Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces

Question

From a bisimplicial space $T$, one can consider the simplicial spaces $p \mapsto T_{pp}$, $p \mapsto | q \mapsto T_{pq}|$, and $q \mapsto |p \mapsto T_{pq}|$, where $| \cdot|$ denotes geometric realisation. In a lemma (used in proving Theorems A and B) on page 10 of '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 proof.

First of all, he claima that any $T$ has a canonical presentation of the form $$ \amalg_{(r,s) \rightarrow (r′,s′)}h^{r′,s′} \times T^{r s} \rightrightarrows \amalg_{(r,s)} h^{r,s} \times T^{r s} \rightarrow T$$, where $h^{rs} =hom_ { \Delta} (-,r-) \times hom_{ \Delta} ( -,s-)$, (with the hom-sets having the discrete topology).

The fact that it is canonical comes from its functoriality in $T$. To show that our diagram is in fact a coequalizer diagram it suffices to show that it is an objectwise coequalizer diagram. But at this point I don't know how to do this.

Once we have the claim, I understand how the proof continues.

I would be very grateful if someone could help me. Quillen gives as a reference a paper from Tornehave, On BSG and the symmetric groups, but I can not find it anywhere. Does someone know if I can find it?

Answer 1

I certainly respect going back to primary sources. But, in this case, it's helpful to remember that a LOT has been written about bisimplicial sets since Quillen's 1973 paper. For example, a reference that spells out the claim about being a coequalizer is Goerss, Jardine Simplicial Homotopy Theory, Lemma 5.17 in Chapter IV (bisimplicial sets). Another reference is Lemma 5.3.1 in Hovey's book (the claim about being a coequalizer is half a page above the proof of the lemma). More references can be found at this nLab page, which also proves the Lemma from Quillen's paper, as Lemma 3.4. While Quillen's proof is hands-on, the nLab proof is nice and clean, using that geometric realization is a co-end (and adjoint to Sing(-)), that the diagonal pops up when you compute the mapping space, and the universal property. See also Jeff Smith's answer at this MO thread, which proves the same result by recognizing things as homotopy colimits (and using the universal property).