N-skeleton
In mathematics, particularly in algebraic topology, the of a topological space "X" presented as a simplicial complex (resp. CW complex) refers to the subspace "X" that is the union of the simplices of "X" (resp. cells of "X") of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the . These subspaces increase with "n". The is a discrete space, and the a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are  particularly important when "X" has infinite dimension, in the sense that the "X" do not become constant as In geometry. In geometry, a of P (functionally represented as skel("P")) consists of all elements of dimension up to "k". For example: For simplicial sets. The above definition of the skeleton of a simplicial complex is a particular case of the notion of skeleton of a simplicial set. Briefly speaking, a simplicial set formula_1 can be described by a collection of sets formula_2, together with face and degeneracy maps between them satisfying a number of equations. The idea of the "n"-skeleton formula_3  is to first discard the sets formula_4 with formula_5 and then to complete the collection of the formula_4 with formula_7 to the "smallest possible" simplicial set so that the resulting simplicial set contains no non-degenerate simplices in degrees formula_5. More precisely, the restriction functor has a left adjoint, denoted formula_10. (The notations formula_11 are comparable with the one of image functors for sheaves.) The "n"-skeleton of some simplicial set formula_1 is defined as Coskeleton. Moreover, formula_14 has a "right" adjoint formula_15. The "n"-coskeleton is defined as For example, the 0-skeleton of "K" is the constant simplicial set defined  by formula_17. The 0-coskeleton is given by the Cech nerve The above constructions work for more general categories (instead of sets) as well, provided that the category has fiber products. The coskeleton is needed to define the concept of hypercovering in homotopical algebra and algebraic geometry.