Editing Simplicial set (section)

==Singular set for a space==
The '''singular set''' of a topological space ''Y'' is the simplicial set ''SY'' defined by 
:(''SY'')([''n'']) = hom<sub>'''T''op'''''</sub>(|Δ<sup>''n''</sup>|, ''Y'') for each object [''n''] ∈ Δ.

Every order-preserving map &phi;:[''n'']→[''m''] induces a continuous map |Δ<sup>''n''</sup>|→|Δ<sup>''m''</sup>| by

:<math>(x_0,...,x_n) \in |\Delta_n| \mapsto (y_j),~~ y_j = \sum_{\phi(i) =j}x_i.</math>

Then, by composition it yields to a map ''SY''(''&phi;'') : ''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 [[adjoint functor|right adjoint]] to the geometric realization functor described above, i.e.:

:hom<sub>'''Top'''</sub>(|''X''|, ''Y'') &cong; hom<sub>'''sSet'''</sub>(''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.