Exponentiable if and only if compactly assembled

For the “only if” direction, assume that \(T\) is exponentiable. To show it is compactly assembled, consider some compactly assembled category \(C\) and consider \(S := \Fun^{\omega}(C,\An)\). By assumption, the exponential \(S^T\) exists. Moreover, we have In particular, taking \(C = \An\), we see that \(S^T\) classifies \(T\)-valued sheaves. By Proposition 6.148, it follows that \(T\) is compactly assembled.For the “if” direction, assume \(T\) is compactly assembled. By Lemma 6.155, we may write \(S\) as a pullback in \(\Topos\) of a square of the form

where \(A_0\), \(B_0\) and \(C_0\) are presheaf topoi. Since exponentiation by \(T\), whenever defined, is right adjoint to \((-)\otimes T\), it preserves limits. It is therefore enough to construct \(S^T\) when \(S\) is a presheaf topos.Assume then that \(S = \PSh(D)\), where \(D\) has finite limits. We have Set \(K:=\Ind(D\catop)\). The category \(K\) is compactly generated, so \(K\otimes T\) is compactly assembled by Lemma 6.154. Its classifying topos \(E\) exists by Proposition 6.148, and for every topos \(U\) we have Thus \(E\simeq S^T\). This proves the theorem; compare [Anel and Lejay 2018, Theorem 4.33].