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SUMMARY:Homotopy Fiber Sequences from a New Perspective
DTSTART:20211123T141500
DTEND:20211123T151500
DTSTAMP:20260916T065216Z
UID:ce0b1db2781370e72c17a0fd77f1d84a4636e3552b927527dc294a12
CATEGORIES:Conferences - Seminars
DESCRIPTION:Alisa Govzmann\, Universitéit Lëtzebuerg\nQuillen introduced
  fibration sequences in the homotopy category of a pointed model category
 . Up to a (non-canonical) Ho(M)-isomorphism a fibration sequence is then a
  Ho(M)–sequence K → F → G that is implemented by the kernel K of a 
 fibration F → G between fibrant objects F and G. Using the fact that th
 e loop space functor Ω^QF of a fibrant object F is a group object in the
  homotopy category Ho(M)\, Quillen shows that there is an action of Ω^QG
  on K which induces a connecting Ho(M)–morphism Ω^QG → K and the seq
 uence Ω^QG → K → F is again a fibration sequence. I want to present 
 an alternative approach to construct homotopy fiber sequences without usin
 g the concept of an action. We define a loop space functor Ω and for eve
 ry morphism f : F → G we define its homotopy fiber K_f such that K_f 
 → F → G is a homotopy fiber sequence. We get a universal connecting m
 orphism ΩF → K_f such that ΩF → K_f → G is also a homotopy fiber 
 sequence. It turns out that the loop space functor we define as well as t
 he connecting homomorphism coincide with the one proposed by Quillen. At 
 the beginning of this talk I will explain how to construct an equivalence 
 of categories between the homotopy category of morphisms (arrows) in M\, 
 denoted by Ho(M^→) and the homotopy category of long homotopy fiber seq
 uences\, denoted by Ho(l(M)). This will lead to a canonical choice of an 
 isomorphism in Ho(M) between two different homotopy fibers of a morphism 
 in Ho(M).
LOCATION:MA A1 12 https://plan.epfl.ch/?room==MA%20A1%2012
STATUS:CONFIRMED
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