[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 &Sigma; of Top we should form the colimit, in the (&infin;,1)-category Pres of presentable categories and colimit-preserving functors, of the sequence Top &rarr; Top &rarr; ... where all the functors in the diagram are &Sigma;.  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 (&infin;,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 &larr; Top &larr; ... where the functors are &Omega;, and the limit of this sequence is precisely the classical definition of (&Omega;-)spectrum: a sequence of spaces X<sub>n</sub> with equivalences X<sub>n</sub> &rarr; &Omega;X<sub>n+1</sub>.