Under what conditions do left adjoints preserve products?

Question

Are there any results concerning under what conditions a left adjoint preserves products?

Answer 1

An interesting situation (and frequent, too) is when your left adjoint is part of an adjunction of the form "left Kan extension-nerve functor". In that situation, if you are extending a flat functor, then your left-adjoint preserves finite limits. An example is given by the geometric realization of simplicial sets followed by the forgetful functor $Top \to Set$. We get the desired result since $[n] \mapsto \Delta^n$ is a flat functor $\Delta \to Set$ (for example because $\Delta$ admits a terminal object).

Answer 2

Examples have already been given when the left adjoint preserves finite limits, in particular finite products. To get all products, the simplest situation is, naturally, when the left adjoint is also a right adjoint, thus, when we have an adjoint triple $f^*\vdash f_*\vdash f^{!}$. Perhaps the most elementary example of this is when we have a homomorphism of commutative rings $f:A\to B$, which induces functors on module categories with $f^*(M)=B\otimes_A M$ extension of scalars, $f_*(N)=N_A$ restriction of scalars, and $f^!(M)=\hom_A(B_A,M)$ coextension of scalars. But if you're looking for an abstract categorical condition, asking for an adjoint triple is about the best you'll do.

Answer 3

If $u : C \to D$ is a functor commuting with finite products, then the left adjoint $u_!$ to the restriction of presheaves $u^* : P(D) \to P(C)$ commutes with finite products.