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3
votes
1answer
30 views

Notions of free (and/or cofree) Hopf algebras?

I don't know if this somewhat vague question has a precise answer, but I'd appreciate thoughts and references. Let $k$ be a field. I am looking at Hopf algebras over $k$ by which I mean an algebra ...
5
votes
1answer
57 views

Universal property of quotient group

Let $H\le G$ be a normal subgroup of $G$. Then we can think a natural projection map $\pi:G \to G/H$. Then this map has the following universal property: Let $\phi:G \to G'$ be a homomorphism. If $H ...
3
votes
1answer
56 views

What's so special about the grounded poset of cardinality $2$?

By a grounded poset, I mean a poset $P$ with a bottom element $0_P$. Arrows between grounded posets are monotone mappings that preserve the basepoint. This gives a self-enriched category; lets call it ...
0
votes
1answer
39 views

Functions in the definition of Universal Mapping Property of a free monoid

In Awodey's Category Theory (http://www.amazon.com/dp/0199587361/?tag=stackoverfl08-20) p.19, what is $|\bar{f}|$? What is its relation to $\bar{f}$, which is in the existence statement? Is it ...
1
vote
1answer
42 views

Unique isomorphisms and universal properties

Having not studied category theory, I'm trying to piece together without using category theory what is meant when an algebraic structure is said to possess a universal property or be unique up to ...
1
vote
1answer
32 views

What does “universal w.r.t. this property” mean? (kernel of a morphism in an additive category)

In an additive category $\mathcal{A}$ a kernel of a morphism $f: B\to C$ is defined to be a map $i : A \to B$ such that $fi = 0$ and that is universal with respect to this property. This is ...
2
votes
1answer
65 views

Universal Property for isomorphic objects

Suppose $A,B$ are isomorphic objects of some category $C$. Suppose that A satisfies a given universal property does B satisfy the same? If not can you provide a counterexample for an elementary ...
6
votes
1answer
57 views

How to construct a tensor product of two preadditive categories in pure categorical fashion?

Let $\mathsf C$ and $\mathsf D$ be two preadditive categories (by an preadditive category I mean a category together with compatible abelian group structure on every hom-set). I thought of ...
2
votes
3answers
63 views

Given a subset $S$, is there any universal construction (property) of $L(S)$, the linear span of $S$?

Given a subset $S$ of a vector space $V$, is there any universal construction (property) of $L(S)$, the linear span of $S$ (like the way we can construct the fraction field of an integral domain)?
2
votes
0answers
96 views

What should this definition be?

This is from Advanced Linear Algebra by S.Roman. Definition (on page 356). Referring to the figure below, let A be a set and let $\mathcal S$ be a family of sets. Let $$ \mathcal F = \{g\colon ...
6
votes
1answer
77 views

Does every continuous map between compact metrizable spaces lift to the Cantor set?

I'm interested in the universal properties of the Cantor set. It is well-know that the Cantor set $2^\mathbb{N}$ is "universal" in the category of metrizable compact spaces, in the sense that every ...
7
votes
1answer
89 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
6
votes
1answer
101 views

Proving that the free group on two generators is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$

I want to show that the free group on two elements $F(\{x,y\})$ is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$. The idea is to use the universal property of free groups to prove that ...
1
vote
2answers
57 views

Uses of Universal Properties

So in reading about category theory I'm starting to see this picture that it is just a higher level of abstraction where we consider similarities between mathematical structures by way of morphisms ...
3
votes
1answer
46 views

Universal property of natural number semi-ring

I asked a question similar to the one I am about to ask, and I think I got a satisfactory answer. However, this time I have some more specific question. Let a semiring $(R,+,\times)$ be an algebraic ...
12
votes
2answers
165 views

Does the forgetful functor from $\mathbf{TopGrp}$ to $\mathbf{Top}$ admit a left adjoint?

Let TopGrp be the category of topological groups (not necessarily $T_0$) and Top the category of topological spaces. Does the forgetful functor $U:\mathbf{TopGrp}\to\mathbf{Top}$ admit a left ...
2
votes
0answers
31 views

Characterization of epic morphisms in the category of rings.

Let $C$ denotes the category of rings (with identities). It's easy to prove that for any $R\in \text{ob }C$, any multiplicative set $S\subset R$ and any ideal $I\leqslant R$, both $R\to S^{-1}R$ and ...
1
vote
2answers
51 views

Prove Kähler Differential is always surjective using universal property.

Let $A$ be an $R$-Algebra. An $R$-linear derivation $d \colon A \to \Omega_{A/R}$ is called universal derivation or Kähler differential if for every $R$-linear derivation $D \colon A \to M$ there is a ...
1
vote
1answer
27 views

Universal property question

I am not sure how I can draw a commutative diagram, so I will do my best to describe it verbally. So, suppose $f_1,f_2:G\to K$ be (group, field, ring)homomorphisms. I want to claim the following: ...
4
votes
2answers
204 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 $\mathbf{Ab}$, $\mathbf{Grp}$, ...
0
votes
0answers
21 views

Description of a universal regular space

tl;dr: Given a topological space $X$, is there a useful description of the universal continuous map from $X$ to a regular (Hausdorff) space? The Čech–Stone compactification $\beta X$, given a ...
1
vote
1answer
48 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
2answers
49 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 ...
0
votes
1answer
80 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 ...
8
votes
1answer
128 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 ...
1
vote
1answer
38 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
0answers
17 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
34 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 ...
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 ...
1
vote
3answers
40 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 ...
2
votes
0answers
79 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 ...
6
votes
1answer
136 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'} ...
5
votes
1answer
231 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} ...
1
vote
1answer
72 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 $V_1, \cdots, V_m$, ...
2
votes
2answers
206 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 ...
0
votes
1answer
25 views

Transition functions of sheaf tensor product

Suppose $\mathcal{L},\mathcal{M}$ are invertible sheafs on a scheme $X$. I've seen an abstract construction of $\mathcal{L}\otimes_X \mathcal{M}$, but I'm having trouble connecting this with a more ...
1
vote
1answer
168 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 ...
0
votes
1answer
24 views

Mapping property of complex fraction field

I recently came across a proof which said that: Suppose $\phi: \mathbb{C}[x]\rightarrow \mathcal{F}$ where $\mathcal{F}$ is a field is a homomorphism. If $ker\phi=0$ then $\phi$ maps isomorphically to ...
1
vote
1answer
91 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 \\ ...
5
votes
1answer
129 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 ...
0
votes
1answer
77 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 ...
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 $X$ ...
0
votes
1answer
41 views

Universal property for vector spaces

QUESTION: I would like some hints for part b.I am unsure of how to show that $\widetilde{T}$ is surjective or injective for that matter. My construction for $\widetilde{T}$ is ...
4
votes
0answers
71 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 ...
2
votes
2answers
89 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
1answer
95 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 ...
0
votes
0answers
35 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 ...
2
votes
0answers
56 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 ...
0
votes
2answers
70 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) ...
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 $f(N)$. ...