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I am looking for a reference, preferably as elementary as possible, for the following statement.

Let $X_{m,n}$ be a bi-simplicial object in an additive category $\mathcal{A}$. Then the complex $|X_{m,n}|$ obtained from iterative realization (i.e. the total complex of the associated bi-complex) is homotopy equivalent to the complex obtained by realizing the diagonal $|X_{n,n}|$.

I know that it should follow from the cofinality of the map $\Delta^{op} \to \Delta^{op} \times \Delta^{op}$ together with the identification of the realization as the colimit of a simplicial object in the $\infty$-category of bounded above complexes in $\mathcal{A}$. However, all this seem to me like a real overkill for this problem, and I would expect a much more elementary explanation can be given. I also remember seeing a proof along this lines in the case where $\mathcal{A}$ is Abelian, but I really need a more general result, since the example I have in mind is not abelian.

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For abelian categories this is known as the Eilenberg–Zilber theorem, see, for instance, Theorem 8.5.1 in Weibel's book. One can write down explicit comparison maps in both directions (namely, the Alexander–Whitney and Eilenberg–Zilber maps) and also write down an explicit chain homotopy that shows these maps to be chain homotopy equivalences. The constructions are explicit and concrete, and perhaps the proof goes through for additive categories, but I have not checked this.

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  • $\begingroup$ Thanks for the references! My feeling is of course the same, that the formulas uses only addition and subtraction so you should not need kernels an cokernels, but I hoped that it is stated in this language somewhere. If not, I guess varifying it myself in the text is the only option :-) $\endgroup$
    – S. carmeli
    Commented Apr 21, 2019 at 15:55
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    $\begingroup$ You just embed the additive category into an abelian category and you get the result you want. $\endgroup$
    – Fernando Muro
    Commented Apr 21, 2019 at 16:23
  • $\begingroup$ @FernandoMuro right! thats so simple yet so cool. And I guess the Yoneda embedding $\mathcal{A} \to Fun^{\oplus}(\mathcal{A}^{op},Ab)$ gives us such an embedding right? $\endgroup$
    – S. carmeli
    Commented Apr 21, 2019 at 19:55
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    $\begingroup$ @S.carmeli: The Yoneda embedding only works for small additive categories. For large categories there is Adelman's embedding theorem. See mathoverflow.net/questions/52881/… $\endgroup$
    – Dmitri Pavlov
    Commented Apr 21, 2019 at 21:03
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    $\begingroup$ The Yoneda embedding suffices. A simplicial object only involves countably many objects of the abelian category. $\endgroup$
    – Fernando Muro
    Commented Apr 22, 2019 at 17:14

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