Classifying space of a category

In mathematics , the classifying space of a category is a term from algebraic topology that generalizes the notion of the classifying space of a discrete group .

Nerve of a category

The nerve of a small category is the simplicial complex , whose - and -implices correspond to the objects or morphisms in and whose -implices correspond to the composable -tuples of morphisms . The edge mapping maps the corresponding -Simplex onto the corresponding -Simplex. ${\displaystyle N({\mathcal {C}})}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle A_{0}\to A_{1}\to \ldots \to A_{k-1}\to A_{k}}$ ${\displaystyle d_{i}}$ ${\displaystyle A_{0}\to A_{1}\to \ldots A_{i-1}\to A_{i}\to A_{i+1}\to \ldots \to A_{k-1}\to A_{k}}$ ${\displaystyle k}$ ${\displaystyle A_{0}\to A_{1}\to \ldots A_{i-1}\to A_{i+1}\to \ldots \to A_{k-1}\to A_{k}}$ ${\displaystyle (k-1)}$

Classifying space of a category

The classifying space of a category is the geometric realization of its nerve . ${\displaystyle B{\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle N({\mathcal {C}})}$

Example: A group is a category with one object, the group elements correspond to the morphisms, the group multiplication to the composition of morphisms. The classifying space of this category is the classifying space of the group with the discrete topology.

Web links

• Nerve (nLab)
• Chapter 7 in Richer: Category Theory with Applications in Topology
• Weiss: What does the classifying space of a category classify?