Questions tagged [limits-colimits]
For questions about categorical limits and colimits, including questions about (co)limits of general diagrams, questions about specific special kinds of (co)limits such as (co)products or (co)equalizers, and questions about generalizations such as weighted (co)limits and (co)ends.
628
questions
1
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1answer
27 views
Limits in functor categories
Let $C,C’,D$ be categories and $u:C\to C’$ be a functor. The functor $u^*:\mathbf{Hom}(C’^\circ,D)\to\mathbf{Hom}(C^\circ,D)$ that sends a functor $G$ to $G\circ u$ commutes with limits and colimits, ...
2
votes
1answer
24 views
Example of non-filtered colimit not commuting with a finite limit
I know that the canonical morphism $\mathrm{Colim}_i\mathrm{Lim}_jD(i,j)\to\mathrm{Lim}_j\mathrm{Colim}_iD(i,j)$ for a diagram $D:\textbf{I}\times\textbf{J}\to \textbf{Set}$ is not in general an ...
1
vote
1answer
16 views
Sums factor as products in internal hom functor
In case this is helpful, I am working in the category of condensed abelian groups, but I think this can be phrased in a more general categorical way.
Suppose we have a closed symmetric monoidal ...
1
vote
1answer
47 views
Does pushout of schemes along formal neighborhoods exist in the category of schemes?
I have a question about gluing specific types of schemes which doesn't fit into any well-known gluing situation. Assume $C$ is an algebraic curve and $p$ a point on it. The formal completion of $C$ at ...
0
votes
0answers
67 views
Why are there so many different notations for limits and colimits?
My question is relatively simple: Why are there so many different notations for limits and colimits? Is there a reason people don't just use a common convention? Is there any benefit to using one ...
1
vote
1answer
41 views
On a definition of Spivak's fuzzy set
In the paper "Metric Realization of Fuzzy Simplicial Sets" of David Spivak it takes $I=(0,1]$ as poset and consider it as a category. He gives it a Grothendieck topology induce it from ...
2
votes
1answer
36 views
Homotopy push-out squares and exact triangles are colimits
I read somewhere that homotopy push-out squares and exact triangles in a triangulated category can both be interpreted as special cases of higher categorical colimits. Why is this true?
Please note ...
2
votes
2answers
59 views
Inverse limit without category theory
In Hirsch's book Differential Topology, he by and large does not use any category theory, with the exception of one passage on pg. 52 which I am trying to understand. It is as follows (paraphrased):
...
6
votes
1answer
53 views
Relationship between two definitions of pro-representable functors
Edit:
I'm pretty sure that my conjecture
$$
\operatorname{Hom}(\varprojlim_i R/\mathfrak{m}^i, A) = \operatorname{colim}_i \operatorname{Hom}(R/\mathfrak{m}^i, A),
$$
is true. To prove it, just use ...
4
votes
1answer
73 views
Right exactness of projective systems
Suppose that we have a systems of exact sequences $(A_{n}\rightarrow B_{n}\rightarrow C_{n}\rightarrow 0)_{n\in \mathbb{N}}$ together with transitions maps $(A_{n+1}\rightarrow A_{n})_{n\in \mathbb{N}}...
5
votes
1answer
54 views
A directed inverse limit of finite connected spaces is connected
Apparently, a directed inverse limit of finite connected spaces is connected. Does anyone have a reference?
1
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0answers
31 views
General Properties of Direct Limits of Groups
Anyone know a decent reference on the basic theory of direct limits of groups, from an elementary (meaning group theoretic, as opposed to categorical) perspective? Finitely generated abelian groups ...
0
votes
0answers
41 views
Calculating finite inverse limits of Abelian groups
If we are given an infinite system of abelian groups $(\dots \xrightarrow{\varphi_2} A_2 \xrightarrow{\varphi_1} A_1 \xrightarrow{\varphi_0} A_0)$ then I know its inverse limit can be found by
$$\lim_{...
1
vote
0answers
31 views
Presheaves are the Free Cocompletion - Proving that the functor preserves colimits
I am trying to understand a proof that, for any small category $\mathcal{C}$, the category $\widehat{C} = [\mathcal{C}^\mathrm{op}, \textbf{Set}]$ is the free cocompletion of $\mathcal{C}$. In ...
6
votes
1answer
76 views
Example of complete category with no initial object
My original question is this. I found Zhen Lin's answer very useful, but I couldn't think of a category which is complete but has no initial object. The first category that I thought that has no ...
2
votes
1answer
126 views
Define a sketch $s_{\mathbf{Grp}}$ such that $\mathbf{Grp}\backsimeq \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$
I have the following
(a) Define a sketch $s_{\mathbf{Grp}}$ and a equivalence functor $$E: \mathbf{Grp}\to \mathbf{Mod}(s_{\mathbf{Grp}},\mathbf{Set})$$ (b) Knowing that finite limits commute with ...
0
votes
0answers
52 views
finite limits and filtered colimits in the category of groups
I have the following problem
Describe the finite limits and the filtered colimits in the category $Grp\ $ and show that they commute.
This is my first "more practical" exercise on ...
0
votes
0answers
23 views
Directed system by inclusion.
My book[Algebra - Lang, pg no 161] says the following, " Let $x$ be a point in a Topological space $X$. The Open neighborhood form directed system by Inclusion.", It went on saying that &...
2
votes
2answers
70 views
$\ker \varphi_p \subset (\ker \varphi)_p$ where $(\cdot)_p$ is taking the stalk of sheaves at the point $p$ (Diagram inside!).
I've already proven that $(\ker \varphi)_p \subset \ker \varphi_p$ using a commutative diagram and the definition $F_p = \lim\limits_{\longrightarrow \\ U \ni p} F(U) = \bigsqcup\limits_{U \ni p} F(U)/...
1
vote
1answer
63 views
Strange inverse limit of infinite direct sums $S_n$
Let $p_j$ be the prime numbers in order, indexed by $j$. Define $S_n$ as follows:
\begin{equation}
\begin{aligned}
S_1 &:= \mathbb Z / 2 \mathbb Z &\oplus &\mathbb Z / 2 \mathbb Z &\...
4
votes
1answer
90 views
Limits in the localization of a category of fibrant objects
Suppose we have category $\mathcal{C}$ which has the structure of a category of fibrant objects, and suppose we have a functor $F:I\to \mathcal{C}$ with a limit $\lim F$ in $\mathcal{C}$.
If we have ...
0
votes
0answers
57 views
Stable homotopy groups as a generalized (reduced) homology theory
It is known that $\pi_*^{st}$ defines a generalized homology theory, where $\pi_n^{st}(X) = \text{colim}_{k \geq 0} \pi_{n+k}(\Sigma^k X)$ is the $n$th stable homotopy group of the based space $X$. ...
0
votes
0answers
40 views
Segre Embedding and Group Actions
The segre embedding takes two schemes $X, Y$ with $G_m$ and (I think) produces an embedding $X/G_m \times Y/G_m \hookrightarrow (X \times Y)/G_m$, with $G_m$ acting diagonally on $X \times Y$.
My ...
1
vote
0answers
26 views
Colimit of a (relatively) complicated diagram
I want to compute the colimit of the following diagram, where $A$ is some set.
I was also wondering about how to approach these complicated diagrams in general.
Thanks!
3
votes
0answers
46 views
Argumentation via “Limits are constructed object-wise”
How do the "limits are constructed objectwise thus a property about limits true in $\rm Set$ is also true in $
{\rm{Set}}^I$" argument works?
For example, I encountered the following two ...
0
votes
1answer
34 views
Limit in $\mathbb{Z} / P^{2} \to \mathbb{Z} / P $
Consider the commutative diagram
in Ab, where the morphisms are the canonical ones. Let us denote the top horizontal sequence by A and the bottom one by B. Then the vertical maps induce a morphism of ...
2
votes
1answer
65 views
Inverse limit of $\left(\mathbb{Z}/p^n\mathbb{Z}\right)_{n \in \mathbb{N}}$
In the wikipedia article about the inverse limit it is stated that for a prime number $p$
$$\varprojlim_{n \in \mathbb{N}} \mathbb{Z}/p^n\mathbb{Z} = \mathbb{R}/\mathbb{Z},$$
where the arrow between ...
0
votes
0answers
32 views
Bounded lattice that's isomorphic to proper subcategories of itself
Suppose I have a bounded lattice $L$ and a full subcategory $C\subseteq L$ containing exactly the elements of $L$ that are both join prime and meet prime.
Now suppose $C$ can be partitioned into $n>...
1
vote
1answer
36 views
Find the product and coproduct of the category of Set with a given set
I am learning Category theory and I've found a problem :
Let $S$ be a fixed set. Define a category $\textbf{Set}_S$ , where collection of object is a set map $ f: X \rightarrow S$. Let $f':X' \to S$ ...
1
vote
1answer
34 views
Vanishing direct limit of vector spaces
Let $k$ be a field. Let $\{V_n\}_{n\in \mathbb{N}}$ be a direct system of (possibly infinite-dimensional) $k$-vector spaces. Is it true that if $\varinjlim_{n\in \mathbb{N}}V_n=0$, then $V_m=0$ for ...
2
votes
1answer
51 views
When is a bounded lattice isomorphic to some $(\mathbf{2}^S,\subseteq)$?
If I have a set $S$, I can turn it into a bounded lattice in any number of ways, the simplest of which is to designate an initial object and a terminal object, adding a single object if necessary. ...
1
vote
1answer
55 views
Inverse limit of (sub)sets
Let $(X_i)_{i\in I}$ be a family of subsets $X_i\subset X$ partially ordered by inclusion. If $X_j\subseteq X_i$ let $\iota_{ij}\colon X_j\hookrightarrow X_i$ be the inclusion and write $i\le j$. This ...
4
votes
5answers
112 views
A category which direct limits but no general colimits
I am looking for a (at best, real life) category that has direct limits, but no general small colimits, or a category that has inverse limits, but no general small limits. Are there any interesting ...
-2
votes
1answer
47 views
Composition of $h_1\cdot h_0$ [closed]
How can I see both directions for the case $\lambda=2$ below ?
A picture would help.
3
votes
0answers
90 views
Inverse Limit of Complexs and Homology
Let $\mathcal{A}$ be an abelian category and denote by $D(\mathcal{A})$ its derived category. Let $K_n\in D(\mathcal{A})$ be an inverse system of complexes in $\mathcal{A}$ viewed as elements of the ...
1
vote
0answers
34 views
Generalization of $\lambda$-directedness
It is well known that objects in $\mathbb{Set}$ whose hom-functors preserve
$\lambda$-directed colimits,i.e. colimits whose schemes are $\lambda$-directed posets are those sets whose cardinality is $&...
0
votes
1answer
18 views
Embedding the vertex of a cone inside inverse limit
Let $(R_i,f_{ij})_{i \in I}$ be an inverse system of rings and ring homomorphisms. Let $(L,p_i)_{i \in I}$ be the inverse limit of $(R_i,f_{ij})_{i \in I}$ and let $(K,f_i)_{i \in I}$ be a cone into $...
5
votes
2answers
154 views
Solving a (fun!) coequalizer problem for $\mathrm{SL}_n(\mathbb{R})\rightarrow\mathrm{SL}_n(\mathbb{C})$ in $\mathbf{Grp}$
First off, the problem posed below is mostly arbitrary; it's just for my own education. (And maybe for yours, as well.)
It's fairly clear to me what the (co)equalizers of abelian groups in $\mathbf{...
0
votes
0answers
62 views
Colimit of coproduct
Let $g := \underset{c\in ob(\mathcal{C})}{\coprod} \frac{\mathcal{F}(c)}{\sim}$
for a small category $\mathcal{C}$ and $g \in Top$ where $\sim$ is the equivalence relation $x\sim\mathcal{F}(f)x$ for $...
0
votes
1answer
52 views
When must a specific product map exist?
Suppose that it's unknown whether a category $\mathcal{C}$ contains all products, but that it happens to have the products $a\times b$ and $c\times d$. Further, $\mathcal{C}$ has morphisms $f:a\...
1
vote
1answer
57 views
Why $S^n$ is the pushout of the inclusion $S^{n-1} \rightarrow D^n$?
What will be the pushout for the following :
where $i:S^{n-1} \rightarrow D^n$ is the inclusion of the boundary $S^{n-1}$ to the n-disk $D^n$.
According to Pg 40 in Julia E. Bergner's The Homotopy ...
1
vote
2answers
54 views
Computing $\Bbb F_p[t]^{perf}$
This is the second example of 1. in Ex. 2.0.3 of Bhatt's notes in perfectoid space.
We define $R^{perf}:= \varprojlim ( \cdots R \xrightarrow{\phi} R)$ where $\phi$ is the Frobenius map.
He claims ...
4
votes
0answers
94 views
Limits as initial objects
I am relatively new to category theory and was wondering about the following problem:
Can I consider a limit as an initial object in some categories?
Let $\mathscr{C}$ be a category and $\mathbf{J}$ a ...
2
votes
1answer
40 views
How do I see that the map $X^n \to X^{n+1}$ of CW-building blocks is an embedding?
In his book “A concise course in algebraic topology”, May defines a CW complex inductively as being the union of increasing subspaces $X^n$, where
$X^0$ is discrete
$X^{n+1}$ is the simultaneous ...
2
votes
1answer
94 views
Every $R$-module is an iterated colimit of $R$
Let $R$ be a commutative algebra over a field $k$. My problem set asks me to
Show that every $R$-module is an iterated colimit of $R$.
My idea
Let $A$ be a free $R$-module with basis $M$, and let $\...
2
votes
0answers
53 views
Is $M_\infty \simeq M_\infty \otimes M_\infty \simeq M_n \otimes M_\infty$?
All matrix rings are over the integers, colimits are colimits of $\mathsf{Rng}$s and $\otimes$ means $\otimes_{\mathbb{Z}}$.
My guess is that yes, indeed, and my reasoning is as follows: since $M_\...
4
votes
2answers
121 views
Whats wrong with this argument that $\operatorname{Spec}(\prod A_i) = \bigsqcup\operatorname{Spec}(A_i)$ infinite product.
We have the spec functor $\text{CRng}^\text{op} \rightarrow \text{Aff}$.$\DeclareMathOperator{\Spec}{Spec}\DeclareMathOperator{\Hom}{Hom}$
Then $$\Hom _{\text{Aff}}(\Spec(\lim A_i), \Spec B) = \Hom_{\...
5
votes
2answers
170 views
Why is $\operatorname{colim} F \cong \pi_0\left (\int F\right )$?
Given a small functor $F:\mathsf{C \to Set}$, I need to prove that $\operatorname{colim} F$ is isomorphic/in bijection with the connected components of the category of elements $\int F$. It's not the ...
11
votes
1answer
106 views
Can the fundamental group and homology of the line with two origins be computed as a direct limit?
Let $X$ be the line with two origins, the result of identifying two lines except their origins. Let $X_n$ be the result of identifying two lines except their intervals $(-\frac{1}{n},\frac{1}{n})$. $...
9
votes
1answer
127 views
Why are (co-)ends called “(co-)ends”?
Briefly put, the (Co-)end is the universal wedge of a diagram.
Why is it called (co-)end? What is it an "end" of? By the universal property, it is in some sense the "universal end/...
