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edited Dec 7, 2014 at 18:20

user62675

When n=∞, this states that the homotopy category of infinite loop spaces is equivalent to the homotopy category of connective spectra; see The geometry of iterated loop spaces by May.

We can also prove this statement in the general case. Let X be a (pointed) n-connected space; then the loop space Ωⁿ⁺¹X is an E_n+1-space, and this is easily checked to be grouplike. So if nConnTop and A_n are the categories of n-connective spaces and E_n+1-spaces, respectively, then the (n+1)-fold loop space functor Ωⁿ⁺¹ induces an equivalence of categories between nConnTop and A_n. (I believe the statement for E_n+1-spaces in general should be somewhere in May's book; I can't find it, though.)

One implication of this, using the Freudenthal suspension theorem, is that if a homotopy n-type has the structure of a grouplike E_n+2-space, then it canonically has the structure of an E_∞-space, which is a special case of the Baez-Dolan stablization hypothesis. (See here, for example.)

RE your edit: a connective spectrum is usually defined as an object of the homotopy limit of the sequence ...→Top_n+1→Top_n→...→Top₁. (See also Denis's comment on your question.)

When n=∞, this states that the homotopy category of infinite loop spaces is equivalent to the homotopy category of connective spectra; see The geometry of iterated loop spaces by May.

We can also prove this statement in the general case. Let X be a (pointed) n-connected space; then the loop space Ωⁿ⁺¹X is an E_n+1-space, and this is easily checked to be grouplike. So if nConnTop and A_n are the categories of n-connective spaces and E_n+1-spaces, respectively, then the (n+1)-fold loop space functor Ωⁿ⁺¹ induces an equivalence of categories between nConnTop and A_n. (I believe the statement for E_n+1-spaces in general should be somewhere in May's book; I can't find it, though.)

One implication of this, using the Freudenthal suspension theorem, is that if a homotopy n-type has the structure of a grouplike E_n+2-space, then it canonically has the structure of an E_∞-space, which is a special case of the Baez-Dolan stablization hypothesis. (See here, for example.)

When n=∞, this states that the homotopy category of infinite loop spaces is equivalent to the homotopy category of connective spectra; see The geometry of iterated loop spaces by May.

We can also prove this statement in the general case. Let X be a (pointed) n-connected space; then the loop space Ωⁿ⁺¹X is an E_n+1-space, and this is easily checked to be grouplike. So if nConnTop and A_n are the categories of n-connective spaces and E_n+1-spaces, respectively, then the (n+1)-fold loop space functor Ωⁿ⁺¹ induces an equivalence of categories between nConnTop and A_n. (I believe the statement for E_n+1-spaces in general should be somewhere in May's book; I can't find it, though.)

One implication of this, using the Freudenthal suspension theorem, is that if a homotopy n-type has the structure of a grouplike E_n+2-space, then it canonically has the structure of an E_∞-space, which is a special case of the Baez-Dolan stablization hypothesis. (See here, for example.)

RE your edit: a connective spectrum is usually defined as an object of the homotopy limit of the sequence ...→Top_n+1→Top_n→...→Top₁. (See also Denis's comment on your question.)

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created Dec 6, 2014 at 19:50

user62675

When n=∞, this states that the homotopy category of infinite loop spaces is equivalent to the homotopy category of connective spectra; see The geometry of iterated loop spaces by May.

We can also prove this statement in the general case. Let X be a (pointed) n-connected space; then the loop space Ωⁿ⁺¹X is an E_n+1-space, and this is easily checked to be grouplike. So if nConnTop and A_n are the categories of n-connective spaces and E_n+1-spaces, respectively, then the (n+1)-fold loop space functor Ωⁿ⁺¹ induces an equivalence of categories between nConnTop and A_n. (I believe the statement for E_n+1-spaces in general should be somewhere in May's book; I can't find it, though.)

One implication of this, using the Freudenthal suspension theorem, is that if a homotopy n-type has the structure of a grouplike E_n+2-space, then it canonically has the structure of an E_∞-space, which is a special case of the Baez-Dolan stablization hypothesis. (See here, for example.)