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simplicial object
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(Definition)
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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
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.
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"simplicial object" is owned by mhale.
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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 MSC: | 18G30 (Category theory; homological algebra :: Homological algebra :: Simplicial sets, simplicial objects ) |
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Pending Errata and Addenda
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