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Questions tagged [universal-property]

For questions about universal properties of various mathematical constructions.

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Universal property definition of an ideal generated by a subset?

I'm puzzled by the definition of ideals generated by a subset of a ring in Aluffi, Algebra: Ch 0. The previous chapter on groups is (for an algebra book of this level) quite categorical in spirit. In ...
David M's user avatar
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4 votes
2 answers
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Universal property of HNN extensions

The amalgamated free product of groups can be defined as a certain presentation; or, one can define it as a pushout where the morphisms are embeddings. I was wondering if the definition of an HNN ...
Mithrandir's user avatar
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Equivalent definitions of tensor power of a vector space

I have two definitions of the tensor power $T^nV$ of a vector space $V\in \bf{kVect}$. For every $n$-multilinear map $f:V\times ...\times V\rightarrow W$, there exists a unique $\bar{f}:T^nV\...
Wyatt Kuehster's user avatar
2 votes
1 answer
130 views

Definite description in homotopy type theory

I asked this question there and I have been suggested to ask it here. In a paper by David Corfield, we have an account of definite description in homotopy type theory. The author gives the following ...
Bruno's user avatar
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why is the universal side divisor called universal?

With respect to the usage of the term "universal" in category theory I struggle to see the connection? In terms of elements, if we were to let divisibility represent the morphism, then ...
DoubleA Batteries's user avatar
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Confused about question on universal property inverse limit

Context: Let $S = \varprojlim S_{i}$ be a projective limit and define for each $j \in I$ the projection map $f_{j} : S \rightarrow S_{j}$ by $f_{j}((x_{i})_{i \in I}) = x_{j}$. Now, I have already ...
ByteBlitzer's user avatar
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45 views

Continuous function from a product topological space to a sum topological space

I denote by $\mathbf{2}$ the set $\{0, 1\}$ that is given the discrete topology. Let $X$ be some topological space. I denote by $\mathbf{2} \times X$ the topological space with the product topology ...
Bruno's user avatar
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Universal property of product

I'm trying to reconcile the commutative diagram for the universal property of the product with my understanding through a concrete example. I'm very confused by all of this, and would appreciate if ...
Tomek Dobrzynski's user avatar
5 votes
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59 views

Can we construct the exterior algebra just from simple multivectors?

$ \newcommand\K{\mathbb K} \newcommand\Ext{\mathop{\textstyle\bigwedge}} \newcommand\Lip{\mathrm{Lip}} \newcommand\ev{\mathrm{ev}} \newcommand\Gr{\mathrm{Gr}} $Let $V$ be a finite-dimensional $\K$-...
Nicholas Todoroff's user avatar
4 votes
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76 views

The pushout $F_\alpha(H)$ of $H \overset{\pi}{\longleftarrow} \mathbb{Z}^{\ast H} \overset{\alpha^{\ast H}}{\longrightarrow} G^{\ast H}$

$\require{AMScd}$ Definition 1: If $G$ is a group and $X$ is a set, define $G^{\ast X} = \ast_{x \in X} G$. This is functorial in both $G$ and $X$: If $G$ and $H$ are groups, $\varphi: G \to H$ is a ...
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Understanding the Product as a Universal Arrow from the Diagonal Functor

This is my first question and it has been asked before, but I'm not comfortable with the answer so I thought I'd raise it again. In Mac Lane's Categories for the Working Mathematician it is claimed ...
Corlio's user avatar
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Serge Lang's Definition of a Free Group

Serge Lang says the following in his "Algebra": We now consider the category $\mathfrak{C}$ whose objects are the maps of $S$ into groups. If $f:S\rightarrow G$ and $f':S\rightarrow G'$ are ...
Aravind Gundakaram's user avatar
2 votes
1 answer
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Why is this localization thin?

I am studying this paper by Malkiewich and Ponto. I am unsure about one claim. Let $\Delta$ be the augmented simplex category. Denote by $\mathfrak{J}$ the wide subcategory of $\Delta$ consisting of ...
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Why the associated graded universal enveloping algebra of a Lie algebra is Poisson?

Let $L$ be a Lie algebra and $U(L)$ be its universal enveloping algebra. How the associated graded algebra $grU(L)$ of the universal enveloping algebra $U(L)$ can be a Poisson algebra? How the Poisson ...
Nil's user avatar
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If A is idempotent then A+AB−ABA isidempotent for any square matrix B with the same dimension as A. [closed]

If A is idempotent then A+AB−ABA is idempotent for any square matrix B with the same dimension as A. I have this question to solve and I tried squaring the entire expression and then simplifying it ...
Valentina Tanguy's user avatar
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1 answer
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Universal commutatization of a ring

Let $R$ be a (not-necessarily-commutative) ring with unity. Say that ${^{\operatorname{ComRing}\rightarrow}}R \overset{\iota}{\longrightarrow} R$ is universal for morphisms from commutative rings with ...
Smiley1000's user avatar
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Should there be the table of triangles as in the table of all elements since there are patterns in the laws of triangles?

I don't know if there are people who recognize the following pattern in Isosceles triangles and they are new to me, it's difficult to understand and I've been working on it and doesn't seem to get ...
Mesfin's user avatar
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2 votes
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About the universal morphism from the pushout of monomorphisms.

Let $A,B,C$ be objects in a category $\mathrm{C}$ and we have the pushout diagram of monomorphisms$\require{AMScd}$ \begin{CD} C @>>> A\\ @VVV @VVV\\ B @>>> A\bigsqcup\limits_C B\end{...
Epsilon's user avatar
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the free product of two presentations is isomorphic to a third presentation using UP of free product.

Here is the question that I want an answer to it using commutative diagrams (as small number of them as possible): Prove that the free product of $ \langle g_1, \dots ,g_m | r_1, \dots ,r_n \rangle$ ...
Intuition's user avatar
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3 votes
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Question on universal property of Clifford algebra.

I have short question regarding the universal property of the Clifford algebra. Suppose $(V,q)$ is a quadratic $\mathbb{F}$-vector space and $(\mathrm{Cl}(V,q),j)$ is the Clifford algebra. Recall that ...
B.Hueber's user avatar
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11 votes
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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 ...
joriki's user avatar
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Extending functors on module categories

Let $A$ be a $k$-algebra for a commutative ring $k$. Denote by $A^e$ the $k$-algebra $A\otimes A^{op}$. The center of a bimodule yields a functor $C$ from the category of right $A^e$-modules $Mod_{A^e}...
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2 votes
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Are (commutative) squares in some sense universal among edge-symmetric double categories?

Definitions: Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$,...
hasManyStupidQuestions's user avatar
4 votes
1 answer
85 views

Using only the universal property, prove that $X$ is dense in its Cauchy completion

Long ago, I learned about the Cauchy completion of metric spaces via the usual explicit construction of quotienting the set of Cauchy sequences. For a metric space $X$, let $\hat X$ denote this Cauchy ...
Atom's user avatar
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On the universal property of Tensor Product

I was readying the construction of the Tensor Product that was made on the book Introduction to Smooth Manifolds, from John M. Lee. In the proposition 12.7 he proved that his construction satisfied ...
Paulo Estêvão's user avatar
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1 answer
58 views

Difference betwen universal properties

I'm studying the construction of the tensor product. To do this, I opted for the quotient space's approach, defining first the free vector space over the cartesian $V\times W$ of two vector spaces. ...
Paulo Estêvão's user avatar
1 vote
1 answer
28 views

Reference for quotient lattices and universal property?

Question: Are there any references explaining the definition (or definitions) of quotient objects in the category of lattices? In particular a characterization in terms of universal properties would ...
hasManyStupidQuestions's user avatar
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Passing an involution to a quotient algebra

This question is inspired by the discussion under this MO answer. I hope I have captured correctly what is going on in the below. Let $A_0$ be a finitely generated universal unital complex algebra $...
JP McCarthy's user avatar
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1 vote
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Four Lemma in an Abelian Category

Let $\mathcal{C}$ be an abelian category. Consider the following commutative diagram: I am trying to prove the following version of the four lemma: if $\alpha$ is an epimorphism and $\beta$ and $\...
user82261's user avatar
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2 votes
1 answer
247 views

Equivalence statement for existence product of modules

The following exercise is taken from T.S. Blyth book, Module Theory, chapter 6, exercise 9: Something does not add up, especially with using (1). It seems that condition (1) is not necessary or ...
User666x's user avatar
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-3 votes
1 answer
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Understanding the Universal Property of Tensor Products for Modules Over Any Ring

I am delving into the concept of tensor products and their universal properties in the context of modules over rings. While the universal property of tensor products is well-established for vector ...
Yang's user avatar
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1 vote
1 answer
143 views

How do I prove $U(\mathfrak{g} \oplus \mathfrak{h})\cong U(\mathfrak{g})\otimes U(\mathfrak{h})$ using the universal properties?

Reading lecture notes on Hopf algebras, I came across a statement (which was heavily used by the author without a proof) that, given Lie algebras $\mathfrak{g}, \mathfrak{h}$, the universal enveloping ...
Daigaku no Baku's user avatar
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1 answer
57 views

Why is the map $\text{Alt}^m V \times \text{Alt}^n V \to \text{Alt}^{m+n} V$ well defined?

In Fulton & Harris's "Representation Theory: A First Course" it is claimed that the wedge product $\wedge$ determines a product $\text{Alt}^m V \times \text{Alt}^n V \to \text{Alt}^{m+n} ...
רוי רז's user avatar
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 ...
puck29's user avatar
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0 votes
0 answers
88 views

Determinant Formula for Wedge Product via Universal Property of Exterior Powers

I'm currently learning about differential forms in my analysis class, and I thought I'd dig a bit more into the linear algebra of exterior powers. I've seen the universal property of $\bigwedge^k(V)$: ...
Joshua Yagupsky's user avatar
5 votes
1 answer
312 views

Does the term "free" in "free ultrafilter" have a meaning related to category theory?

I know that free ultrafilters are defined in contrast to principal/fixed ultrafilters. Nonetheless, is there some categorical way to view the use of the word "free" here (e.g. some pair of ...
Kellen Brosnahan's user avatar
1 vote
1 answer
69 views

Prove by universal property that epimorphism onto quotient group kills the subgroup

Today I'm obsessed with universal properties and I'm trying to prove that the universal property of quotient groups implies that the subgroup is killed by the cannonical epimorphism. I hope this ...
Al.G.'s user avatar
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4 votes
1 answer
251 views

Why do universal properties require a unique isomorphism?

I am a novice in the field of category theory, and one of the things I struggle to wrap my head around is the notion of universal properties. Precisely, I struggle to understand why universal ...
Thomas.M's user avatar
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2 votes
1 answer
74 views

A Lipschitz map universal with respect to factorization through it of Lipschitz functions.

Let $(X,d)$ be a metric space. Given a metric space $(Y,\rho)$ and a $1$-Lipschitz map $\varphi\colon X\to Y$, we say that a $1$-Lipschitz function $f\colon X\to \mathbb{R}$ factors through $\varphi$ ...
Rafael's user avatar
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0 votes
1 answer
204 views

Show that: exists unique ring homomorphism $\varphi : \mathbb{Z} [ i ] \longrightarrow R$ with $\varphi ( i ) = a$, where $a^{2} =-1_{R}$

Problem: Let R be a ring which has an element $a \in R$ such that $a^{2}=-1_{R}$. Prove that: there exists a unique ring homomorphism $\varphi : \mathbb{Z} [ i ] \longrightarrow R$ such that $\varphi (...
TrItOs's user avatar
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0 answers
68 views

Is universal object unique up to unique isomorphism?

If $F:C \to D$ is functor between categories $C$ and $D$ and $X$ is an object of $D$, then a universal map from $X$ to $F$ is a pair $(A_X,u)$ where $A_X \in \mathrm{ob}(C)$ and $u \in D(X,F(A_X))$ ...
TheWildPalms's user avatar
3 votes
1 answer
154 views

Does a product need to be unique up to unique isomorphism?

I’m re-reading Bartosz Milewski’s Category Theory for Programmers and I am a bit confused. As I understand it, the universal construction (apparently called universal property elsewhere) is unique up ...
schuelermine's user avatar
14 votes
2 answers
626 views

Show a free group has no relations directly from the universal property

The free group is often defined by its universal property. A group $F$ is said to be free on a subset $S$ with inclusion map $\iota : S \rightarrow F$ if for every group $G$ and set map $\phi:S \...
cede's user avatar
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0 votes
2 answers
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every bilinear map is a tensor product if the dimensions matche

In the following every vector spaces are real and finite dimensional. The following statement seems too good to be true: Let $V,W$ be vector spaces. If $Z$ is a vector space of dimension $dim(V)dim(W)...
Laurent Claessens's user avatar
1 vote
0 answers
74 views

Categorical notion of finiteness

TLWR : I'd like to define either $\operatorname{Hom}_F : \text{Set} \times \text{Set}_* \to \text{Set}_*$ (set of functions with finite support) or simply $\mathcal{P}_F : \text{Set} \to \text{Set}$ (...
B. Pillet's user avatar
0 votes
1 answer
47 views

Proving that 2 formulations of the Universal Property of Free Groups are equivalent

The present question is inspired by an answer to this question of mine on applications of Category Theory in Abstract Algebra. One of the answers stated that the universal property of free groups ...
Acharyachakit's user avatar
3 votes
0 answers
36 views

Characterization of the completion of a first-countable topological abelian group

$\def\codom{\operatorname{codom}}$If $G$ is a first-countable abelian topological group, one can find a morphism $G\to\hat{G}$ of abelian topological groups, with $\hat{G}$ first-countable and ...
Elías Guisado Villalgordo's user avatar
2 votes
0 answers
38 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: $...
Physicist in disguise's user avatar
1 vote
1 answer
102 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 ...
HapeFS's user avatar
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0 votes
1 answer
65 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-...
Elías Guisado Villalgordo's user avatar

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