A simplicial set is a functor from the opposite of the simplex category to sets. Its face and degeneracy maps satisfy the simplicial identities.
The standard simplicial -simplex is the representable simplicial set
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The th horn is the union of all codimension-one faces except the face opposite vertex .
A horn is inner when and outer when or .
A quasicategory is a simplicial set in which every inner horn has a filler.
A Kan complex is a simplicial set in which every simplicial horn, including every outer horn, has a filler.
The nerve of a category has
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so an -simplex is a string of composable morphisms. Every inner horn in a nerve has a unique filler.
The simplicial mapping space is defined by
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If is a monomorphism and is a Kan complex, restriction is a Kan fibration.

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Simplicial set by Wikipedia Bot 1
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A simplicial set is a fundamental concept in algebraic topology and category theory that generalizes the notion of a topological space. It is a combinatorial structure used to study objects in homotopy theory and other areas of mathematics. ### Definition A **simplicial set** consists of: 1. **Sets of n-simplices**: For each non-negative integer \( n \), there is a set \( S_n \) which consists of n-simplices.
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