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Let $X$ be a CW-spectrum. It is well-known that $[\_ ,X]$ is a generalized cohomology theory and, by Brown's representability theorem, every generalized theory is $H$ represented by a spectrum (namely, $H$ has this form).

What about $[X, \_]$? Is it a homology theory? (I do not claim every homology is corepresented by a spectrum.)

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    $\begingroup$ @LSpice thank you. $\endgroup$
    – Victor TC
    May 21, 2020 at 23:36
  • $\begingroup$ No worries. You can also rollback an edit, rather than making the opposite edit. $\endgroup$
    – LSpice
    May 21, 2020 at 23:39

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This holds only for compact objects (i.e. finite CW spectra), since it is easy to see that additivity fails otherwise (the other axioms of homology theories are satisfied). The usual way to obtain a homology theory from a spectrum $X$ is to consider $\pi_*(X\otimes -)$, note that for compact $X$ your $[X,-]$ is of this form as well, since $$ [X,-] = \pi_*(DX\otimes -) $$ by Spanier-Whitehead duality.

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  • $\begingroup$ I also suspected that some conditions on X were necessary, but these notes left me puzzled (see page 4), thank you!. $\endgroup$
    – Victor TC
    May 22, 2020 at 14:43

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