Editing Spectrum (topology) (section)

== Technical complexities with spectra ==
One of the canonical complexities while working with spectra and defining a category of spectra comes from the fact each of these categories cannot satisfy five seemingly obvious axioms concerning the infinite loop space of a spectrum <math>Q</math><blockquote><math>Q: \text{Top}_* \to \text{Top}_*</math></blockquote>sending<blockquote><math>QX = \mathop{\text{colim}}_{\to n}\Omega^n\Sigma^n X</math></blockquote>, a pair of adjoint functors <math>\Sigma^\infty: \text{Top}_* \leftrightarrows \text{Spectra}_* : \Omega^\infty</math>, and the smash product <math>\wedge</math> in both the category of spaces and the category of spectra. If we let <math>\text{Top}_*</math> denote the category of based, compactly generated, weak Hausdorff spaces, and <math>\text{Spectra}_*</math> denote a category of spectra, the following five axioms can never be satisfied by the specific model of spectra:<ref name=":0" />

# <math>\text{Spectra}_*</math> is a symmetric monoidal category with respect to the smash product <math>\wedge</math>
# The functor <math>\Sigma^\infty</math> is left-adjoint to <math>\Omega^\infty</math>
# The unit for the smash product <math>\wedge</math> is the sphere spectrum <math>\Sigma^\infty S^0 = \mathbb{S}</math>
# Either there is a natural transformation <math>\phi: \left(\Omega^\infty E\right) \wedge \left(\Omega^\infty E'\right) \to \Omega^\infty\left(E \wedge E'\right)</math> or a natural transformation <math>\gamma: \left(\Sigma^\infty E\right) \wedge \left(\Sigma^\infty E'\right) \to \Sigma^\infty\left(E \wedge E'\right)</math> which commutes with the unit object in both categories, and the commutative and associative isomorphisms in both categories.
# There is a natural weak equivalence <math>\theta: \Omega^\infty\Sigma^\infty X \to QX</math> for <math>X \in \operatorname{Ob}(\text{Top}_*)</math> which means that there is a commuting diagram:<blockquote><math>\begin{matrix}
  X & \xrightarrow{\eta} & \Omega^\infty\Sigma^\infty X \\
  \mathord{=} \downarrow & & \downarrow \theta \\
  X & \xrightarrow{i} & QX
\end{matrix}</math></blockquote>where <math>\eta</math> is the unit map in the adjunction.

Because of this, the study of spectra is fractured based upon the model being used. For an overview, check out the article cited above.