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We prove two representability theorems, up to homotopy, for presheaves taking values in a closed symmetric combinatorial model category \cat V. The first theorem resembles the Freyd representability theorem, the second theorem is closer to the Brown representability theorem. As an application we discuss a recognition principle for mapping spaces., Comment: V2: minor improvements suggested by the referee; 10 pages

Let L 2 L_2 denote the Bousfield localization functor with respect to the Johnson-Wilson spectrum E ( 2 ) E(2) . A spectrum L 2 X L_2X is called invertible if there is a spectrum Y Y such that L 2 X ∧ Y = L 2 S 0 L_2X\wedge Y=L_2S^0 . Hovey and Sadofsky, Invertible spectra in the E ( n ) E(n) -local stable homotopy category, showed that every invertible spectrum is homotopy equivalent to a suspension of the E ( 2 ) E(2) -local sphere L 2 S 0 L_2S^0 at a prime p > 3 p>3 . At the prime 3 3 , it is shown, A relation between the Picard group of the E ( n ) E(n) -local homotopy category and E ( n ) E(n) -based Adams spectral sequence, that there exists an invertible spectrum X X that is not homotopy equivalent to a suspension of L 2 S 0 L_2S^0 . In this paper, we show the homotopy equivalence v 2 3 : Σ 48 L 2 V ( 1 ) ≃ V ( 1 ) ∧ X v_2^3\colon \Sigma ^{48}L_2V(1)\simeq V(1)\wedge X for the Smith-Toda spectrum V ( 1 ) V(1) . In the same manner as this, we also show the existence of the self-map β : Σ 144 L 2 V ( 1 ) → L 2 V ( 1 ) \beta \colon \Sigma ^{144}L_2V(1)\to L_2V(1) that induces v 2 9 v_2^9 on the E ( 2 ) ∗ E(2)_* -homology.

Let G G be a finitely generated nilpotent group. The object of this paper is to identify the Bousfield localization L h B G L_hBG of the classifying space B G BG with respect to a multiplicative complex oriented homology theory h ∗ h_{*} . We show that L h B G L_hBG is the same as the localization of B G BG with respect to the ordinary homology theory determined by the ring h 0 h_0 .