I do think we need an independent reference for this usage and even then it deserve disambiguation. Because, the references that I am aware of use “environment” to refer not to the co-reader but for the reader monad, e.g.:

Also I think the term “indexed monad” needs some substantiation, and of course “simple fibration”.

It seems like you read about these things somewhere, before remarking on them on the nLab. So if you don’t want to spell out details yourself, just add a pointer to where you read about them, so that readers can see what is behind your paragraph.

]]>hyperlink the environment comonad

]]>I have added more hyperlinks, to this example and the following one.

But more substantially I’d urge again to provide more details or else a pointer to the literature. (I assume by the “environment comonad” you mean the one whose coproduct is induced from the diagonal map on the given object.)

]]>Add the simple fibration as an example

]]>have tried to brush-up the list of references (publication data, order of and comments around citations)

]]>added the DOI-link for

- Jean Bénabou,
*Fibered Categories and the Foundations of Naive Category Theory*, The Journal of Symbolic Logic, Vol.**50**1 (1985) 10-37 [doi:10.2307/2273784]

and deleted the whimsical commentary around it; instead moved it way up the list to after the Grothendieck references

and added

- Jean Bénabou,
*Fibrations petites et localement petites*, C. R. Acad. Sci. Paris**281**Série A (1975) 897-900 [gallica]

added pointer to:

- Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Chapter 3 of:
*Fundamental algebraic geometry. Grothendieck’s FGA explained*, Mathematical Surveys and Monographs**123**, Amer. Math. Soc. (2005) [MR2007f:14001, ISBN:978-0-8218-4245-4, lecture notes]

have hyperlinked “fibered limit” in the paragraph here (in order to create that entry now)

]]>added pointer to:

- Bart Jacobs, Chapters 1 and 9 in:
*Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics**141**, Elsevier (1998) [ISBN:978-0-444-50170-7, pdf]

I previously added an example (vector bundles) and forgot to fill out this box!

Emilio Verdooren

]]>Mention “total category” and “base category” terminology.

]]>added pointer to:

- Niles Johnson, Donald Yau, Chapter 9 of:
*2-Dimensional Categories*, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)

elaboration of the properties section – limits and cartesian closed fibrations. Hope I didn’t make a mistake.

]]>Add a bit more language to the “idea” section to ensure the reader doesn’t start off with the mistaken assumption that $Fib(B)$ is a full subcategory of $Cat/B$.

]]>Added a paragraph explaining Grothendieck fibrations versus presheaves of categories.

]]>Sharon Hollander’s thesis constructs a Quillen equivalence between three model categories: Grothendieck fibrations in groupoids, pseudofunctors valued in groupoids, and presheaves valued in groupoids.

Is there a written reference for the same statement with groupoids replaced by categories?

There are in fact two ways to rectify a Grothendieck fibration: the left adjoint “adds formal pullbacks” and the right adjoint “chooses pullbacks along all possible morphisms”. Ideally, both would be discussed in such a source.

I am only aware of a quasicategorical analog, as explained in the work of Joyal, Lurie, and Heuts-Moerdijk. But these sources do not talk about the middle category (the one with pseudofunctors), which, of course, can also be treated using available rectification tools. In any case, quasicategories are a bit of an overkill here.

]]>Thanks, I agree; I added more links and made the terminology align.

]]>made “weakly cartesian” link to prefibered category since these seem like they should either be the same thing or are closely related. Would be interested to know if I’m wrong.

]]>Alright, I added a summary of Mike’s hint that I think would have been enough to get me by. (I also explicitly introduced the terminology “cartesian lifting of f to e”, which I hope is okay with you all.)

]]>Please be so kind to add a remark on this to the entry, so that the next reader stumbling over this will know.

]]>Yes, I think that works!

]]>I *thought* the following argument works. Suppose $\phi:e'\to e$ is weakly cartesian for $p:E\to B$. Since $p$ is a fibration, there is a cartesian arrow $\psi:e''\to e$ with $p(\psi) = p(\phi)$. Then the universal properties of $\phi$ and $\psi$ give inverse isomorphisms $e'\to e''$ and $e''\to e'$ lying over the identity. Thus, since $\phi$ is isomorphic to a cartesian arrow, it is also cartesian.

I have a small technical question here. It it true (as the article says) that “In a fibration, every weakly cartesian arrow is cartesian”? If so, can someone explain how to demonstrate this? I can see how to show that in an *opfibration*, every weakly cartesian arrow is cartesian, and dually, in a fibration that every weakly opcarteesian arrow is opcartesian, but I can’t see how to show the quoted statement.

I added a section “Properties” to Grothendieck fibration, as a place to put remarks about various things one can lift along a fibration (limits, colimits, factorizations).

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