[Turkmath:5552] Feza Gursey Center for Physics and Mathematics-Higher Structures Seminars [no.7][04.12.2022]::M. Akif Erdal(Yeditepe University)

[kazimilhan.ikeda]

Dear all,

We would like to announce that Mehmet Akif ERDAL (Yeditepe University, 
Istanbul)
will give a talk in the "Higher Structures Webinars"  of  Feza Gürsey 
Center for
Physics and Mathematics

TITLE: "Homotopy theory of monoid actions via group actions and an 
Elmendorf style
theorem"

ABSTRACT: For a group $G$ and a collection of subgroups $Y$ of $G$, the 
orbit category
$O_{Y}$ is the category whose objects are $G$-orbits $G/H$ for each 
$H\in Y$ and whose
morphisms are  $G$-equivariant maps in between. Due to Elmendorf's 
Theorem that the
category $G$-spaces and the category of contravariant  $O_{Y}$-diagrams 
of spaces have
equivalent homotopy theories. This provides a great convenience when 
studying $G$-equivariant
homotopy theory since one can reduce it to non-equivariant homotopy 
theory of associated
fixed point systems. In this talk, we describe a non-trivial extension 
of  this idea to
the actions of monoids. Let $M$ be a monoid and $G(M)$ be its group 
completion, $Z$ be a
collection of submonoids of $M$ and for each $N\in Z$, $Y_N$ be a 
collection of subgroups
of $G(N)$. First we will show that the category of $M$-spaces and 
$M$-equivariant maps
admits a model structure in which weak equivalences and fibrations are 
determined by the
standard equivariant homotopy theory of $G(N)$-spaces for each $N \in 
Z$. Then, we will
show that this model structure is Quillen equivalent to the projective 
model structure
on the category of contravariant $O_{Z,Y}$-diagrams of spaces, where $ 
O_{Z,Y}$ is the
category whose objects are induced orbits $Mtimes_N G(N)/H$ for each 
$N\in Z$ and
$H \in Y_N$ and morphisms are $M$-equivariant maps. Finally, if time 
permits, we will
state some applications.

DATE & TIME: Tuesday, April 12, 2022, 14:30 Istanbul local time.

Zoom link details:

Kazim Ilhan Ikeda is inviting you to a scheduled Zoom meeting.

Topic: FGC-Higher Structures Seminars [No.7][12 April::Mehmet Akif 
Erdal]
Time:  Tuesday, April 12, 2022, 14:30 Istanbul local time.

Join Zoom Meeting
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Meeting ID: 943 0442 7243
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