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There are two ways to define a stack.

The first one is that the presheaf of sets Isom (a,b) is a sheaf and that every descent data is effective.

The second one says that a stack is a homotopy sheaf of groupoids.

How does the homotopy sheaf condition imply that every descent data is effective? Does it follow from the simplicial identities? What is the idea behind it?

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    $\begingroup$ The homotopy sheaf condition is literally the effective descent condition. (Unless you have substituted (co)monadic descent for the traditional definition.) You just write down a concrete model for the limit of the diagram and find that it is the same as the category (groupoid) of descent data. $\endgroup$
    – Zhen Lin
    Commented Mar 13, 2023 at 22:53
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    $\begingroup$ @ZhenLin I agree, but I tried working it out by hand and got stuck (because I'm not fluent at model categories, I suppose). A first mistake you can make is to take $\mathscr F(U) \to \prod \mathscr F(U_i) \rightrightarrows \prod \mathscr F(U_{ij})$ for the sheaf condition (in the $\infty$-categorical setting you need the entire nerve, or least one more term for $1$-truncated spaces = $1$-groupoids), and the second question is at what point do you start taking fibrant (and cofibrant?) replacements (notably, do you need to do it for presheaves, or is it enough after plugging in various opens?). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Mar 13, 2023 at 23:10
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    $\begingroup$ In other words, it would be great if you could turn your comment into an answer! $\endgroup$
    – R. van Dobben de Bruyn
    Commented Mar 13, 2023 at 23:12
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    $\begingroup$ Er, that sounds to me more like a failure to find the correct definition of “homotopy sheaf condition” than not understanding it. More importantly the OP has not given any indication of the framework they want to work in. Vague question gets vague answer! $\endgroup$
    – Zhen Lin
    Commented Mar 14, 2023 at 0:19
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    $\begingroup$ Please point to a reference for the homotopy sheaf condition. :-) The first definition is rather well-known, but the second needs spelling out to pin down precisely what you mean. $\endgroup$
    – David Roberts ♦
    Commented Mar 14, 2023 at 1:15

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