realization of X is the topological space (in fact a CW complex) obtained if every n-simplex of X is replaced by a topological n-simplex (a certain n-dimensional subset of (n + 1)-dimensional Euclidean space defined below) and these topological simplices are glued together in the fashion the simplices of X hang together. In this process the orientation of the simplices of X is lost. To define the realization functor, we first define it on standard n-simplices Δn as follows: the geometric realization |Δn| is the standard topological n-simplex in general position given by The definition then naturally extends to any simplicial set X by setting |X| = limΔn → X | Δn| where the colimit is taken over the n-simplex category of X. The geometric realization is functorial on sSet. It is significant that we use the category CGHaus of compactly-generated Hausdorff spaces, rather than the category Top of topological spaces, as the target category of geometric realization: like sSet and unlike Top, the category CGHaus is cartesian closed; the categorical product is defined differently in the categories Top and CGHaus, and the one in CGHaus corresponds to the one in sSet via geometric realization. Singular set for a space The singular set of a topological space Y is the simplicial set SY defined by (SY)([n]) = homTop(|Δn|, Y) for each object [n] ∈ Δ. Every order-preserving map φ:[n]→[m] induces a continuous map |Δn|→|Δm| in a natural way, which by composition yields SY(φ) : SY([m]) → SY([n]). This definition is analogous to a standard idea in singular homology of "probing" a target topological space with standard topological n-simplices. Furthermore, the singular functor S is right adjoint to the geometric realization functor described above, i.e.: homTop(|X|, Y) ≅ homsSet(X, SY) for any simplicial set X and any topological space Y. Intuitively, this adjunction can be understood as follows: a continuous map from the geometric realization of X to a space Y is uniquely specified if we associate to every simplex of X a continuous map from the corresponding standard topological simplex to Y, in such a fashion that these maps are compatible with the way the simplices in X hang together. Homotopy theory of simplicial sets In order to define a model structure on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences. One can define fibrations to be Kan fibrations. A map of simplicial sets