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.