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Although Σ and Ω are autoequivalences of Spectra, so that the maps X → ΩΣX and ΣΩX → X are equivalences, that does not mean that Σ^∞ sends maps of these forms in Top to equivalences in Spectra, because Σ^∞ does not commute with Ω. For instance, if X is the space S⁰, then ΣΩX is a point, and ΣΩX → X is clearly not sent to an equivalence by Σ^∞. So, you are not going to get such a functor F which Σ^∞ factors through.

You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top as a presentable (∞,1)-category. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming ℤ[1/p] and ℤ/(p).

Here is one way to verify the claim. To invert the endomorphism Σ of Top we should form the colimit, in the (∞,1)-category Pres of presentable categories and colimit-preserving functors, of the sequence Top → Top → ... where all the functors in the diagram are Σ. A basic fact about Pres is that we can compute such a colimit by forming the diagram (on the opposite index category) formed by the right adjoints of these functors, and taking its limit as a diagram of underlying (∞,1)-categories [HTT 5.5.3.18]. The functors in the limit cone will have left adjoints which are the functors to the colimit in Pres. In our case we obtain the sequence Top ← Top ← ... where the functors are Ω, and the limit of this sequence is precisely the classical definition of (Ω-)spectrum: a sequence of spaces X_n with equivalences X_n → ΩX_n+1.

[Removed a paragraph relating to an earlier version of the question]

You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top as a presentable (∞,1)-category. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming ℤ[1/p] and ℤ/(p).

Here is one way to verify the claim. To invert the endomorphism Σ of Top we should form the colimit, in the (∞,1)-category Pres of presentable categories and colimit-preserving functors, of the sequence Top → Top → ... where all the functors in the diagram are Σ. A basic fact about Pres is that we can compute such a colimit by forming the diagram (on the opposite index category) formed by the right adjoints of these functors, and taking its limit as a diagram of underlying (∞,1)-categories [HTT 5.5.3.18]. The functors in the limit cone will have left adjoints which are the functors to the colimit in Pres. In our case we obtain the sequence Top ← Top ← ... where the functors are Ω, and the limit of this sequence is precisely the classical definition of (Ω-)spectrum: a sequence of spaces X_n with equivalences X_n → ΩX_n+1.

Although Σ and Ω are autoequivalences of Spectra, so that the maps X → ΩΣX and ΣΩX → X are equivalences, that does not mean that Σ^∞ sends maps of these forms in Top to equivalences in Spectra, because Σ^∞ does not commute with Ω. For instance, if X is the space S⁰, then ΣΩX is a point, and ΣΩX → X is clearly not sent to an equivalence by Σ^∞. So, you are not going to get such a functor F which Σ^∞ factors through.

You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top as a presentable (∞,1)-category. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming ℤ[1/p] and ℤ/(p).

Although Σ and Ω are autoequivalences of Spectra, so that the maps X → ΩΣX and ΣΩX → X are equivalences, that does not mean that Σ^∞ sends maps of these forms in Top to equivalences in Spectra, because Σ^∞ does not commute with Ω. For instance, if X is the space S⁰, then ΣΩX is a point, and ΣΩX → X is clearly not sent to an equivalence by Σ^∞. So, you are not going to get such a functor F which Σ^∞ factors through.

You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top as a presentable (∞,1)-category. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming ℤ[1/p] and ℤ/(p).

Here is one way to verify the claim. To invert the endomorphism Σ of Top we should form the colimit, in the (∞,1)-category Pres of presentable categories and colimit-preserving functors, of the sequence Top → Top → ... where all the functors in the diagram are Σ. A basic fact about Pres is that we can compute such a colimit by forming the diagram (on the opposite index category) formed by the right adjoints of these functors, and taking its limit as a diagram of underlying (∞,1)-categories [HTT 5.5.3.18]. The functors in the limit cone will have left adjoints which are the functors to the colimit in Pres. In our case we obtain the sequence Top ← Top ← ... where the functors are Ω, and the limit of this sequence is precisely the classical definition of (Ω-)spectrum: a sequence of spaces X_n with equivalences X_n → ΩX_n+1.

Although Σ and Ω are autoequivalences of Spectra, so that the maps X → ΩΣX and ΣΩX → X are equivalences, that does not mean that Σ^∞ sends maps of these forms in Top to equivalences in Spectra, because Σ^∞ does not commute with Ω. For instance, if X is the space S⁰, then ΣΩX is a point, and ΣΩX → X is clearly not sent to an equivalence by Σ^∞. So, you are not going to get such a functor F which Σ^∞ factors through.

You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming ℤ[1/p] and ℤ/(p).

Although Σ and Ω are autoequivalences of Spectra, so that the maps X → ΩΣX and ΣΩX → X are equivalences, that does not mean that Σ^∞ sends maps of these forms in Top to equivalences in Spectra, because Σ^∞ does not commute with Ω. For instance, if X is the space S⁰, then ΣΩX is a point, and ΣΩX → X is clearly not sent to an equivalence by Σ^∞. So, you are not going to get such a functor F which Σ^∞ factors through.

You can construct Spectra categorically by adjoining an inverse to the endofunctor Σ of Top as a presentable (∞,1)-category. Inverting an endofunctor is a very different operation than inverting maps! It's like the difference between forming ℤ[1/p] and ℤ/(p).