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

 
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.