coskeleton

Coskeleton is a construction in the theory of simplicial objects that serves as the dual notion to the skeleton. Given a category C with limits and a simplicial object X: Δ^op → C, the n-coskeleton of X, denoted cosk_n X, is defined as the right Kan extension of X along the inclusion of the full subcategory Δ_{\le n}^op into Δ^op. Equivalently, cosk_n X is the right adjoint to the restriction functor that forgets all simplices above dimension n.

Intuitively, cosk_n X is the best possible way to extend the data of X from dimensions up to n to higher dimensions, such that all higher simplices are determined by their compatible faces in dimensions ≤ n. Concretely, for each level k ≥ 0, the k-simplices of cosk_n X are obtained by a limit over all ways of assembling lower-dimensional data from X along maps [m] → [k] with m ≤ n. In particular, if k ≤ n then cosk_n X_k ≅ X_k, so the coskeleton agrees with X in the low dimensions, while for k > n the higher simplices are constrained by the lower-dimensional information.

Key properties include that cosk_n is a right adjoint to the restriction functor from simplicial objects to their values on the ordinal numbers up to n, hence it preserves limits. If X is already n-coskeletal, the natural map cosk_n X → X is an isomorphism. The dual notion, skeleton Sk_n, is left adjoint to the same restriction and is used to truncate information from higher dimensions.

Coskeleton constructions are used in areas such as algebraic topology, higher category theory, descent theory, and the theory of simplicial objects and hypercovers, where one studies how higher-dimensional data is determined by, or reconstructed from, lower-dimensional information.