Questions tagged [universal-property]
For questions about universal properties of various mathematical constructions.
0
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1answer
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Generalising idea of projection of continuous laticces into any category
Definition: A continuous lattice $D$ is said to be a projection of a continuous lattice $D'$ if and only if there are a pair of continuous maps
$$i:D\rightarrow D'$$
and
$$j:D'\rightarrow D$$
...
4
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0answers
28 views
No canonical correspondance between bilinear forms $b:V\times V \rightarrow \mathbb{R}$ and linear forms $\hat{b}:V\otimes V \rightarrow \mathbb{R}$?
In the wiki on bilinear forms, the universal product says that to each bilnear map $b : V\times V \rightarrow \mathbb{R}$ we can associate a linear map $\hat{b} : V\otimes V \rightarrow \mathbb{R}$, ...
1
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0answers
18 views
Show that if a submonoid $S$ of a commutative monoid $M$ contains an absorbing element, then the localization of $M$ by $S$ contains only one element.
Here is the full question:
If $M$ is a commutative monoid, $S\subset M$ is a submonoid and there is a $z\in S$ that is absorbing in $M$ (i.e. $zm=z$ for all $m\in M$), then show that $M_S$ has only ...
1
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1answer
44 views
Proving disjoint union is defined up to an isomorphism using coproducts in $\mathbf{Set}$
The book on abstract algebra I'm going through has the following exercise. Let $A' \cong A'', B' \cong B''$ be two pairs of isomorphic sets. Further, let them be pairwise disjoint: $A' \cap B' = \...
1
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2answers
94 views
Prove the uniqueness of function $X \rightarrow A \times B$
Let A and B be sets and consider their Cartesian product $A\times B\rightarrow A$.
Give surjective functions $\pi_A:A\times B\rightarrow A$ and $\pi_B:A\times B\rightarrow B$.
Suppose we are given a ...
1
vote
0answers
31 views
When, if ever, is this a valid universal property for $A \times (B_1 + B_2)$? (Category Theory)
Notation: Let $\pi$'s denote canonical projections, $j$'s denote canonical injections, given a pair of maps $X \overset{f_1}{\to} Y_1$ and $X \overset{f_2}{\to} Y_2$ denote the unique map $X \overset{...
1
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1answer
33 views
Universal property of the direct product, proof verification
Let $I$ be a set. For every $i\in I$ let a set $X_i$ be given, and for every $j\in I$ denote $p_j:\prod_{i\in I} X_i\to X_j$, $p_j((x_i)_{i\in I}):= x_j$, the j-th projection.
If $X,Y$ are sets ...
2
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0answers
31 views
Products, coproducts and morphisms
The universal properties of products and coproducts "amount" to the statements
$$ \hom(\coprod_i X_i , Y) = \prod_i \hom(X_i, Y) \quad \text{and} \quad \hom(X,\prod_i Y_i) = \prod_i \hom(X,Y_i)$$
...
0
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2answers
55 views
Universal Property…The Map is not Well-defined?
How does the following proposition from this book make sense:
Proposition 2.8 Let $G$ be the group defined by the presentation $(X,R)$. For any group $H$ and map of sets $\alpha : X \to H$ sending ...
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0answers
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A construction of the Hodge Dual operator
This question about showing that an alternative construction of the Hodge dual operator satisfies to the universal property through which the Hodge dual is usually defined.
Let me give the ...
1
vote
1answer
26 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 ...
3
votes
1answer
45 views
Proving the Free Abelian Group is Free Abelian…?
On page 40 of these notes is the following exercise:
Prove that the group with generators $a_1,...,a_n$ and relations $[a_i,a_j]=1$, $i \neq j$, is the free abelian group on $a_1,...,a_n$.
On ...
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0answers
46 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 \...
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0answers
37 views
Meaning of “universal” in category theory (equalizer definition)
In Awodey's Category Theory on page 62, equalizers are defined like so:
In any category C, given parallel arrows $f, g : A \rightarrow B$, an equalizer of $f$ and $g$ consists of an object $E$ and ...
3
votes
1answer
88 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 $(X,\varphi_i)$.
...
12
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0answers
159 views
Does the box topology have a universal property?
Given a set of topological spaces $\{X_\alpha\}$, there are two main topologies we can give to the Cartesian product $\Pi_\alpha X_\alpha$: the product topology and the box topology. The product ...
0
votes
1answer
40 views
Uniqueness in the Universal Property of Quotient Maps
Here is Munkres' way of phrasing the universal property of quotient maps:
Let $p : X \to Y$ be a quotient map. Let $Z$ be a space and let $g : X \to Z$ be a map that is constant on each set $p^{-1}...
0
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2answers
27 views
Product property [closed]
What is the name of the following property of two equal products?
If $ab = cd$, then $a(b-d)=(c-a)d$
2
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0answers
74 views
Is there a “universal family of hypersurfaces”?
Consider the category of (flat) families of degree $d$ hypersurfaces in $\mathbb P^n$, i.e. objects are $X \to Y$ whose fibers are degree $d$ hypersurfaces in $\mathbb P^n$, and morphisms are the ...
1
vote
1answer
58 views
Calculating a presentation of $\mathbb{Z}_{3}$ in detail.
Theorem: Let $G$ groups and $S\subset G$ such that
$\langle S\rangle =G$. (Here $G=\left\{s_1\ldots, s_n:s_i\in S\cup S^{-1}, n\in\mathbb{N}\right\}$.) Let $\varphi:S\to G$ with $\varphi(s)=s$. By ...
2
votes
1answer
84 views
Can it be useful to think of functors as representing themselves ?
Here's a thought I had and I wonder if it can be of any use, for instance has it ever helped proving any result (however minor the result).
Say you're in a situation where you have some objects in ...
0
votes
1answer
54 views
Direct sum of non-abelian groups doesn't satisfy the universal property of direct sum.
Let $C$ and $D$ be non-abelian groups. Show that $C\oplus D$ doesn't satisfy the universal property of direct sum.
I think I must assume that the universal property is true and then use the free ...
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0answers
46 views
Universal Property of the Ring of Quotients.
Let $R$, $R'$ be commutative rings with multiplicative subsets $S, S'$ and the corresponding ring homomorphisms $\phi, \phi'$ into their respective ring of fractions.
Prove that for any ring ...
1
vote
2answers
114 views
The peculiar properties of 14732
About a year ago, I delved into the properties of the number 14732. A number that have popped into my brain at random times for well over a decade.
Through a bit of experimenting with base10 and ...
0
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0answers
43 views
The Universal Mapping Property of a free vector space.
Definition: Let $X$ be a non-empty set. A free vector space on $X$ is a pair $(V,i)$ consisting of a vector space $V$ and a function $i:X\to V$ satisfying the following universal mapping property.
I ...
0
votes
2answers
21 views
Inequality properties between their multiplications [closed]
If $A>B>C>D$ then will $$(A/B * A/C * A/D) > (B/A * B/C * B/D) > (C/A * C/B * C/D) > (D/A * D/B * D/C)$$ be always true? If not, in what intervals will it not be true?
Obviously $A,...
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0answers
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Find the properties of the function f(x) which fulfills the following inequality
Let us assume that $f:\mathbb{R}^+ \rightarrow \mathbb{R}$ fulfills the following inequality:
$$
\exists_{C \in \mathbb{R^+}} \quad|f'(x) \cdot x| < C.
$$
In other words we can say that $f'(x)\...
0
votes
1answer
46 views
A topological space with the Universal Extension Property which is not homeomorphic to a retract of $\mathbb{R}^J$?
A topological space $Y$ has the universal extension property if for every normal space $X$, every closed subset $A$ of $X$, and every continuous function $f:A\rightarrow Y$, we can extend $f$ to a ...
1
vote
1answer
77 views
Free monoids as universal constructions (example)
I'm a beginner and I'm desperately trying to understand free monoids as universal constructions using an example.
What is the relation between the following?
a generator set (e.g. $\mathbb{N}$ (...
1
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1answer
45 views
Does universal construction apply to any arbitrary category? [closed]
Does universal construction presuppose a category equipped with morphisms that implement an order?
Otherwise how does the universal construction pick the most suitable element for the given purpose? ...
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0answers
67 views
Looking for a pair of adjoint functors
I'm struggling with finding the left adjoint to a functor, and the right adjoint to another one. Here's some context.
Given any ring $R$, we can associate two categories to it. The first one is a ...
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0answers
69 views
How is functor with “image” unique up to a unique isomorphism defined exactly?
In an abelian category $\mathscr A$ we encounters the notions of kernel, cokernel, chain homology, derived functors, etc. These notions are frequently referred to as functors, and yes, they actually ...
1
vote
1answer
233 views
The universal property of polynomial rings and universal identities
To verify some identity involving say two variables $x, y \in R$ for any commutative ring $R$ it suffices to verify this identity in $\mathbb{Z}[x,y]$. This is just the sort of trick that should be ...
3
votes
1answer
74 views
Being universal with respect to a property
I have always concerned universal properties under a statement of the form
Universal Property of the Quotient Group: Let $G$ be a group and $H$ a subgroup of $G$. If $f: G \to G'$ is a group ...
3
votes
2answers
98 views
Soft question on universal properties
My question heads towards an intuition of universal properties. Take for example a group $G$ and a subset $M$. Then $F_M$, the free group on $M$ together with a map of sets $i:M\to U(F_M)$ where $U$ ...
0
votes
1answer
88 views
Universal Property of free vector space
Let $X$ be any set and $F$ a field. Let $V_{X}$ be set of functions $f:X \to F$ for which $f(x)$ is nonzero for finitely many $x \in X$. Note that $V_{X}$ is a vector space.
Call $\sigma_{x}$ a ...
3
votes
0answers
50 views
Is there a category theoretic characterisation of the exponential map from differential geometry?
While I have only a shallow understanding, I like category theory. I find definitions and proofs in terms of category theoretic concepts to be very clean and deep, often cutting to the core of a ...
1
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1answer
83 views
Dummit and Foote Universal Property of Tensor Products
Here is the theorem giving me trouble:
Here is theorem 6, which is invoked in the proof of theorem 6:
Here is the proof of theorem 8:
Here is the sentence giving me trouble so far:
"Since $\...
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0answers
30 views
Functional property name?
I would like to know if this functional property has a name.
Definition. We say that a functional $L$ has a property if and only if $L(f) \leq L(g)$ implies
$$
L(f) \leq L\left(\frac{f+g}{2}\right) \...
0
votes
1answer
47 views
Compact open topology on $\operatorname{GL}(n, \mathbb{R})$ coincides with Euclidean topology.
There are two ways to assign $\operatorname{GL}(n,\mathbb{R})$ topologies: as subspace of $\mathbb{R}^{n^2}$, or subspace of $\operatorname{Maps}(\mathbb{R}^n, \mathbb{R}^n)$ where the latter is ...
8
votes
1answer
200 views
Why do exponential objects (in category theory) require currying?
I'm a bit confused about exponential objects in category theory.
I find them pretty intuitive when I think about them as "arbitrary-arity cartesian products"; in the sense that if I had never seen ...
3
votes
2answers
70 views
Universal property of quotient group to get epimorphism
I know there are already answers to the question of the universal property of quotient groups. For example here: Universal property of quotient group.
My question is now: If I have the homomorphism: $...
1
vote
2answers
68 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 ...
3
votes
1answer
85 views
What makes an universal mapping unique?
Maybe the title doesn't explain very well my question. I was wondering why if some object satisfies an universal property, then for any other object there is one and only one mapping between them. Let ...
0
votes
1answer
72 views
Isomorphic Free $R$-module has Universal Property
Here is corollary 7 in chapter 10.3 of Dummit and Foote book on Algebra:
If $F_1$ and $F_2$ are free modules on the same set $A$, there is a unique isomorphism between $F_1$ and $F_2$ which is ...
0
votes
1answer
51 views
on the maps to moduli spaces
Let $M$ be a coarse moduli scheme for some moduli functor $\mathcal{F}$( from the category of schemes to the category of sets). Let $V$ be a scheme such that for each point of $v \in V$ we can ...
0
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1answer
172 views
Proving the universal mapping property for polynomial rings
I am studying from Patrick Morandi's Field and Galois Theory, and I am stuck on Problem 6 from Section 1 (on page 13).
Verify the following universal mapping property for polynomial rings:
...
1
vote
1answer
36 views
Sum of two isomorphic objects and relation between the inclusions
Consider objects $A, B$ in some category $C$. Let $S\in C$ to be the sum of $A$ and $B$ with inclusions as $i_A:A\rightarrow S$ and $i_B:B\rightarrow S$.
If $A\cong B$ with isomorphisms $f:A\...
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0answers
74 views
Universal Finitely Presented Group
Higman's embedding theorem states every finitely generated, recursively presented group embeds into some finitely presented group. A further result of Higman, Neumann and Neumann shows that every ...
1
vote
2answers
158 views
Defining Groups via the Universal Property
The snippets below on group presentation led to the following questions:
What is the explanation/intuition of the universal property. Basically what (2) is saying. I understand free groups (though ...
