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I originally posted it on stackexchange, but did not get any comments or answer, so I try my luck here.

I am looking for the naive notion of homotopy.

For maps $f, g: A\to B$ of simplicial sets $\Delta^{op}\to Sets$, a homotopy $H$ is a simplicial map $H:A\times\Delta^{1}\to B$ such that $H|_{A\times\{0\}}=f$ and $H|_{A\times\{1\}}=g$, or equivalently a collection of maps $H_i: X_n\to Y_{n+1}, 0\le i\le n$ subject to a bunch of identities, see encyclopedia of math.

Now for maps $f, g:A\to B$ of bisimplicial sets $\Delta^{op}\times\Delta^{op}\to Sets$, what is a homotopy between them?

Is it a map $A\times\Delta^{1,1}\to B$ such that a suitable boudary condition hold? But now there are four vertexes of $\Delta^{1,1}$.

Obviously, we may define a map $A\times\Delta^{0,1}\to B$ or $A\times\Delta^{1,0}\to B$ to be vertical/horizontal homotopy.

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    $\begingroup$ I guess there are many possible answers. From a more general point this question is answered by giving a model structure on the category of bisimplicial sets. Then you could decipher the explicit notion of homotopy. A quick search gave me math.uwo.ca/~jardine/papers/HomTh/lecture008.pdf It contains a list with some model structures.. $\endgroup$ Jan 27, 2014 at 13:46
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    $\begingroup$ I think, depending on the context, one just defines it to be a vertical or horizontal homotopy. There is no reason, why there should be a single definition (unless you view in the context of model categories where you have an exact notion of a cylinder object, which leaves you a lot of freedom to choose it). $\endgroup$ Jan 27, 2014 at 16:05

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Following Stephan's comment, we can define model structures on s2Set, using which we can define left and right homotopies. However, if we look at your maps A×Δ0,1→B and A×Δ1,0→B, drawing a simple picture allows us to think of A×Δ1,0→B as a horizonal homotopy and A×Δ0,1→B as vertical homotopy with a suitable orientation of the Cartesian space. A general homotopy can then naively be considered some sort of "composition" of the two - first vertical then horizontal homotopy, or first horizontal then vertical homotopy. But the best one to go with would be the one defined by a chosen model structure on s2Set.

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