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What you could be looking for is Theorem 6 of J.H.C. Whitehead's "Combinatorial Homotopy II" Bull. Amer. Math. Soc. 55 (1949) 453-496. For a CW-complex K he defines what he calls a "homotopy system" $\rho (K)$ and the theorem says that for a map $f:K \to L$ of CW-complexes, $f$ is an equivalence if and only if $\rho(f)$ is an equivalence. However to get "only if $\rho(f)$ is an isomorphism" is unlikely for this functor.

We have come to call $\rho(K)$ the fundamental crossed complex $\Pi(K_*)$ of the filtered space $K_*$, the filtration being by skeleta. Many aspects of Whitehead's work are dealt with in the book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tract Vol 15, 2011.

To add to the above, a recent preprint available from Graham Ellis' preprint page at Galway

G. Ellis and Le Van Luyen, "Homotopy 2-types of low order", preprint.

lists isomorphism classes and quasi-isomorphism classes of some crossed modules of low order.

This answer is related to my answer to this mathoverflow question.

What you could be looking for is Theorem 6 of J.H.C. Whitehead's "Combinatorial Homotopy II" Bull. Amer. Math. Soc. 55 (1949) 453-496. For a CW-complex K he defines what he calls a "homotopy system" $\rho (K)$ and the theorem says that for a map $f:K \to L$ of CW-complexes, $f$ is an equivalence if and only if $\rho(f)$ is an equivalence. However to get "only if $\rho(f)$ is an isomorphism" is unlikely for this functor.

We have come to call $\rho(K)$ the fundamental crossed complex $\Pi(K_*)$ of the filtered space $K_*$, the filtration being by skeleta. Many aspects of Whitehead's work are dealt with in the book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tract Vol 15, 2011.

To add to the above, a recent preprint available from Graham Ellis' preprint page at Galway

G. Ellis and Le Van Luyen, "Homotopy 2-types of low order", preprint.

lists isomorphism classes and quasi-isomorphism classes of some crossed modules of low order.

This answer is related to my answer to this mathoverflow question.

What you could be looking for is Theorem 6 of J.H.C. Whitehead's "Combinatorial Homotopy II" Bull. Amer. Math. Soc. 55 (1949) 453-496. For a CW-complex K he defines what he calls a "homotopy system" $\rho (K)$ and the theorem says that for a map $f:K \to L$ of CW-complexes, $f$ is an equivalence if and only if $\rho(f)$ is an equivalence. However to get "only if $\rho(f)$ is an isomorphism" is unlikely for this functor.

We have come to call $\rho(K)$ the fundamental crossed complex $\Pi(K_*)$ of the filtered space $K_*$, the filtration being by skeleta. Many aspects of Whitehead's work are dealt with in the book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tract Vol 15, 2011.

To add to the above, a recent preprint available from Graham Ellis' preprint page at Galway

G. Ellis and Le Van Luyen, "Homotopy 2-types of low order", preprint.

lists isomorphism classes and quasi-isomorphism classes of some crossed modules of low order.

What you could be looking for is Theorem 6 of J.H.C. Whitehead's "Combinatorial Homotopy II" Bull. Amer. Math. Soc. 55 (1949) 453-496. For a CW-complex K he defines what he calls a "homotopy system" $\rho (K)$ and the theorem says that for a map $f:K \to L$ of CW-complexes, $f$ is an equivalence if and only if $\rho(f)$ is an equivalence. However to get "only if $\rho(f)$ is an isomorphism" is unlikely for this functor.

We have come to call $\rho(K)$ the fundamental crossed complex $\Pi(K_*)$ of the filtered space $K_*$, the filtration being by skeleta. Many aspects of Whitehead's work are dealt with in the book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tract Vol 15, 2011.

To add to the above, a recent preprint available from Graham Ellis' preprint page at Galway

G. Ellis and Le Van Luyen, "Homotopy 2-types of low order", preprint.

lists isomorphism classes and quasi-isomorphism classes of some crossed modules of low order.

This answer is related to my answer to this mathoverflow question.

What you could be looking for is Theorem 6 of J.H.C. Whitehead's "Combinatorial Homotopy II" Bull. Amer. Math. Soc. 55 (1949) 453-496. For a CW-complex K he defines what he calls a "homotopy system" $\rho (K)$ and the theorem says that for a map $f:K \to L$ of CW-complexes, $f$ is an equivalence if and only if $\rho(f)$ is an equivalence. However to get "only if $\rho(f)$ is an isomorphism" is unlikely for this functor.

We have come to call $\rho(K)$ the fundamental crossed complex $\Pi(K_*)$ of the filtered space $K_*$, the filtration being by skeleta. Many aspects of Whitehead's work are dealt with in the book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tract Vol 15, 2011.

What you could be looking for is Theorem 6 of J.H.C. Whitehead's "Combinatorial Homotopy II" Bull. Amer. Math. Soc. 55 (1949) 453-496. For a CW-complex K he defines what he calls a "homotopy system" $\rho (K)$ and the theorem says that for a map $f:K \to L$ of CW-complexes, $f$ is an equivalence if and only if $\rho(f)$ is an equivalence. However to get "only if $\rho(f)$ is an isomorphism" is unlikely for this functor.

We have come to call $\rho(K)$ the fundamental crossed complex $\Pi(K_*)$ of the filtered space $K_*$, the filtration being by skeleta. Many aspects of Whitehead's work are dealt with in the book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tract Vol 15, 2011.

To add to the above, a recent preprint available from Graham Ellis' preprint page at Galway

G. Ellis and Le Van Luyen, "Homotopy 2-types of low order", preprint.

lists isomorphism classes and quasi-isomorphism classes of some crossed modules of low order.