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15
votes
1answer
181 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
138 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 ...
10
votes
2answers
139 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
106 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
292 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
219 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
93 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
0answers
64 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 ...
3
votes
3answers
110 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
54 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
48 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
1answer
58 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
53 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
1answer
66 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
48 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
106 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
0answers
45 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
37 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
1answer
53 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
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
44 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
3answers
45 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
2answers
86 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
29 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
239 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
238 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
1answer
56 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
102 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
2answers
54 views

How is this an example of a Universal Property?

I'm totally lost on this problem, which is from a set of notes by Tom Leinster on Category Theory: Let $\phi: G \to H$ be a homomorphism of groups. Associated with $\phi$ are the diagrams $ker(\phi) ...
0
votes
1answer
51 views

Continuity of sum/product using characteristic property

I'm self studying Lee's Introduction to Topological Manifolds, and I'm familiarising myself to the characteristic/universal property view of topology. One of the exercises in chapter 3 goes as ...
0
votes
1answer
43 views

Existence of Initial Object implies Existence of Terminal Object? [closed]

Consider an initial object always exists in a category. Reversing all arrows now. Does this guarantee the existence of terminal objects in that category?
0
votes
1answer
179 views

Quotient function [closed]

Let $f : X \to Y$ be a quotient map, and $g : X \to Z$ a continuous function such that $g(x_1) = g(x_2)$ whenever $f(x_1) = f(x_2)$. Show that there exists a unique continuous function $h : Y \to Z$ ...
0
votes
0answers
14 views

Injectivity and uniqueness of some maps from tensor products

Let $V_A$, $V_B$, $W$ be real vector spaces, let $V_A^*$, $V_B^*$, $W^*$ be their dual spaces. Let \begin{align*} &\phi: V_A \times V_B \rightarrow W \\ &\psi: V_A^* \times V_B^* \rightarrow ...
0
votes
1answer
55 views

Tensor Algebra = Universal Property of FORGETFUL FUNCTOR?

Hi there in wiki the tensor algebra is defined w.r.t. the adjoint of the forgetful functor rather than the forgetful functor itself - why so? Besides, does the existence of such algebras for every ...
0
votes
1answer
47 views

Characteristic Property = Universal Property?

Problem They seem to be the same -almost! But are they really or is it just unlucky accident that they look so similar however describe totally different notions? Example I was trying to set the ...
0
votes
0answers
11 views

Universal property of tensor products of real Hilbert spaces

I have the following exercise where I could need some hints: Let H1, H2 be real Hilbert spaces. Prove that there is a weak Hilbert-Schmidt mapping $$ p: H_1 \times H_2 \rightarrow H_1 \otimes H_2 ...
0
votes
0answers
17 views

Subset/subgroups/etc as Equalizers using Pushouts

I am trying to show that in a given category $\mathcal{C}$ which has pushouts, we can show that any morphism $f:X\rightarrow Y$ is an equalizer of some pair of morphisms. My solution to this is by ...
0
votes
0answers
61 views

Universal property of free spaces

In class we had developed the notion of a free space in preparation for our study of tensor product. There is one statement in my notes about the universal property of free spaces that has been ...