Cosk×

Cosk× is a generalized construction in category theory and algebraic topology, described as a variant of the coskeleton functor that incorporates a fixed object into the extension process. It is used when one wishes to extend a diagram not only by filling in higher dimensional data from lower dimensions, but also in a way that is coordinated with a product or tensoring by a chosen object.

Intuitively, given a category with finite products, a fixed object P, and a simplicial object X, cosk×P(X) produces a new simplicial object by adding higher dimensional simplices in a manner determined by the existing lower dimensional data together with the product with P. The construction aims to preserve compatibility with face and degeneracy maps while ensuring that every compatible family of lower dimensional data extended via P determines a unique higher simplex. In many setups the operation is presented as a right Kan extension that has been enriched by the product with P.

Key properties include functoriality in both X and P under appropriate conditions, and compatibility with product structures: under suitable limits, cosk×(P × Q)(X) is expected to correspond to cosk×P(X) and cosk×Q(X) in a coordinated way. When P is a terminal object or when P is trivial, cosk×P reduces to the standard coskeleton construction.

Cosk× sits alongside the standard coskeleton and skeleton constructions and is relevant in contexts such as enriched or parametrized homotopy theory, simplicial methods, and the study of higher categorical structures. See also coskeleton, skeleton, right Kan extension, and simplicial objects.