All Questions
Tagged with limits-colimits universal-property
15
questions
11
votes
2
answers
473
views
Do Wikipedia, nLab and several books give a wrong definition of categorical limits?
It seems unlikely that all these sources are wrong about the same thing, but I can’t find a flaw in my reasoning – I hope that either someone will point out my error or I can go fix Wikipedia and ...
- 239k
1
vote
1
answer
87
views
Properties of the Projective Limit (in $\textbf{Set}$)
Let $I$ be an index set with a preorder $\leq$ and let $(G_i)_{i \in I}$ be a family of sets. Furthermore, for all $i,j \in I$ with $i \leq j$ let $f_{ij} \colon G_i \longrightarrow G_j $ be maps such ...
- 438
3
votes
1
answer
63
views
Confusion on Exercise 3.6.ii in Riehl: showing the product is associative via a unique nat. isomorphism
This is an exercise (I believe) to practice methods using the functoriality of limits. The exercise is as follows:
For any pair of objects $X,Y,Z$ in a category $\mathsf{C}$ with binary products, ...
- 2,504
4
votes
3
answers
147
views
Constructing the topological product space from categorical universal property
Let $X,Y$ be topological spaces. I want to construct the categorical product, i.e. the product space, $X\times Y$ using the universal property of products. To be clear, I don't want to construct the ...
- 7,021
6
votes
0
answers
228
views
How to understand the effect of adjoint functors?
I have a good grasp of all different definitions/interpretations of adjoint functors, but still do not know have to interpret the left or right adjoint of a give functor, when it exist. It would be a ...
- 18.4k
2
votes
1
answer
1k
views
Trying to understand the definition of a universal property
Here's the definition of a universal property in Wikipedia:
(where $U:D\to C$ is a functor and $X$ is an object in $C$)
A terminal morphism from $U$ to $X$ is a final object in the category $(...
- 3,362
1
vote
1
answer
151
views
A question about limit cones and isomorphism
A consider a limit cone on a diagram $D: \mathbf{I} \rightarrow \mathbf{C}$:
$$
\left( L \xrightarrow{p_i} D(I) \right)_{I \in \mathbf{I}}
$$
Now suppose that $L' \in \mathbf{C}$ is some object ...
- 928
1
vote
1
answer
320
views
Showing that the projections of a product are jointly monic
Suppose that we have a category $\mathcal{C}$ which "has binary products" (full definition provided here: https://en.wikipedia.org/wiki/Product_(category_theory)). We want to show that given $A, B, C \...
- 155
5
votes
1
answer
1k
views
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
3
votes
2
answers
392
views
Equalizer of reflexive pair
A reflexive pair is a pair of parallel morphisms $f,g:X\to Y$ having a common section, i.e. a map $s:Y\to X$ such that $f\circ s = g\circ s = id_Y$.
Reflexive maps are famous because their ...
- 8,127
1
vote
1
answer
123
views
Flip map on a direct limit of tensor products
Let $$G_1\xrightarrow{f_1} G_2\xrightarrow{f_2}G_2\to\cdots$$ be an direct system of abelian groups with direct limit $(G,\{f_{n,\infty}\}_n)$.
For every $n\in\mathbb{N}$, let $\tau_n:G_n\otimes _\...
user421621
3
votes
2
answers
272
views
Union in an abelian category
I'm working on the following exercise. I have most of it, but there's one detail at the end that I can't work out.
Let $\{A_i\}$ be a family of subobjects of an object $A$. Show that if $\mathcal{A}...
- 307
3
votes
1
answer
105
views
Limit of product diagrams
Suppose we are in a "familiar" category, like sets, groups, or topological spaces.
Consider the diagram:
$$
X\times X \overset{p_1}{\underset{p_2}{\rightrightarrows}} X
$$
where $p_1,p_2$ are the ...
- 8,127
1
vote
1
answer
69
views
The existence of the product of a morphism from the quotient of an equivalence relation.
In Topologies et Faisceaux by Demazure, Proposition 3.3.4. p.180 (or p.192 in the linked pdf), states the following:
Let $R$ be a universal effective relation in $X$. Let $f:X\rightarrow Z$ be a ...
- 16.6k
0
votes
1
answer
73
views
$\{\alpha_i : A_i \to C_i \}$ family of maps $\implies \exists ! \alpha : \prod_i A_i \to \prod_i C_i$ such that $\pi_i^C \alpha \to \alpha_i \pi_i^A$
Suppose that $\prod_i A_i, \prod_i C_i$ exist in a category, and that there is a family of maps $\{\alpha_i : A_i \to C_i\}$. There exists a unique $\alpha : \prod_i A_i \to \prod_i C_i$ such that $\...
