All Questions
Tagged with limits-colimits proof-explanation
19
questions
4
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3
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190
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What are the inclusion arrows in the coproducts of the category of algebras for a monad?
$\newcommand{\A}{\mathscr{A}}\newcommand{\C}{\mathsf{C}}\newcommand{\T}{\mathcal{T}}\newcommand{\id}{\operatorname{id}}$Riehl, proposition $5.6.11$, from Category Theory in Context:
Suppose $\C$ is a ...
- 42.7k
0
votes
1
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90
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Showing the limit of the discrete category defines the usual product.
How can we see that the categorical limit of the diagram $F$ from the discrete 2 object category to sets is isomorphic to the usual product $\{(x,y)\mid x\in X, y\in Y\}$ defined elementwise on sets. ...
- 423
4
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2
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164
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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 $$\Hom _{\text{Aff}}(\Spec(\lim A_i), \Spec B) = \Hom_{\...
- 9,618
1
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1
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84
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Mappings out of quotient set by a generated equivalence relation (Colimits in Set)
Let $X$ be a set and $R\subset X\times X$. Let
$$\mathcal{G}:=\{G\subset X\times X\ |\ R\subset G\ \land\ G=G^{-1}\ \land\ G=G\circ G\ \land\ pr_1(G)=X\}.$$ Then $\bigcap_{G\in\mathcal{G}}G\in\...
- 733
1
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0
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51
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Every finite diagram in a filtered category has a cocone
Let $\mathcal{C}$ be a filtered category, $\mathcal{D}$ a finite category and $F:\mathcal{D}\rightarrow\mathcal{C}$ a functor.
The family $(F(D))_{D\in\mathcal{D}}$ is finite; thus, there exists $C\in\...
- 733
0
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1
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473
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On the proof of the density theorem
I'm trying to understand Leinster's proof of the density theorem. Here's the terminology and the statement.
Below is his proof. Here are some things that I don't understand:
This must be silly, but ...
- 12k
3
votes
1
answer
277
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Adjoint functors preserve limits/colimits
Here's another theorem from Leinster's book (p. 159) where I got stuck:
Just as in my previous question, I don't see how this sequence of isomorphisms establishes the claimed result. To prove the ...
- 12k
0
votes
1
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88
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General description of colimits in $\mathbf{Set}$ - 2
I've previously posted a question about the example below, but this question is different.
Example 5.2.16.
The colimit of a diagram $D \colon \mathbf{I} \to \mathbf{Set}$ is given by
$$
\lim_{\to \...
- 12k
0
votes
0
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211
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Forgetful functors from categories of algebra create limits?
After Leinster states the following lemma (on p. 139):
Let $F:\mathscr A\to\mathscr B$ be a functor and $I$ a small category. Suppose that $\mathscr B$ has, and $F$ creates, limits of shape $I$. ...
- 12k
1
vote
1
answer
140
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Equivalent definitions of preserving limits
On page 137, Leinster gives two equivalent characterizations of limit preservation:
Is it supposed to be obvious that they are the equivalent? If so, how to see that? (When I tried to prove that, I ...
- 12k
0
votes
1
answer
117
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General description of colimits in $\mathbf{Set}$
I'm not sure I can match the statement given here (from https://arxiv.org/abs/1612.09375) with the real results:
Example 5.2.16.
The colimit of a diagram $D \colon \mathbf{I} \to \mathbf{Set}$ is ...
- 12k
1
vote
2
answers
133
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Limits in $\mathbf {Set}$
I'm a little confused by this example from Leinster's book:
The $x_I$, I suppose, are strictly speaking arrows $x_I: \{\ast\}=1\to D(I)$, and he identifies them with elements of $D(I)$ -- is that ...
- 12k
2
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0
answers
146
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inverse limit and graded functor commute?
I am trying to understand a proof from Okounkov-Olshanski, Shifted Schur functions's paper. Exactly Proposition 1.5. which says
PROPOSITION 1.5. The graded algebra $gr(\Lambda^*)$ corresponding to ...
- 1,428
0
votes
1
answer
84
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On why categories with coseparating sets and intersections have initial objects [Proof Explanation]
I'm trying to understand the proof of the following result from Category Theory in Context, p. 148:
Lemma 4.6.11. Suppose $C$ is locally small, complete, has a small coseparating set $\Phi$, and ...
- 17.2k
5
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1
answer
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Understanding the inverse limit and universal property of topological spaces
Let $\{X_i, \varphi_{ij},I\}$ be an inverse system of topological space index by a directed poset $I$.
Now I would like to understand the proof for the existence of an inverse limit $(X,\varphi_i)$.
...
- 3,149