Timeline for "Boundaries" in Free Simplicial Monoids
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 19, 2021 at 18:22 | comment | added | Dmitri Pavlov | There is more than one suspension and loop functor in simplicial homotopy theory, but none of them simply shifts simplicial degrees by one. Additionally, for simplicial monoids the suspension functor must be derived, if you are to get a correct answer. If you simply shift simplicial degrees by one, how is the resulting sequence of sets turned into a simplicial set? | |
| Feb 19, 2021 at 10:20 | comment | added | Mathemologist | It's just a map into the suspension. I am sorry if my terminology is incorrect. The n-cells of $X^{*}$ are lists of n-cells in $X$, and the suspension shifts the grading by one. | |
| Feb 19, 2021 at 2:20 | comment | added | Dmitri Pavlov | Okay, how is a "shifted homomorphism of simplicial monoids" defined? | |
| Feb 19, 2021 at 2:09 | comment | added | Mathemologist | Every n-cell in $X$ has a list of faces of length n+1, which is an (n-1)-cell in the free simplicial monoid $X^{*}$. This defines a map $X\to \Sigma X^{*}$, which is the adjunct to a map $X^{*}\to \Sigma X^{*}$. This operation is defined on lists of n-cells, giving you the result of first replacing each list element with its list of faces, and then passing this list-of-lists through the component for the multiplication of the free monoid monad, ie just concatenate the finite list of finite lists (all of whose elements are faces of n-cells). | |
| Feb 18, 2021 at 18:50 | comment | added | Dmitri Pavlov | How is the "list of lists of faces of each n-cell" defined? | |
| Feb 18, 2021 at 12:10 | history | asked | Mathemologist | CC BY-SA 4.0 |