nontrivial p-Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG.

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