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I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fourth page.

Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its nerve. By applying the $k$-differential forms (contravariant) functor $\Omega^k$ to the nerve $\mathcal{N}$ we get a semicosimplicial abelian group

$$\Omega^q(X_0)\rightrightarrows\Omega^q(X_1) \rightarrow\rightrightarrows \cdots$$

(sorry I can't stack three arrows)

that induces a cochain complex in the usual way. The author calls this cochain complex the "Cěch complex" and its cohomologies are called the "Cěch cohomologies".

A $2$-isomorphism $\theta:f\Rightarrow g$ between two morphisms of Lie groupoids $f,g:X_\bullet\to Y_\bullet$ induces an homotopy between the Cěch complexes $$\theta^\ast:\Omega^q(Y_{p+1})\to \Omega^q(X_{p})$$ defined as follows:

$$(\theta^\ast\omega)(\phi_1,...,\phi_{p}):=\sum_{i=0}^p (-)^i\omega((f(\phi_1),...,f(\phi_{i-1}),f(\phi_{i-1})\theta(\phi_i)g(\phi_{i+1}),g(\phi_{i+1}),..,g(\phi_p))$$

I want to stress that in this definition, we are evaluating the differential forms at some precise points.

Until now everything is clear. Then the author basically says that the contravariant functor $\Omega^k$ has nothing special, and any contravariant functor $F:\text{Manifolds}\to \text{Abelian Groups}$ will do the job!

I really cannot understand how to define $\theta^\ast:F(Y_{p+1})\to F(X_p)$ for a generic contravariant functor $F$, because when $F=\Omega^q$ we use a really specific property of differential forms to define $\theta^\ast$: the fact that differential forms can be evaluated at a point!

Thank you in advance for your help!

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    $\begingroup$ May be I am misunderstanding your question.. The data of a $2$-morphism from $f$ to $g$ comes with a collection of maps $\{X_p\rightarrow Y_{p+1}\}_{p\geq 0}$.. As $F$ is a contra variant functor, these would give a collection of maps $\{F(Y_{p+1})\rightarrow F(X_p)\}_{p\geq 0}$... In that sense this may be done for any contravariant functor.. As $F$ is any random functor, that $\theta^*$ may not be as nice as what we have in the case of $\Omega^q$.. but, it exists.. because of functoriality of $F$... Is that what you wanted to ask? $\endgroup$
    – Praphulla Koushik
    Commented Apr 29 at 5:35
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    $\begingroup$ @PraphullaKoushik Thank you, I got it! $\endgroup$
    – Kandinskij
    Commented Apr 29 at 15:10
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    $\begingroup$ Your questions are good.. your questions are making me to question myself of what I read sometime ago :D.. $\endgroup$
    – Praphulla Koushik
    Commented Apr 29 at 16:34
  • $\begingroup$ I'm not a professional mathematician. I'm asking here, just because these topics are (relatively) advanced. I hope my doubt wasn't too stupid. $\endgroup$
    – Kandinskij
    Commented Apr 29 at 17:00
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    $\begingroup$ Not at all… Your style of asking questions is really good, where you mention everything relevant to the question.. hope to see more such nicely framed questions from you… $\endgroup$
    – Praphulla Koushik
    Commented Apr 30 at 2:19

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