36
votes
Accepted
Is every commutative ring a limit of noetherian rings?
The answer is no to all questions except 4.
Negative answers to 1,2 and 3:
It is easy to construct a ring $A$ with an element $a$ satisfying:
(i) $a≠0$,
(ii) $a$ is nilpotent,
(iii) for each $n≥1$...
- 19.6k
25
votes
Accepted
Shapes for category theory
I think focusing on graphs is not a good idea. We focus on functors for very good reasons. Here are a few:
Many diagrams which are used in practice are functors between categories, and forgetting ...
23
votes
Is there a useful limit or co-limit of a diagram that has only a single object?
One example: Let $X$ be a set with a group $G$ acting on it. Consider the diagram, in the category of sets, with $X$ as its only object, but with all the elements of $G$ (considered as permutations of ...
- 73.1k
23
votes
Conceptual reason that monadic functors create limits?
The heart of the diagram chase is: All the operations and equations involved in an algebra structure on X have target X. (More generally: the target of each operation/equation could be any limit-...
- 21.3k
22
votes
Accepted
Does a fully faithful functor always preserve limits and colimits?
The forgetful functor from abelian groups to groups is fully faithful, and does not preserve coproducts. For example, in abelian groups, $\mathbb Z\coprod \mathbb Z=\mathbb Z\times \mathbb Z$, but in ...
- 476
17
votes
Accepted
Defining abstract varieties and their morphisms over a finitely generated subfield of the base field
This is treated (in much greater generality) in EGA IV$_3$, Théorème 8.8.2. The existence of $X_0$ and $Y_0$ follows from part (ii), whereas the existence of $f_0$ is part (i).
References.
[EGA IV$...
- 28.7k
17
votes
Accepted
Do filtered colimits commute with finite limits in the category of pointed sets?
Yes, filtered colimits commute with finite limits in the category of pointed sets. This is because the forgetful functor from the category of pointed sets to the category of sets creates finite limits ...
- 5,539
15
votes
Cofinality for coends?
A sufficient condition for a functor $u:I\to J$ to induce a cofinal functor $Tw(I)\to Tw(J)$ is that $u$ is universally cofinal (i.e. any base change of $u$ is cofinal). Another sufficient condition ...
- 13.6k
15
votes
Accepted
Is there a large colimit-sketch for topological spaces?
The answer is no. I think this example illustrates why pure colimit sketches are rarely studied; one generally allows limits into the sketch before generalizing to allow colimits into the sketch. That ...
- 63.9k
14
votes
Accepted
Calculating limits progressively
Let me try to turn my comments into an answer (I think it's also essentially what Vladimir was saying). Suppose you have some diagram $F: K \to \mathcal{C}$. To compute the limit of $F$ is the same as ...
- 13.5k
14
votes
Shapes for category theory
On questions 1 and 2: how would you handle reflexive coequalizers with the graphs-as-shapes approach? I agree that it can be done, but isn't it more natural with categories as shapes?
On question 3: ...
13
votes
Is every commutative ring a limit of noetherian rings?
Binary coproducts do not always exist in $\textrm{Noeth}$.
Assume that the coproduct $C=\overline{\mathbb{Q}} \sqcup \overline{\mathbb{Q}}$ exists in $\textrm{Noeth}$. Letting $A=\overline{\mathbb{Q}}...
- 20.8k
13
votes
Do filtered colimits commute with finite limits in the category of pointed sets?
I find it sometimes useful to remember that the stated commutation property holds in any ind-category. Pointed sets is ind-(finite pointed sets).
Of course, to verify your particular case by hand, ...
- 6,162
13
votes
Accepted
Original reference for categories of presheaves as free cocompletions of small categories
The earliest reference I can find to the universal property of the presheaf construction is Proposition 9.1 of André's Categories of Functors and Adjoint Functors (1966).
There is an earlier reference ...
- 10.6k
13
votes
Accepted
Homotopy groups of categories of elements as higher colimits
To answer these questions, the best is to note that $|\int_C D|$, the geometric realization of this total category, is equivalently the colimit of $D$, viewed as a functor with values in the $\infty$-...
- 15.8k
12
votes
Accepted
Existence of homotopy limits and colimits in model categories
Let me start first answering your second question (this is quite close to the idea of John Klein).
Definition.
A pair $(\mathbf C,\mathcal W)$, with $\mathbf C$ a category and $\mathcal W$ a class of ...
- 2,452
12
votes
Accepted
What's the intuition for weighted limits?
In enriched category theory, weighted limits may be strictly more general than conical limits, in the sense that an enriched category with all conical limits may fail to have all weighted limits.
...
- 15.9k
12
votes
Accepted
Why isn't pullback-stability defined for individual colimits but for colimits with the same shape?
Pullback-stability is sometimes considered for individual colimits, or at least, smaller classes than “all colimits of shape $D$”. However, it’s most often used in settings where it holds for large ...
- 21.3k
11
votes
Is there a tricategory of bicategories and biprofunctors?
For those coming across this question more recently, there is now an answer to the original question. In fact, the tricategory of pseudoprofunctors has been defined twice, independently, via different ...
- 10.6k
11
votes
Accepted
Cocomplete and finitely complete category with nice pullbacks that is not locally presentable
If you look at the article on quasitoposes, which are locally cartesian closed and therefore satisfy your exactness condition, you'll find a number of examples that are not locally presentable. For ...
- 53.3k
11
votes
Accepted
Ferrand pushouts for algebraic stacks
Yes, this is exactly Theorem A.4 in my old preprint Compactification of tame Deligne–Mumford stacks which is long overdue to appear on the arXiv. The proof is rather terse but fairly standard (compare ...
- 5,039
11
votes
Accepted
Does Grothendieck's algebraization imply existence of colimits of schemes?
Here's one way to see what's going on. I will use the Tannakian duality theorem of Hall and Rydh (see Theorem 1.1 here). It is stated for algebraic stacks, but if you replace the word "algebraic ...
- 2,607
11
votes
Filling square to push-out in abelian category
If there is such a pushout, then $B\to D$ is also a monomorphism, i.e. $B$ is a subobject of $D$.
Phrased more concretely, you're asking when there is a subobject $B$ of $D$ such that $B+C = D$ and $B\...
- 15.8k
11
votes
Shapes for category theory
Here are some leads for 1. and 2. (this is far from a full answer, but slightly too long for a comment). I only went in the "disadvantages of this approach" direction, too so that's another ...
11
votes
Accepted
Finite coproducts commute with which limits in Set?
Indeed, let $D$ be a category. The canonical functor $D \to \pi_0(D)$ is both cofinal and coinitial. Therefore, if finite coproducts commute with $D$-limits in a category $\mathcal C$, then finite ...
- 63.9k
11
votes
Accepted
Does the category of locally compact Hausdorff spaces with proper maps have products?
Suppose $X,Y$ are locally compact Hausdorff spaces admitting a product $X\otimes Y$ in the category $\mathcal{H}$ of all such spaces. If one of $X,Y$ is empty, then $X\otimes Y$ exists, so I'll assume ...
- 5,596
10
votes
Defining abstract varieties and their morphisms over a finitely generated subfield of the base field
A variety $V$ is a finite union of open affine varieties $V_i$, and because $V$ is separated (usually part of the definition of variety) the intersections $V_i\cap V_j$ are also affine. Now $V$ can be ...
- 101
10
votes
Accepted
Behaviour of direct limit with matrices
You’re asking whether the functor $M_2$ on Banach spaces preserves colimits of direct sequences.
(In case you’re not familiar with the categorical terminology I’m using here, don’t be put off — it’s ...
- 21.3k
10
votes
Colimits in the category of (not necessarily locally convex) topological vector spaces
I found a construction of the coproduct in the category of topological abelian groups
from this reference: https://core.ac.uk/download/pdf/82771298.pdf,
which can be also applied to the category of ...
- 419
10
votes
Accepted
Colimits in the category of (not necessarily locally convex) topological vector spaces
The Springer Lecture Notes 639 Topological Vector Spaces of Adasch, Ernst, and Keim contain in § 4 a more or less explicit construction of inductive (=co-) limits in the category of topological vector ...
- 16.4k
Only top scored, non community-wiki answers of a minimum length are eligible