Phantom map

In homotopy theory, phantom maps are continuous maps $f: X \to Y$ of CW-complexes for which the restriction of $f$  to any finite subcomplex $Z \subset X$  is inessential (i.e., nullhomotopic). produced the first known nontrivial example of such a map with $Y$ finite-dimensional (answering a question of Paul Olum). Shortly thereafter, the terminology of "phantom map" was coined by, who constructed a stably essential phantom map from infinite-dimensional complex projective space to $S^3$. The subject was analysed in the thesis of Gray, much of which was elaborated and later published in. Similar constructions are defined for maps of spectra.

Definition
Let $$\alpha$$ be a regular cardinal. A morphism $$f: x \longrightarrow y$$ in the homotopy category of spectra is called an $$\alpha$$-phantom map if, for any spectrum s with fewer than $$\alpha$$ cells, any composite $$s \longrightarrow x \xrightarrow{f} y$$ vanishes.