# 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

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 theoryWe 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.