condition; it is also sufficient. Indeed, let X be the geometric realization of a simplicial set K. The set of simplices in K is partially ordered, by the relation x ≤ y if and only if x is a face of y. We may consider this partially ordered set as a category. The nerve of this category is the barycentric subdivision of K, and thus its realization is homeomorphic to X, because X is the realization of K by hypothesis and barycentric subdivision does not change the homeomorphism type of the realization. The nerve of an open covering If X is a topological space with open cover Ui, the nerve of the cover is obtained from the above definitions by replacing the cover with the category obtained by regarding the cover as a partially ordered set with relation that of set inclusion. Note that the realization of this nerve is not generally homeomorphic to X (or even homotopy equivalent). A moduli example One can use the nerve construction to recover mapping spaces, and even get "higher-homotopical" information about maps. Let D be a category, and let X and Y be objects of D. One is often interested in computing the set of morphisms X → Y. We can use a nerve construction to recover this set. Let C = C(X,Y) be the category whose objects are diagrams such that the morphisms U → X and Y → V are isomorphisms in D. Morphisms in C(X, Y) are diagrams of the following shape: Here, the indicated maps are to be isomorphisms or identities. The nerve of C(X, Y) is the moduli space of maps X → Y. In the appropriate model category setting, this moduli space is weak homotopy equivalent to the simplicial set of morphisms of D from X to Y. References Blanc, D., W. G. Dwyer, and P.G. Goerss. "The realization space of a -algebra: a moduli problem in algebraic topology." Topology 43 (2004), no. 4, 857–892. Goerss, P. G., and M. J. Hopkins. "Moduli spaces of commutative ring spectra." Structured ring spectra, 151–200, London Math. Soc. Lecture Note Ser., 315, Cambridge Univ. Press, Cambridge, 2004. Segal, Graeme. "Classifying spaces and spectral sequences." Inst. Hautes Études Sci. Publ. Math. No. 34 (1968) 105–112. Category theory Simplicial sets