Free loop spaces and cyclohedra

Markl, Martin. "Free loop spaces and cyclohedra." Proceedings of the 22nd Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 2003. 151-157. <http://eudml.org/doc/220881>.

@inProceedings{Markl2003,
abstract = {It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an $A_\infty$-operad. The classical model for such an operad consists of Stasheff's associahedra. The present paper describes a similar recognition principle for free loop spaces. Let $\{\cal P\}$ be an operad, $M$ a $\{\cal P\}$-module and $U$ a $\{\cal P\}$-algebra. An $M$-trace over $U$ consists of a space $V$ and a module homomorphism $T:M\to\text\{End\}_\{U,V\}$ over the operad homomorphism $\{\cal P\}\to\text\{End\}_U$ given by the algebra structure on $U$. Let $\{\cal C\}_1$ be the little 1-cubes operad.\par The author shows that the free loop space $\wedge X$ is a trace over the $\{\cal C\}_1$-space $\Omega X$. This trace is related to the cyclohedra in a way similar to the relation of $\{\cal C\}_1$ to the associahedra. Given a $\{\cal P\}$-module $M$ and a $\{\cal P\}$-algebra $U$ one can define the free $M$-trace over $U$ like one can construct free $\{\cal P\}$-al!},
author = {Markl, Martin},
booktitle = {Proceedings of the 22nd Winter School "Geometry and Physics"},
keywords = {Winter school; Geometry; Physics; Srní (Czech Republic)},
location = {Palermo},
pages = {151-157},
publisher = {Circolo Matematico di Palermo},
title = {Free loop spaces and cyclohedra},
url = {http://eudml.org/doc/220881},
year = {2003},
}

TY - CLSWK
AU - Markl, Martin
TI - Free loop spaces and cyclohedra
T2 - Proceedings of the 22nd Winter School "Geometry and Physics"
PY - 2003
CY - Palermo
PB - Circolo Matematico di Palermo
SP - 151
EP - 157
AB - It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an $A_\infty$-operad. The classical model for such an operad consists of Stasheff's associahedra. The present paper describes a similar recognition principle for free loop spaces. Let ${\cal P}$ be an operad, $M$ a ${\cal P}$-module and $U$ a ${\cal P}$-algebra. An $M$-trace over $U$ consists of a space $V$ and a module homomorphism $T:M\to\text{End}_{U,V}$ over the operad homomorphism ${\cal P}\to\text{End}_U$ given by the algebra structure on $U$. Let ${\cal C}_1$ be the little 1-cubes operad.\par The author shows that the free loop space $\wedge X$ is a trace over the ${\cal C}_1$-space $\Omega X$. This trace is related to the cyclohedra in a way similar to the relation of ${\cal C}_1$ to the associahedra. Given a ${\cal P}$-module $M$ and a ${\cal P}$-algebra $U$ one can define the free $M$-trace over $U$ like one can construct free ${\cal P}$-al!
KW - Winter school; Geometry; Physics; Srní (Czech Republic)
UR - http://eudml.org/doc/220881
ER -