what means 'the realization of a topological category'

Question

In the paper Homology Fibrations and the "Group-Completion" Theorem. page 280 bottom line 10-bottom line 12, what means 'the realization of a topological category'?

Answer 1

A topological category is an internal category in topological spaces, i.e. a space of objects $C_0$ and one of morphisms $C_1$ with maps $i:C_0\to C_1$ and $s,t:C_1\to C_0$ satisfying the axioms of a category. A topological category gives rise to a single space via a geometric realization that's inspired by the geometric realization of abstract simplicial sets, or similarly by the nerve of a category, etc. Anyway, the realization is like a CW complex but with a space of 0-cells and one of 1-cells: the 0-cells are $C_0$, the $1$-cells are $I\times C_1$, and we quotient as follows: glue $0\times C_1$ to $C_0$ along $s$, $1\times C_1$ to $C_0$ along $t$, and identify $C_0$ with its image under $i$.

There are some issues with this naive definition in certain cases, as addressed in the appendix to [9] cited in your quote. I won't go into that stuff here.