Does F_* have a description in terms of the Grothendieck construction?

Question

Let $F\colon C\to D$ be a functor. Given a functor $\delta\colon D\to Sets$, one can compose with $F$ to get $\delta\circ F\colon C\to Sets$. This process is functorial in the category of $D$-sets, and we denote it $$F^\ast\colon D-set\to C-set.$$ The functor $F^\ast$ has both a left and a right adjoint, denoted $F_!$ and $F_\ast$ respectively.

It turns out that the two left adjoints in this picture, $F^\ast$ and $F_!$ have nice descriptions in terms of the Grothendieck construction. $F^\ast$ is given by pullback: given a functor $\delta\colon C\to Sets$, take its Grothendieck construction $\int\delta\to D$ and form the product with $C\to D$ over $D$; this is $F^\ast\delta$. The "left pushforward" functor $F_!$ is given by a factorization system on $Cat$; given $\gamma\colon C\to Sets$, we have a composition $$\int\gamma\to C\to D$$ which we can factor as an initial functor followed by a discrete op-fibration. The discrete op-fibration is the Grothendieck construction applied to $F_!\gamma$.

Does the "right pushforward" functor $F_\ast$ have any kind of description in terms of Grothendieck constructions? If not, is there a nice reason?

Answer 1

I guess it depends what you would accept as a "description in terms of the Grothendieck construction."

For each $d \in D$, we have the projection $\pi_d : d/F \rightarrow C$. Given some $\gamma : C \rightarrow Set$, I can form the composite $$ \gamma \circ \pi_d : d/F \rightarrow C \rightarrow Set$$

This determines a discrete opfibration $\int \gamma \circ \pi_d$ over $d/F$. Let $\Gamma_{d/F}$ denote the set of sections of the projection $\int \gamma \circ \pi_d \rightarrow d/F$. Then I believe you will find that $F_* \gamma$ is the Grothendieck construction on the functor $d \mapsto \Gamma_{d/F}$.

Of course, this is not really that interesting, since it is really nothing more than the observation that the limit of a functor $F : C \rightarrow Set$ can be calculated as the set of sections of the natural projection $\int F \rightarrow C$, together with definition of the right Kan extension.

On the other hand, it suggests (to me at least) that there is unlikely to be a more "global" description in terms of the Grothendieck construction since the object we are trying to describe is like a "union of a collection of maps," and these to operations tend not to commute (maps from a union is the product of the maps on each component). Probably you knew all this, but maybe somebody will find it useful . . .