Questions tagged [universal-property]
For questions about universal properties of various mathematical constructions.
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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:
$...
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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 ...
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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 $x_n-...
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What are colored morphisms/arrows intended to mean in these diagrams?
I've been reading more category theory as a prerequisite to understanding some more complicated theorems, and for the first time I'm running into arrows that are distinctly colored. Examples include ...
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Looking for properties of a pseudo-logarithm where you add all the (prime factor$\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 &...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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 $U$
...
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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 ...
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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)...
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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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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&...
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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 $i : V \times W \rightarrow V \otimes W : (v,w) \mapsto v \otimes w$....
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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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$
\bar{\mathscr M} = \{{E\cup F : \text{$ E\in \...
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If $R$ is an integral domain, then $Frac(R[x_1,…,x_n])= Frac(K[x_1,…,x_n])$ where $K=Frac{(R)}$
I’m trying to show that if $R$ is an integral domain, then $Frac(R[x_1,…,x_n])= Frac(K[x_1,…,x_n])$ where $K=Frac{(R)}$ (here $Frac{(R)}$ denotes “field of fractions of $R$”).
I think I can use the ...
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Complexification of $O(n,\mathbb{R})$
The definition of complexification for a Lie group $G$ is a complex Lie group $G_C$ with a continuous homomorphism $\phi: G\to G_C$ with the universal property that, if $f: G → H$ is an arbitrary ...
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Commutative monoid "generating" algebraic number field
Let's say that we have an algebraic number field, which is also an ordered field (e.g. no roots of -1), and whose ring of integers is a UFD. We would like to form a multiplicative, commutative free ...
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Dependence of strict transform on the subscheme along which we blow up
It is stated in https://stacks.math.columbia.edu/tag/080C that
Note that taking the strict transform along a blowup depends on the closed subscheme used for the blowup (and not just on the morphism $...
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Can a product map factor up to homotopy through the fat wedge
For $n \geq 2$, for $i = 1,\dots,n$, let $f_i \colon X_i \longrightarrow Y_i$ be maps of based $CW$-complexes, and consider a map
$$f \colon \prod_{i=1}^n X_i \longrightarrow \prod_{i=1}^n Y_i$$ such ...
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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 $\oplus f_i \colon \oplus M_i \to \oplus N_i$ ...
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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}...
3
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1
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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
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0
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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 ...
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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 ...
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2
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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 $...