Domov > Obvestila > Matija Bašić, Combinatorial models for stable homotopy theory, Boštjan Lemež, An uncountable family of upper semicontinuous functions...

Matija Bašić, Combinatorial models for stable homotopy theory, Boštjan Lemež, An uncountable family of upper semicontinuous functions...

Datum objave: 23. 5. 2018
Vir: Mednarodni topološki seminar Ljubljana-Maribor-Zagreb
Sobota 26. 5. 2018, ob 10. uri, soba 2.02, Jadranska 21
 
Matija Bašić (Zagreb), Combinatorial models for stable homotopy theory 
 
We will recall the definition of dendroidal sets as a generalization of simplicial sets, and present 
 
the connection (Quillen equivalence) to connective spectra which gives a factorization of the so-called 
 
K-theory spectrum functor from symmetric monoidal categories to spectra. We will present a common 
 
generalization of two results of Thomason: 1) posets model all homotopy types; 2) symmetric monoidal 
 
categories model all connective spectra. We will introduce a notion of multiposets (special type of 
 
coloured operads) and of the subdivision of dendroidal sets which can be used to show that multiposets 
 
model all connective spectra. If time permits we will mention homology of dendroidal sets as it provides 
 
means to define equivalences of multiposets in an internal combinatorial way. 
 

  
 

Boštjan Lemež (Maribor), An uncountable family of upper semicontinuous functions $F$ such that  
 
the graph of $F$ is homeomorphic to the inverse limit of closed unit intervals with $F$ as the only  
 
bonding function 
 
There are many examples of upper semicontinuous functions $f:[0, 1]\rightarrow 2^{[0, 1]}$ such 
 
that both, the graph $\Gamma(f)$ and the inverse limit $\varprojlim\{[0,1],f\}_{i=1} ^{\infty}$ are 
 
arcs, hence the the graph $\Gamma(f)$ and the inverse limit $\varprojlim\{[0,1],f\}_{i=1} ^{\infty}$ 
 
are homeomorphic. We construct a nontrivial family of upper semicontinuous functions 
 
$F:[0, 1]\rightarrow 2^{[0, 1]}$ with the property that the graph of $F$ is homeomorphic to the inverse 
 
limit of the inverse sequence of closed unit intervals $[0, 1]$ with $F$ as the bonding function. 
 
As a special case, we use this construction to produce the Gehman dendrite as the graph of such function.