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The nLab article Grothendieck construction states that the left adjoint to the Grothendieck construction can be described as a certain construction with comma categories.
Is there a citeable reference for this?
Do I understand it correctly that the adjoint is understood in the bicategorical (or ∞-categorical) sense, and not in the sense of 1-categories?
The reason for my asking this is that Heuts and Moerdijk in arXiv:1308.0704v5 described two adjunction between right fibrations in simplicial sets and simplicial presheaves, and in their notation the pair h_! ⊣ h^* is the Grothendieck construction and its right adjoint (the traditional rectification functor that uses cleavages), whereas r_! ⊣ r^* is the generalization of the left adjoint described in nLab and Lurie’s relative nerve functor.
Heuts and Moerdijk show that there is a zigzag of weak equivalences between h_! and r^*, which would seem to validate nLab’s claim as long as the left adjoint is taken in the bicategorical sense. The functors h_! and r^* are, however, not isomorphic, which would seem to imply that there is no left adjoint to the Grothendieck construction in the 1-categorical sense.
The 2-categorical case is discussed in
(He refers to
as well.)
Addendum: I had in mind in particular section 1 of the first paper but on closer inspection Street gives only a right adjoint to $\mathcal{G}$ there!
Dimitri: the Grothendieck construction, viewed, for any fixed category $I$, as a functor $Cat^I \rightarrow Cat / I$, does have a strict left adjoint, namely the functor $Cat / I \rightarrow Cat^I$ given on objects by $(F : A \rightarrow I) \rightarrow (i \mapsto A / i)$, where $A / i$ is the comma category defined using $F$. A reference is Proposition 3.1.2 of this paper of Maltsinitiois, but the result certainly is from long before this. It is important in the construction of derivators. Maltsiniotis’ discussion is based on that of Grothendieck in Pursuing Stacks, in the section where he introduces derivators.
The result is actually straightforward, but the nLab article Grothendieck construction gives some weird construction of the left adjoint using quasi-categories which may not give that impression.
Thanks for the references!
I guess my confusion was caused by the fact that r^* and h_! are isomorphic in the 1-categorical case.
Thus for 1-categories we actually have an adjoint triple r_! ⊣ r^* = h_! ⊣ h^*, whereas in for simplicial sets we can only hope for a pair of adjoint functors r_! ⊣ r^*, h_! ⊣ h^*, where the derived functors of r^* and h_! are weakly equivalent (Theorem C in Heuts-Moerdijk), but no longer isomorphic.
I added a new section in the article: https://ncatlab.org/nlab/show/Grothendieck+construction#_2
I also added references to Maltsiniotis and Heuts-Moerdijk.
Dmitri,
thanks for your additions!
I went and added some more hyperlinks to your text, and added arXiv links for the references.
Also, I took the liberty of giving your sub-section an anchor name. It’s more robust to do this before linking to any subsection. So now your section is behind this link.
@Urs: Thanks! I wasn’t aware about the anchor name stuff.
After revisiting this topic, I discovered that r^* and h_! are not isomorphic in the 1-categorical case. In fact, r^* recovers the classical Grothendieck construction, whereas h_! lands in nonfibrant objects of the Joyal model structure, so clearly it cannot be isomorphic to r^*.
I fixed the article accordingly.
My new point of confusion is that r^* seems to have a right adjoint in the classical case (or at least that’s what the Grothendieck construction article claims), but Heuts and Moerdijk make no mention of the right adjoint of r^*.
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