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A few questions about the equivalence between 2-types and crossed modules. For simplicity, assume everything is connected.

  1. What is the precise statement? Is there an equivalence of categories (or at least a bijection of isomorphism classes) of the form

    $$\{\text{2-truncated spaces}\}[\text{weak homotopy equivalence}^{-1}] \simeq \{\text{crossed modules}\}[\mathcal{W}^{-1}]$$

    for some class of morphisms $\mathcal W$? If so, what is $\mathcal W$? Is it exactly the isomorphisms? What is a precise reference?

  2. Crossed modules $1 \to A \to H_2 \to H_1 \to G \to 1$ are classified by $H^3(G;A)$ up to zigzags of morphisms of extensions. Does this translate to a classification of connected homotopy 2-types $X$ with $\pi_1(X) = G$ and $\pi_2(X) = A$ up to homotopy equivalence, or is it only up to a coarser equivalence relation? If this is a classification up to homotopy equivalence, then how does one see that a non-invertible morphism of crossed modules induces a homotopy equivalence of classifying spaces?

  3. In any event, from every crossed module $1 \to A \to H_2 \to H_1 \to G \to 1$, I can extract an element of $H^3(G;A)$. Homotopically, this corresponds to a Postnikov invariant for a possibly non-principal Postnikov tower. Where is the theory of non-principal Postnikov invariants written, and in particular does this invariant (exist and) completely classify a 2-type?

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  • $\begingroup$ There's certainly a functor $(\text{crossed modules})\to (\text{simplicial sets})$, which should realize the equivalence of (1) (turn the crossed module into a 2-groupoid, then take its 2-nerve). The paper arxiv.org/abs/math/0512106 by Noohi looks like it might have some of what you want? $\endgroup$ – Charles Rezk Apr 19 at 15:43
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    $\begingroup$ 1. Yes, you get that equivalence of categories. Weak equivalences of crossed modules are quasi-isomorphisms. 2. Up to homotopy equivalence. The classifying space functor from crossed modules to spaces preserves weak equivalences. 3. Isn't this in Hatcher? It's quite classical. See also Baues obstruction theory book. $\endgroup$ – Fernando Muro Apr 20 at 11:41
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The book Nonabelian Algebraic Topology discusses the classifying space $BC$ of a crossed complex $C$ and the homotopy classification $$[X,BC] \cong [\Pi X_* , C] $$ for a CW-complex $X$ with skeletal filtration $X_*$. (Theorem 11.4.19). The functor $\Pi$ from filtered spaces to crossed complexes goes back in essence to Blakers (1948) and to Whitehead (1949).

The paper Modelling and Computing homotopy types:I explains something of the origin and methodology lying behind the above work, including how we should represent cohomology classes.

Maybe helpful in relation to the question of dealing with finite crossed modules is the paper

"Homotopy 2-Types of Low Order" G Ellis, LV Le - Experimental Mathematics, 2014 - Taylor & Francis

April 23,2019: I mention also that p.428ff of the NAT book discusses the notion of crossed $n$-fold extension and the bijection $$\text{OpExt}^n(G,M) \cong H^{n+1}(G,M) .$$ An advantage of the approach is that we work in the realm of free crossed resolutions, and that we can use a standard free crossed resolution $SF(G)$ which is related to the crossed complex of the Nerve of the group $G$ and so to more classical cocycle conditions; thus conditions in several dimensions are replaced by the condition of giving a morphism of crossed complexes $SF(G) \to C$. See also the paper Bullejos, M., Faro, E., and García-Muñoz, M.A., Postnikov invariants of crossed complexes. J. Algebra 285 (1) (2005) 238-291 for a different approach.

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