15
votes
Accepted
Is the Cartesian product of two finitely presented objects finitely presentable?
No. For counterexamples, see Theorems 3.8, 3.9, and 3.10 of
Finiteness properties of direct products
of algebraic structures
Peter Mayr, Nik Ruškuc
Journal of Algebra 494 (2018) 167-187.
These ...
- 10.7k
14
votes
Accepted
Category of small categories is not adhesive?
Consider the following commutative cube of posets and inclusions:
The bottom face is a pushout in both $\mathbf{Cat}$ and $\mathbf{Poset}$. The vertical faces are all pullbacks in both $\mathbf{Cat}$ ...
- 256
6
votes
Who introduced the notion of 2-categories?
It appears that the definition of 2-category was introduced independently by two authors, both of whom independently introduced the modern notion of enriched category, for which 2-categories appeared ...
- 6,592
4
votes
(When) do filtered colimits exist in the effective topos?
There are probably easier ways to see this, but my favourite example is to look at the filtered colimit over all finite coproducts of $1$ with inclusions. We can also view this as a countable sequence ...
- 3,461
4
votes
2-completeness of stacks
I am very interested in this question. I can only write a partial answer, hoping that someone can complete it or suggest other approaches.
As Kevin Arlin pointed out, Street's papers "2-...
- 41
3
votes
What does it mean for a category to be generated under (some) colimits?
It seems that dense generation does not imply 1-naïve generation in general. For a counterexample it is enough to consider the class $\Phi$ for small (or just finite) coproducts.
Indeed, given any non ...
- 179
2
votes
Pushforward of cocartesian fibrations
Let me try to expand on what I wrote in the comments. I'll focus on left Kan extensions- I think the story for right Kan extensions is a bit more subtle, for reasons I'll ty to mention at the end.
A (...
- 11.3k
1
vote
Accepted
Determinant line of Fredholm operators and composition of morphisms
As mentioned in the comments the first part of the question is in Abbonandolo and Majers "Infinite dimensional Grassmannians". My Hilbert spaces are real and separable and infinite ...
- 6,965
1
vote
Commuting filtered colimits & finite limits in infinitary theories
The category $sSet$ of simplicial sets fits the bill.
As $sSet$ is a topos, it is Barr-exact. Moreover, the coproduct of representables $D = \amalg_{n \in \mathbb N} \Delta^n$ is a projective ...
- 55.4k
1
vote
Is Spec of a ring monoidal or anti-monoidal?
Just because $Spec$ is contravariant on 1-morphisms doesn't mean it has to be contravariant on 0-morphisms.
As indicated in the comments, $Spec$ is naturally monoidal. You can make it anti-monoidal if ...
- 55.4k
1
vote
A bestiary of topologies on Sch
Since it is a long time down the road, it turns out that now Wikipedia has its own page giving examples and links to dedicated pages for each (except the canonical topology):
https://en.wikipedia.org/...
- 32.2k
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