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15
votes
1answer
207 views

Universal property of de Rham differential.

Suppose $A$ is a commutative algebra over a field $k$. It is well known that there is a module that generalizes the notion of differential $1$-forms. It is denoted $\Omega^1_{k}(A)$ and is called the ...
11
votes
2answers
152 views

Is there a concept of a “free Hilbert space on a set”?

I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set $X$. Before ...
8
votes
1answer
114 views

Are there any non-obvious colimits of finite abelian groups?

Does the forgetful functor $U : \mathsf{FinAb} \to \mathsf{Ab}$ from finite abelian groups to abelian groups preserve colimits? Morally this should be true, but it is not so easy (for me) to come up ...
7
votes
2answers
159 views

How to get used to commutative diagrams? (the case of products).

I've returned to Aluffi's book (after getting the basics of groups from Herstein's) and I hit the same brick wall that made me put it aside. My general problem is this: I can't seem to get used to ...
7
votes
1answer
112 views

Universal properties of “interesting” families of graphs

Can cycles – and/or other "interesting" families of graphs like paths, trees, hypercubes, etc. – be characterized by a universal property in the category of graphs? I admit that there is ...
6
votes
2answers
298 views

Why is category theory not just another theory? [closed]

Consider category theory as one theory among many others: with a simple signature and some simple axioms. Compare it with - e.g. - group theory as another theory with a simple signature and some ...
6
votes
1answer
118 views

Help with diagram chasing

Given the diagram $\require{AMScd}$ \begin{CD} 0 @>>> A @>f>> B @>g>> C @>>> 0 \\ @. @V\alpha VV \#@V\beta V V\# @VV\gamma V @. \\ 0 @>>> {A'} ...
6
votes
1answer
290 views

Axioms vs. Universal Constructions/Properties

What (exactly) is the difference between defining a mathematical object by it's axioms and by a universal construction ? Please take my 3 opinions into consideration, as they also contain more ...
5
votes
2answers
109 views

Is there a logical interpretation for equalizer and co-equalizer?

I know the logical equivalent to several universal constructions. For example product $\times$ is $\land$ and co-product $+$ is $\lor$. The associated arrows are projection and inclusion. The ...
5
votes
1answer
120 views

Universal Property: do people study terminal objects in $(X\downarrow U)$?

Suppose that $U:D\to C$ is a functor and $X$ is an object of $C$. In defining Universal property, Wikipedia writes about terminal objects in $(U\downarrow X)$ and initial objects in the category ...
5
votes
0answers
105 views

Universal Property for Tangent Space

As far as I know there are basically three different approaches to the tangent space -all of them coming with advantages and disadvantages: Though the oldest one precisely represents the derivative ...
4
votes
2answers
176 views

Identify the universal property of kernels

I'm reading Mac Lane's "Categories for the Working Mathematician". I found the following sentence in page 59 of it: Similarly, the kernel of a homomorphism (in $\mathbf{Ab}$, $\mathbf{Grp}$, ...
4
votes
1answer
184 views

The Kahler differential is zero

While studying for my commutative algebra exam, I have come across this problem. Let $K$ be a field. Let $A$ be a $K-$algebra, finitely generated as a $K$-vector space. Prove that $\Omega^1_{A/K} ...
4
votes
0answers
68 views

Using the universal property of tensor product to show that $(\Bbb{Z}/4\Bbb{Z}) \otimes (\Bbb{Z}/6\Bbb{Z})\cong \Bbb{Z}/2\Bbb{Z}$ [duplicate]

In the algebra lecture i need to solve the following exercise Use the universal property of the tensor product to show that $$(\Bbb{Z}/4\Bbb{Z}) \otimes (\Bbb{Z}/6\Bbb{Z})\cong ...
3
votes
3answers
123 views

Can an object be universal with respect to several properties?

Does anyone know an example of an object $A$ in some category $\mathcal C$ such that $A$ satisfies at the same time more than one universal property? Of course, when I say that $A$ is universal with ...
3
votes
1answer
65 views

About the Stone-Čech universal property

There's something I am missing here and I don't know what it is. I understand that the Stone-Čech compactification of $X$ satisfies the property that for every continuous map $f: X \rightarrow K$ ...
3
votes
0answers
55 views

A mathematical object that can be characterised by axioms as well as by universal constructions

This question (more specifically one of the comments) prompted me to ask this question: Can someone give me an example of a mathematical object that one can characterize both via axioms and universal ...
2
votes
2answers
108 views

“Canonical” symmetrization/skew-symmetrization/alternation of multilinear functions

Is there some precise sense in which the "alternation" functor $A$ that maps a multilinear function $f\colon M^d\to N$ to the alternating multilinear function $A(f)\colon M^d\to N$ defined by $$ ...
2
votes
1answer
65 views

How to use universal constructions to create this category/object?

Given an a category $\mathcal{C}$ what is the most concise way to construct the smallest one object category $\mathcal{D}$ such that there exists a faithful functor from $\mathcal{C} \to \mathcal{D}$? ...
2
votes
1answer
56 views

Distinguishing characteristic and spurious properties

When working in set theory with definitions of somehow primitive notions like ordered pairs (Kuratowski) or ordinal numbers (von Neumann) one is supposed to use only the characteristic properties of ...
2
votes
2answers
194 views

Construction of Yoneda extension

In "Category Theory" by Steve Awodey, there is a suggestion for the reader to construct a left adjoint in the proof of the UMP of the Yoneda embedding. Namely, for any small category $\mathbb{C}$, the ...
2
votes
1answer
90 views

Two Definitions of Infinite Cartesian Product

In my one of lecture notes, there are two definitions of infinite Cartesian product, and it reads that we can construct a unique bijection between them. One way to define an infinite Cartesian ...
2
votes
1answer
120 views

$F$ entire with $\lim_{k \rightarrow \infty} F(z + N_{k}) = h(z)$ for every $h$ entire

Prove that there exists an entire function with the following "universal" property: given any entire function $h$, there is an increasing sequence $\{ N_{k}\}_{k=1}^{\infty}$ of positive integers ...
2
votes
1answer
180 views

Why does the fiber coproduct in $\mathbf{Set}$ actually satisfy the universal property?

Suppose you have two morphisms $f\colon A\to B$ and $g\colon A\to C$. Then the fiber coproduct of $B$ and $C$ over $A$ exists and is the disjoint union $B\coprod C$ where we identify $fx$ and $gx$ for ...
2
votes
1answer
38 views

Viewing the universal property of rings of fractions as a universal arrow

For a multiplicatively closed subset $S$ of $A$, we have a functor $S^{-1}: A-Mod \rightarrow S^{-1}A-Mod$. I am trying to understand this functor a little bit better and I was thinking about the ...
2
votes
2answers
83 views

relationship between Exact sequences and Universal mapping property

I am stuck in the following question. Show that for any two short exact sequences. $0\overset{}{\rightarrow}K\overset{i}{\rightarrow}V\overset{T}{\rightarrow}U$ and ...
2
votes
0answers
70 views

Does zero-kernel imply monic in Abelian categories?

I'm trying to learn how to perform diagram-chasing in abstract Abelian categories. Instead of an approach with some elements one have to use universal properties somehow in the proof. But I reckon the ...
2
votes
0answers
50 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
2
votes
0answers
43 views

Show that $H *_{G} (G/N) \cong H/N'$ where $f(N)=N' \subset H$.

I was doing some group theory and this exercise came up. Let $f: G \to H$ be a group morphism and $N$ a normal subgroup of $G$. Denote $\pi: G \to G/N$ and let $N'$ be the normal closure of $f(N)$. ...
2
votes
0answers
25 views

Disjoint Union Topology universal property as instance of final topology's universal property [duplicate]

I'm studying the final topology on a set $X$ induced by a family $\{f_\alpha:X_\alpha\rightarrow X\mid \alpha\in A\}$ and know of its universal property, namely that given a function $g:X\rightarrow ...
1
vote
1answer
37 views

Find the unique lifting for $f:S^1 \to S^1$ given by $f(z)=z^2$, where $z=x+iy, x^2+y^2=1$

Let $f:S^1 \to S^1$ be given by $f(z)=z^2$, where $z=x+iy, x^2+y^2=1$. Then find the unique lift $\bar f: S^1 \to \mathbb{R}$ with the properties that (i) $\bar f(1)=0$ and (ii) $E \circ \bar f=f$, ...
1
vote
1answer
44 views

Limit as universal arrow

I was interested in the definition of limit as universal arrow from $\Delta$ to $F$ given in Mac Lane's CWM book. Let us begin with some notation: Let $C$ be a category, and $J$ be an index ...
1
vote
3answers
35 views

Basic Question Regarding the Universal Property of the Tensor Product.

(All vector spaces are over a fixed field $F$). Universal Property of Tensor Product. Given two finite dimensional vector spaces $V$ and $W$, the tensor product of $V$ and $W$ is a vector space ...
1
vote
1answer
50 views

Dual notion of the subspace topology

Let $X$ and $Y$ be sets with $\iota:X\to Y$ an injection. If $Y$ is a topological space, we define the subspace topology on $X$ as the initial topology induced by this diagram. Analogously, if $X$ ...
1
vote
1answer
50 views

Lifts of embeddings of Lie algebras to their universal enveloping algebras

Let $k$ be an algebraically closed field, and let $(\mathfrak{h},[\;,\;])$ be a finite dimensional abelian Lie algebra $k$. Let $(\mathfrak{g},[\;,\;])$ be a finite dimensional Lie algebra over $k$ ...
1
vote
2answers
43 views

A universal construction of the field of fractions of an integral domain?

Let $R$ be an integral domain and For a field $\hat R$ consider the following : There is an injective ring homomorphism $i:R \to \hat R$ such that for any field $F$ and any injective ring homomorphism ...
1
vote
1answer
128 views

Universal object

I'm trying to define the concept of universal object in category theory, but I failed to find a universal way to do it. Maybe someone can help ? THe idea is quite simple. Let $\mathcal{C}$ be a ...
1
vote
3answers
47 views

Would this proof be considered true.Proving a property of a operation

Ok so the operation [x] is defined to be equal to the integer such that it is $\leq x$ From this definition it holds that : $$ [x] \leq x $$ I need to prove that $$ [x+n] = [x] + n $$ My proof ...
1
vote
1answer
28 views

Direct sum is isomorphic?

Given R, a PID, and two finitely generated R-modules A and B, suppose $\varphi: A \rightarrow B$ is a homomorphism. If B is a free module, show that $A \cong \ker(\varphi) \oplus ...
1
vote
1answer
55 views

The use of universal properties to prove the existence of isomorphism

I just start self learning tensor and I find the universal property is difficult to use. I think I understand the basic concept of the universal property. The tensor product of $V_1, \cdots, V_m$, ...
1
vote
1answer
72 views

Tensor product of function space and vector space

In one book I found that they treat vector valued functions as tensor product of some function space and vector space. So for example $$ C(\mathbb{R},V) \simeq C(\mathbb{R})\otimes V \\ ...
1
vote
2answers
151 views

Binary number theory

I want to know if there exists some book about "Binary number theory", I'm interesting in this because there many problems of ICPC about this topic. Thanks in advance
1
vote
1answer
33 views

Universal property of sum of ideals

If $A$ is a commutative algebra over the field $K$, and $I,J$ are ideals of $A$ (in particular, they are also algebras over $K$), is it possible to characterize the ideal they generate together $I + J ...
1
vote
1answer
343 views

Direct (Inductive) limit of (locally convex) TVEs and universal property

This is not really a question, I'd just like to discuss a little about universal properties (more specifically, the direct limit) in TVEs. I'm trying to work with universal properties in Topological ...
1
vote
1answer
368 views

Anti-derivation of an exterior algebra

Let $K$ be a commutative ring. $M$ is a free $K$-module of rank $m$. $E(M)$ is the exterior algebra defined by $M$. $\iota$ is a $K$-homomorphism from $M$ to $E(M)$ such that $\iota(x)=-x$. Since ...
1
vote
0answers
10 views

“Universal solution” to factorization of arithmetic mean through a semigroup homomorphism.

Headnote A: my apologies for the long winded exposition; I couldn't find a more compact way to convey what I'm looking for, why I'm looking for it, and what have I tried myself. Suggestions for ...
1
vote
1answer
64 views

Extending the universal property of tensor product

Suppose that we defined a tensor product of vector spaces $U$ and $V$ as a quotient of a vector space with basis $V \times W$ by the vector space spanned by $-(u_1+u_2,v)+(u_1,v)+(u_2,v), ...
0
votes
2answers
222 views

Universal Property of Localization

Let $R$ be a commutative ring with $1\not =0$, and let $D\ni 1$ be a multiplicative subset of $R$. Consider the universal characterization of $D^{-1}R$: There is a morphism $\pi\colon R\to D^{-1}R$ ...
0
votes
1answer
62 views

Universal Properties and Isomorphisms

If two objects satisfy the same universal property, we know that they are isomorphic in that category. Is the converse true? That is, if two objects are isomorphic in some category, can we construct ...
0
votes
1answer
67 views

Definition of a coproduct and its universal property - connection?

I have a problem connecting the definition of a coproduct with its often mentionend universal property. Let's start with the definition (just for two objects): Let $A_1$ and $A_2$ be objects of a ...