# Simplicial sets from bisimplicial sets, and their realisations.

From a bisimplicial space $T$, one can consider the simplicial spaces $\underline p \mapsto T_{pp}$, $\underline p \mapsto |\underline q \mapsto T_{pq} |$, and $\underline q \mapsto |\underline p \mapsto T_{pq} |$, where $| \cdot |$ denotes geometric realisation. In a lemma (used in proving Theorems A and B, in 'Higher Algebraic K-theory I'), Quillen gives a proof that these three simplicial sets all have coincidental realisations, but I'm having touble understanding the reasoning.

The idea of the proof is to first prove it in the special case of when $T$ is of the form $h^{rs} \times S$, where $h^{rs} = \hom_{\Delta}( -, \underline{r}) \times \hom_{\Delta}( -, \underline{s})$ (with the hom-sets having the discrete topology) and $S$ denotes the constant simplicial space $\underline p \mapsto S$, for some space $S$. This is easy to do. Next is where I get lost, (although I do understand that we're trying to write a general bisimplicial space $T$ is a direct limit of the special cases). He writes

"But any $T$ has a canonical presentation

$\coprod_{(r,s) \to (r',s')} h^{r',s'} > \times T_{rs} \rightrightarrows > \coprod_{(r,s)} h^{r,s} \times T_{rs} > \to T$

which is exact in the sense that the right arrow is the cokernel of the pair of arrows. Since the three functors from bisimplicial spaces to spaces under consideration commute with inductive limits, the lemma follows."

I am confused on a few points:

What is meant by 'canonical presentation'?

By 'cokernel', was 'coequaliser' meant?

In any case, how does one get an inductive limit from that diagram?

May someone please spell out what is going on here?

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I should note that the paper of Tornehave that Quillen referenced in relation to the above seemingly never appeared. – Joshua Seaton Feb 20 '13 at 15:31

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