PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Medium Entry average rating: No information on entry rating
simplicial object (Definition)

A simplicial object in a category $C$ is a contravariant functor from the simplicial category $\Delta$ to $C$ . Such a functor $X$ is uniquely specified by the morphisms $X(\delta^n_i)\colon X([n]) \to X([n-1])$ and $X(\sigma^n_i)\colon X([n]) \to X([n+1])$ , which satisfy

$\displaystyle X(\delta^{n-1}_i)\,X(\delta^n_j)$ $\displaystyle =$ $\displaystyle X(\delta^{n-1}_{j-1})\,X(\delta^n_i)$   for $\displaystyle i<j,$ (1)
$\displaystyle X(\sigma^{n+1}_i)\,X(\sigma^n_j)$ $\displaystyle =$ $\displaystyle X(\sigma^{n+1}_{j+1})\,X(\sigma^n_i)$   for $\displaystyle i\leq j,$ (2)
$\displaystyle X(\delta^{n+1}_i)\,X(\sigma^n_j)$ $\displaystyle =$ \begin{displaymath}\left\{ \begin{array}{ll} X(\sigma^{n-1}_{j-1})\,X(\delta^n_i... ...1}_j)\,X(\delta^n_{i-1}) & \mbox{if\ } i>j+1. \end{array}right.\end{displaymath} (3)

In particular, a simplicial set is a simplicial object in $\mathcat{Set}$ . Equivalently, one could say that a simplicial set is a presheaf on $\Delta$ . The object $X([n])$ of a simplicial set is a set of $n$ -simplices, and is called the $n$ -skeleton.




"simplicial object" is owned by mhale.
(view preamble | get metadata)

View style:

See Also: simplicial category

Also defines:  simplicial set
Log in to rate this entry.
(view current ratings)

Cross-references: object, presheaf, morphisms, simplicial category, contravariant functor, category
There are 3 references to this entry.

This is version 4 of simplicial object, born on 2002-08-27, modified 2005-10-28.
Object id is 3368, canonical name is SimplicialObject.
Accessed 4862 times total.

Classification:
AMS MSC18G30 (Category theory; homological algebra :: Homological algebra :: Simplicial sets, simplicial objects )

Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)