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15 votes

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 ...
Keith Kearnes's user avatar
14 votes

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 $\mathbf{Cat}$ ...
Peter Haine's user avatar
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 ...
varkor's user avatar
  • 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 ...
aws's user avatar
  • 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-...
Luca Mesiti's user avatar
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 ...
Giacomo's user avatar
  • 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 (...
Maxime Ramzi's user avatar
  • 11.3k
1 vote

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 ...
Thomas Rot's user avatar
  • 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 ...
Tim Campion's user avatar
  • 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 ...
Tim Campion's user avatar
  • 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):
David Roberts's user avatar
  • 32.2k

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