Algebraic Topology: Lecture 13

In the lecture today, we discussed the classifying space of a small category C and the homotopy colimit of a functor X from a small category C to the category U of (unbased) spaces. The standard reference is:

A. K. Bousfield, D. M. Kan: Homotopy limits, completions and localizations, Lecture Notes in Mathematics 304, Springer-Verlag, 1972.

A small category C defines a simplicial set NC[-] called the nerve of C. The k-simplices is the set of k-tuples of composable maps in C,

    NC[k] = { c₀ ← c₁ ← ... ← c_k }

The face map d_i deletes c₀ and the first map, if i = 0, deletes c_i and composes the two maps c_i-1 &larr c_i and c_i &larr c_i+1, if 0 < i < k, and deletes c_k and the last map, if i = k. The face map s_i repeats c_i and inserts an identity map. Hence, a k-simplex c₀ &larr c₁ &larr ... &larr c_k is degenerate if and only if one of the maps is an identity map. The geometric realization

    |C| = | [k] |-> NC[k] |

is called the classifying space of C.