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A simplicial set X is a contravariant functor. Given a simplicial set X, we often write X n instead of X [ n ]. Simplicial sets form a category, usually denoted sSet , whose objects are simplicial sets and whose morphisms are natural transformations between them. The corresponding category of cosimplicial sets is denoted by cSet. So the identities provide an alternative way to define simplicial sets. Concretely, the n -simplices of the nerve NS , i. The face map d i drops the i -th element from such a list, and the degeneracy maps s i duplicates the i -th element. A similar construction can be performed for every category C , to obtain the nerve NC of C.
Here, NC [ n ] is the set of all functors from [ n ] to C , where we consider [ n ] as a category with objects 0,1, In particular, the 0-simplices are the objects of C and the 1-simplices are the morphisms of C.
The degeneracy maps s i lengthen the sequence by inserting an identity morphism at position i. We can recover the poset S from the nerve NS and the category C from the nerve NC ; in this sense simplicial sets generalize posets and categories. Another important class of examples of simplicial sets is given by the singular set SY of a topological space Y.
Here SY n consists of all the continuous maps from the standard topological n -simplex to Y. The singular set is further explained below. The following isomorphism shows that a simplicial set X is a colimit of its simplices: . In this process the orientation of the simplices of X is lost. 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.
The singular set of a topological space Y is the simplicial set SY defined by.
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.www.emlaklobisi.com/wp-includes/61/
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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. 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. Homotopy quantum field theory. Syzygies and homotopy theory.
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