Lukáš Vokřínek
Homotopy weighted colimits.
Tue, 01/17/2012 - 15:01 — SomNambulAuthors: Lukáš Vokřínek
Subjects: Category Theory
AbstractLet V be a cofibrantly generated closed symmetric monoidal model category and
M a model V-category. We say that a weighted colimit W*D of a diagram D
weighted by W is a homotopy weighted colimit if the diagram D is pointwise
cofibrant and the weight W is cofibrant in the projective model structure on
[C^op,V]. We then proceed to describe such homotopy weighted colimits through
homotopy tensors and ordinary (conical) homotopy colimits. This is a homotopy
version of the well known isomorphism W*D=\int^C(W\tensor D).- Read more
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A generalization of Thom's transversality theorem.
Thu, 01/14/2010 - 16:01 — SomNambulAuthors: Lukáš Vokřínek
Subjects: Differential Geometry
AbstractWe prove a generalization of Thom's transversality theorem. It gives
conditions under which the jet map $f_*|_Y:Y\subseteq J^r(D,M)\ra J^r(D,N)$ is
generically (for $f:M\ra N$) transverse to a submanifold $Z\subseteq J^r(D,N)$.
We apply this to study transversality properties of a restriction of a fixed
map $g:M\ra P$ to the preimage $(j^sf)^{-1}(A)$ of a submanifold $A\subseteq
J^s(M,N)$ in terms of transversality properties of the original map $f$.Fibrations up to an equivalence, homotopy colimits and pullbacks.
Thu, 01/14/2010 - 16:01 — SomNambulAuthors: Lukáš Vokřínek
Subjects: Algebraic Topology
AbstractWe gather conditions on a class H of continuous maps of topological spaces
that allow a reasonable theory of fibrations up to an equivalence (a map from
this class) which we call H-fibrations. The weak homotopy equivalences recover
quasifibrations and homology equivalences yield homology fibrations. We study
local H-fibrations that behave nicely with respect to homotopy colimits
together with universal H-fibrations that behave nicely with respect to
pullbacks. We then proceed to classify H-fibrations up to a natural notion of
equivalence.
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