17
votes
What is the geometric significance of fibered category theory in topos theory?
My short answer, is that in the great majority of situation where this $\mathcal{Y}/f^*$ plays a role, it is a mistake to see it as a topos. There are exceptions, but most of the time it is not meant ...
- 42.4k
11
votes
Accepted
Yoneda Lemma for internal presheaves
The Yoneda lemma holds in any finitely complete $2$-category $\mathscr{K}$, such as the $2$-category $\text{Cat}(\mathscr{C})$ of internal categories in a finitely complete category $\mathscr{C}$.
...
- 5,539
9
votes
Accepted
Can we show that a functor is a fibration without choosing a cleavage?
Just as an example, given a category $\mathcal{C}$ with finite limits, showing $\mathrm{cod}\colon \mathcal{C}^\mathbf{2}\to \mathcal{C}$ is a fibration does not involve choosing a cleaving. All that ...
- 35.4k
8
votes
English Reference for the Bénabou-Roubaud theorem
The following is a reasonably literal translation (made by me from the French original) of the published article: Jean Bénabou, Jacques Roubaud. Monades et descente. C. R. Acad. Sc. Paris, t. 270 (12 ...
- 6,051
8
votes
Relationship between enriched, internal, and fibered categories
Too long for a comment. Thanks for compiling this bibliography, Ivan! It seems you've got the literature at your fingertips to get a sense for the available tools.
One thing to say is that in contrast ...
- 63.9k
6
votes
Accepted
Schemes as categories fibered in thin groupoids
This is essentially the functor of points approach to schemes.
Define the site AffSch of affine schemes as the opposite
category of commutative rings, equipped with the Zariski Grothendieck topology.
...
- 37.8k
6
votes
English Reference for the Bénabou-Roubaud theorem
This is incoporated into an existing answer for several reasons, e.g. to avoid moving it to the front page. I only noticed this reference now, when I continued to work on this. While the 'translation' ...
- 6,051
5
votes
English Reference for the Bénabou-Roubaud theorem
I cooked up a detailed proof in this file, even if it is not original, because I wanted to understand everything.
About Zoran Škoda's blog: it is factually false. Fibred categories appear right at the ...
- 532
4
votes
What is the geometric significance of fibered category theory in topos theory?
One interesting perspective is given by Spencer Breiner's thesis. Here, the codomain fibration $\mathcal Y^{[1]} \to \mathcal Y$ of a topos $\mathcal Y$, viewed as a sheaf of "local" logoi on $\...
- 63.9k
4
votes
English Reference for the Bénabou-Roubaud theorem
Here is the copy of The Bénabou-Roubaud monadic descent theorem via string diagrams paper:
https://www.dropbox.com/s/236deicmr3636d5/draft.pdf?dl=0
Disclaimer: the papers was, and still is, just a ...
- 41
3
votes
Internal equality for Eq-fibrations' morphisms
Yes, I agree with your argument that any two equivalent morphisms in this sense must be equal, if all fibers of $q$ are non-empty. To present it a bit more concisely: Any fibration with non-empty ...
- 21.3k
3
votes
Accepted
What is a correct notion of an internal pseudofunctor?
Let $\mathcal{C}$ be a category with pullbacks, with ${\sf C}$ an internal category in $\mathcal{C}$. We define a category $${\sf Ext( C})$$ called the externalization of ${\sf C}$ as follows:
The ...
- 10.1k
3
votes
Relationship between enriched, internal, and fibered categories
Earlier this year Lyne Moser gave a talk at an online workshop for double categories which discussed some recent (joint) work that I think is relevant to this question (if I understand it correctly). ...
- 1,109
3
votes
What is the geometric significance of fibered category theory in topos theory?
See my notes on fibered categories (arXiv:1801.02927) where in the second part I expose the fibered view of gemetric morphisms explaining that bounded geometric morphisms to a base topos SS are ...
- 121
3
votes
Accepted
Fiberwise skeleton vs. category of isomorphism types
As you say, for a general category it is not possible to form an equivalent category from its isomorphism classes. However, if the category is thin (i.e. a preorder) then this is possible, and gives ...
- 66.7k
2
votes
Accepted
Closure properties of fibrations
Your proof is nice, except for the first piece of $(2)$ where the lemma asks you to show that all $F_I$ are fibrations, but you begin by assuming that $F_J$ is. Rather, proceed as follows:
We wish to ...
- 10.1k
2
votes
Morphisms of fibered categories which are compatible with the chosen cleavages
I'm sure there are better references, but if you have none my notes on researchgate address these topics from pages 193 onward. They aren't complete in any sense of the word, but should provide enough ...
- 10.1k
2
votes
Accepted
Is there any relation between two pseudofunctors associated to two different cleavages of the same fibered category?
Two different cleavages produce isomorphic pseudofunctors.
This follows immediately from Theorem 8.3.1
in Borceux's Handbook of Categorical Algebra 2.
Specifically, part (1) of this theorem states
...
- 37.8k
2
votes
English Reference for the Bénabou-Roubaud theorem
There is an excellent translation of the French article presenting the
Benabou–Roubaud Theorem to English by Zoran Škoda which he certainly will provide on request.
On p.101 of Fibered Categories (à ...
- 121
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