Questions tagged [universal-property]
For questions about universal properties of various mathematical constructions.
491
questions
2
votes
0
answers
32
views
Universality of correlators in $\beta$ Hermite ensembles
Recently I got interested in the world of Random Matrix Models and I bumped into some generalizations of the usual random matrix theories classified by Dyson whose probability density functions are:
$...
1
vote
1
answer
48
views
Universal Property of tensor product of $\mathbb Z_2$-graded algebras.
If $A$ and $B$ are two $k$ algebra's with a $\mathbb{Z}_2$-grading then I know that a $\mathbb{Z}_2$-graded structure can be defined on their tensor product. One does this by altering the ...
- 13
0
votes
1
answer
34
views
Universal property of the completion of a first-countable abelian topological group
Let $G$ be a first-countable abelian topological group. Its completion $\hat{G}$, set-theoretically, is the family of Cauchy sequences in $G$ modulo the equivalence relation $(x_n)\sim (y_n)$ iff $x_n-...
0
votes
0
answers
37
views
What are colored morphisms/arrows intended to mean in these diagrams?
I've been reading more category theory as a prerequisite to understanding some more complicated theorems, and for the first time I'm running into arrows that are distinctly colored. Examples include ...
- 545
2
votes
1
answer
52
views
Looking for properties of a pseudo-logarithm where you add all the (prime factor$\cdot$exponent) terms in an integer's prime factorization?
If, for positive integer $n=p_1^{q_1}p_2^{q_2}\dots p_k^{q_k}$ with the right side the prime factorization of $n$, we define $J(n)$ as $p_1q_1+p_2q_2+\dots+p_kq_k$, does $J(n)$ have any generally &...
- 1,725
2
votes
1
answer
110
views
Is this diagram of the free product of two groups correct?
From Wikipedia:
"In mathematics, specifically group theory, the free product is an operation that takes two groups $G$ and $H$ and constructs a new group $G ∗ H$. The result contains both $G$ and ...
- 545
1
vote
1
answer
57
views
Wikipedia definition of Image in Category theory applied to Set
According to Wikipedia, the image of a morphism $f:A \to B$ is a monomorphism $m:I \to B$ for which there is some $e:A \to I$ with $f=m\circ e$, and such that the following universal property is ...
8
votes
1
answer
224
views
Objects of categories with no morphisms from them other than endomorphisms
Let $\mathcal{C}$ be a category. My question is the following. Is there a name for an object $\zeta \in \mathrm{ob}(\mathcal{C})$ such that, for every object $X \neq \zeta$, there exists no morphism $...
- 1,322
0
votes
1
answer
52
views
Continuous maps between covering spaces
Here is the problem statement:
Let $p:X\to S^1$, $q:Y\to S^1$ be covering maps, and $s_0=1\in S^1\subseteq \mathbb C$ is a basepoint for the unit circle. Let $M$ be the set of continuous maps $f:X\to ...
- 1,001
1
vote
1
answer
45
views
Do commutative diagrams on projections lift to a commutative diagram on a product category?
I have a family of categories $\{A_i\}_I$, their product category $\Pi A_i$, a category $B$ and a family of functors $F_i:B\rightarrow A_i$. Then there is also a unique functor $F_I:B\rightarrow \Pi ...
- 13
1
vote
0
answers
141
views
If $a^{p_1} b^{q_1}\ldots a^{p_n} b^{q_n} = e$ then $S = \{a,b\}$ is not a free generating set of $G = \langle S \rangle$
Let $S = \{a,b\}$ be a generating set for a group $G$. If a non-trivial word in $a, b \in S$ equals the identity $e$ of $G$, i.e.,
$$a^{p_1} b^{q_1} a^{p_2} b^{q_2} \ldots a^{p_n} b^{q_n} = e$$
for ...
- 13.1k
3
votes
1
answer
52
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, ...
- 2,169
0
votes
2
answers
35
views
The "characteristic property" of the tensor space
This relates to page 309 of John Lee's textbook on Smooth Manifolds. Specifically Proposition 12.7, the characteristic property of the tensor product space.
I fundamentally fail to see how this would ...
- 665
1
vote
1
answer
75
views
Associativity of tensor product using the universal property (and not elements)
My question
I thought the author of this question was really pushing for what I'm asking here and I don't think it was ever fully addressed in other questions (happy to be wrong!).
My main question is ...
- 307
2
votes
0
answers
64
views
Are pseudometric completions canonical in any sense?
Let $P$ be a pseudometric space, and let $M_P$ be the metric space obtained by quotienting out points separated by zero distance. We can always complete $M_P$ to $\overline{M_P}$ by forming the ...
- 6,420
1
vote
1
answer
74
views
Show that $C^*(\mathbb Z)$ is the universal $C^*$-algebra generated by a unitary.
Show that $C^*(\mathbb Z)$ is the universal $C^*$-algebra generated by a unitary. I.e.,show that for any unital $C^*$-algebra $A$ containing a unitary $u$, there is a unique unital *-homomorphism $\...
1
vote
1
answer
37
views
Localising a ring twice gives ring isomorphic to localising by each subset in turn
Given a ring $R$, suppose we have two multiplicative subsets $S,U \subseteq R$ (which contain 1). Write $US = \{us \mid u \in U, s \in S\}$ also a multiplicative subset, containing both $S$ and $U$
...
- 431
0
votes
1
answer
46
views
How to derive this from the universal property of the ring of fractions?
(All the involved rings are commutative with identity.)
The "universal" definition that I follow is this:
Let $S$ be a multiplicative subset of a ring $A$ which also contains $1_A$. Then a ...
- 2,894
1
vote
1
answer
106
views
Is there any way to prove$\{v_i \otimes w_j\}$ (with$(i,j) \in I \times J)$are basis for tensor product without using universal property?
Let $V,W$ be a vector space over $K$ with basis $\{v_i\}(i\in I), \{w_j\}(w\in J)$
$\begin{eqnarray} S_1 &=& \{(v_1 + v_2,w)-(v_1,w)-(v_2,w)|v_1,v_2\in V,w\in W\}\\
S_2&=&\{(v,w_1+w_2)...
- 23
1
vote
0
answers
45
views
Is this the correct formulation for the universal property and pushout diagram for cokernel?
The following question is taken from $\textit{Handbook of Mathematics}$ by Thierry Vialar,page 608
I consulted the Vialar's handbook and hoping to see what the dual formulation of cokernel in terms of ...
- 2,715
0
votes
0
answers
90
views
Universal property of multilinear maps (how to make them a tensor product)
context
A tensor product of $V$ and $W$ is a couple $(T,h)$ where $T$ is a vector space and $h:V\oplus W\to T$ is a linear surjective map satisfying the following universal property :
Let $f:V\oplus W\...
3
votes
1
answer
151
views
Does the direct sum have a universal property in the category of groups?
In other categories, like modules, the direct sum is the coproduct, giving it a neat category theory description. The direct product of groups, which is very similar to it, is the product in the ...
- 3,684
2
votes
1
answer
93
views
In category theory, does a definition via a universal property induce a functor?
In category theory one frequently defines concepts via universal properties. Examples are
(co)equalizers of pairs of morphisms having the same domain and codomain
(co)kernels of morphisms in ...
- 3,585
2
votes
1
answer
57
views
Question about the meaning of a "region" in mathemathics.
I am currently going over double integral. One concept that seems to appear a lot is "region". I have consulted different resources and it seems like there is no objetive definition of what ...
- 1,496
0
votes
0
answers
62
views
Topical example of a universal arrow to and from a functor
If $ S\colon \mathsf D\to \mathsf C $ is a functor between categories and $ c\in \mathsf C $, Mac Lane in his Categories for the Working Mathematician defines a universal arrow from $ c $ to $ S $ as ...
0
votes
0
answers
66
views
AF-Algebras and Matrix Algebras
I have been having some trouble with an exercise from Murphy's "$C^*$-Algebras and Operator Theory" recently. Chapter 6 Exercise 6 is as follows:
If $A$ is an AF-Algebra (i.e. a direct limit ...
- 595
3
votes
1
answer
57
views
Embedding a free algebra in a product.
Let $A$ be an algebra; set $V=HSP(A)$. Then for any $X\neq\emptyset$, there is an embedding of the free algebra $F_{V}(X)$ into the algebra $A^{A^{X}}$, $\Psi\colon F_V(X)\rightarrow A^{A^X}$ given by ...
0
votes
0
answers
35
views
Has this constructible property of structures in model theory been studied?
I was looking into model theory where a first-order language can be interpreted with a structure and a variable assignment.
Here is an example of a FOL with interpretation:
We have terms:
Atomic ...
5
votes
1
answer
148
views
Uniqueness of the morphism in the universal property
If we want to define a product of objects in a category (cf. Wikipedia Page), we need to find a morphism and it must be unique.
In our notes, the professor gives some examples such as the product of ...
- 2,195
2
votes
1
answer
108
views
Definition of the evaluation map as a universal element of a representation
This is exercise Exercise 2.3.iii from Riehl's "Category Theory in Context":
The set $B^A$ of functions from a set $A$ to a set $B$ represents the contravariant functor ${\rm Set}(-\times A,...
- 105
1
vote
1
answer
74
views
Proving the existence of an homomorphism $k[x,x^{-1}] \to R$ where $R$ is a ring containing $k$ and a unit
I’d like to understand how to prove the following statement that I found in this answer:
For any commutative ring $k$, the ring $k[x; 1/x]$ is characterized by
the following "mapping property&...
- 83
2
votes
1
answer
70
views
The tensor product as a universal morphism
Given two vector spaces $V,W$ (say, over $\mathbb{C}$), we can form their tensor product $V \otimes W$, and we get a (bilinear) map $i : V \times W \rightarrow V \otimes W : (v,w) \mapsto v \otimes w$....
- 5,478
1
vote
1
answer
51
views
Reference Request: Adjoint Functors and Universal Morphism
I currently work on non-category theoretical research but use some tools
from category theory.
I am not well-versed in category theory and also don't want to expand the preliminaries
of my work too ...
0
votes
0
answers
33
views
How to construct the map which verifies that the tensor product of algebras is the fibered coproduct using universal properties?
Let $A \rightarrow B$ and $A \rightarrow C$ be commutative ring morphisms. I am trying to verify the universal property of the fibered coproduct for the ring $B \otimes_A C$ with maps $i_B: B \...
- 1,525
2
votes
2
answers
150
views
Characterizations of product topology and box topology
Given a collection of topological spaces $X_i$ indexed by the elements $i$ of a set $I$, we consider the set product $P = \prod_{i \in I} X_i$ with projections $p_i : P \to X_i$. There are two methods ...
- 71.4k
2
votes
1
answer
127
views
Convenience of using universal property of tensor product
On the wikipedia page of universal property, it says one of the motivations of universal property is
The concrete details of a given construction may be messy, but if the construction satisfies a ...
- 671
0
votes
0
answers
59
views
Direct limit characterized up to isomorphism
This is from Atiyah & Macdonald - Exercise 2.16:
We are given a direct system $(M_i,\mu_{ij})$ with direct limit $M=\underset{\longrightarrow}{\lim} M_i$. Show that the direct limit is ...
- 21
0
votes
0
answers
32
views
Show that two tensor products are isomorphic.
Let $M$ and $N$ be two $R$-modules,then a pair $(T,\varphi)$ where $T$ is an $R$-module and $\varphi:M\times N\to T$ is a bilinear map is called a tensor product if for any pair $(P,f)$ where $P$ is ...
- 3,194
0
votes
0
answers
49
views
Universal property of completion of a measure space
Let $ (X,\mathscr M,\mu) $ be a measure space. The completion of $ (X,\mathscr M,\mu) $ is the measure space $ (X,\bar{\mathscr M},\bar\mu) $ where
$$
\bar{\mathscr M} = \{{E\cup F : \text{$ E\in \...
- 103
1
vote
1
answer
64
views
If $R$ is an integral domain, then $Frac(R[x_1,…,x_n])= Frac(K[x_1,…,x_n])$ where $K=Frac{(R)}$
I’m trying to show that if $R$ is an integral domain, then $Frac(R[x_1,…,x_n])= Frac(K[x_1,…,x_n])$ where $K=Frac{(R)}$ (here $Frac{(R)}$ denotes “field of fractions of $R$”).
I think I can use the ...
- 1,454
2
votes
1
answer
175
views
Complexification of $O(n,\mathbb{R})$
The definition of complexification for a Lie group $G$ is a complex Lie group $G_C$ with a continuous homomorphism $\phi: G\to G_C$ with the universal property that, if $f: G → H$ is an arbitrary ...
- 514
1
vote
0
answers
40
views
Commutative monoid "generating" algebraic number field
Let's say that we have an algebraic number field, which is also an ordered field (e.g. no roots of -1), and whose ring of integers is a UFD. We would like to form a multiplicative, commutative free ...
- 7,624
1
vote
1
answer
105
views
Dependence of strict transform on the subscheme along which we blow up
It is stated in https://stacks.math.columbia.edu/tag/080C that
Note that taking the strict transform along a blowup depends on the closed subscheme used for the blowup (and not just on the morphism $...
- 128
3
votes
1
answer
73
views
Can a product map factor up to homotopy through the fat wedge
For $n \geq 2$, for $i = 1,\dots,n$, let $f_i \colon X_i \longrightarrow Y_i$ be maps of based $CW$-complexes, and consider a map
$$f \colon \prod_{i=1}^n X_i \longrightarrow \prod_{i=1}^n Y_i$$ such ...
- 3,206
3
votes
1
answer
118
views
Direct sum of injections is injection, categorically
Given $R$-module maps $\{f_i \colon M_i \to N_i\}_{i \in I}$, the universal properties of the direct sums $\oplus M_i$ and $\oplus N_i$ give a unique map $\oplus f_i \colon \oplus M_i \to \oplus N_i$ ...
1
vote
1
answer
68
views
Proving the universal property for the localization functor $L_E$
I am trying to prove the following statement:
If the functor $L_E$ exists,
$(iii)$ for any map $g: X \to Y$ where $Y$ is $E_*$-local, there is a unique map $\tilde{g}: L_E X \to Y$ such that $\tilde{g}...
- 19
3
votes
1
answer
116
views
Formalizing the universal property of the tensor product
Introduction (a bit long)
I am very new to category theory and algebra, in general.
I recently saw the formalized concept of a universal morphism, here in WikipediA.
I will repeat the definition with ...
2
votes
0
answers
92
views
Universal property of the (Vistoli-)sheafification
Given a presheaf $F$, in Notes on Grothendieck topologies,
fibered categories
and descent theory
there is a construction of the sheafification (Proof for theorem 2.64).
The first part is the ...
1
vote
0
answers
63
views
Missing universal properties
Given a category $\mathcal{C}$, we can use copresheaves $H \in [\mathcal{C},Set]$ or presehaves $H \in [\mathcal{C}^{\mathrm{op}},Set]$ to state left or right universal properties. Existence of a ...
- 41
3
votes
2
answers
118
views
Functor demonstrating universal property of quotient spaces
The universal property of quotient spaces is, as quoted from wiki:
The quotient space $X/\sim$ together with the quotient map $q:X\to X/\sim$ is characterized by the following universal property: if $...
- 6,838