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2
votes
1answer
37 views

Proving a set is open in a locally ringed space $(X,\mathscr O_X)$

Let $(X, \mathscr O_X)$ be a locally ringed space and let $A = \Gamma(X,\mathscr O_X)$ be the global sections. For $f\in A$, define the "distinguished open base" as: $$D(f) = \{x\in X : \pi_x(f) ...
4
votes
0answers
48 views

Monomorphisms of monoids are stable under coproducts

Let $M,N,K$ be three monoids (or even groups, if you like) and let $N \to K$ be an injective homomorphism. Then, the induced morphism $M \sqcup N \to M \sqcup K$ is also injective. This is easy to ...
2
votes
0answers
57 views

Grothendieck's definition of a universal problem

In EGA I, Grothendieck writes (I'm paraphrasing): Let $\mathbf{K}$ be a category, $(A_{\alpha})_{\alpha \in I}, (A_{\alpha \beta})_{(\alpha,\beta) \in I \times I}$ two families of objects of ...
0
votes
0answers
11 views

What does “4-universal hash function” mean?

I encountered the notion of 4-universal hash function and I cannot understand what exactly it means. This article https://en.wikipedia.org/wiki/Universal_hashing did not really help to clarify it. ...
1
vote
1answer
44 views

universal properties of dependent types

What is the universal property of dependent product / dependent sum? (I want to see a diagrams) They are must be different from usual ones, aren't they? (i'm trying to understand category theory ...
-1
votes
1answer
45 views

$\{\alpha_i : A_i \to C_i \}$ family of maps $\implies \exists ! \alpha : \prod_i A_i \to \prod_i C_i$ such that $\pi_i^C \alpha \to \alpha_i \pi_i^A$

Suppose that $\prod_i A_i, \prod_i C_i$ exist in a category, and that there is a family of maps $\{\alpha_i : A_i \to C_i\}$. There exists a unique $\alpha : \prod_i A_i \to \prod_i C_i$ such that ...
1
vote
1answer
38 views

proof that there is a unique isomorphism between two free modules

I'm trying to figure out if there's any way to shorten the following proof of Corollary 7 in Dummit & Foote (p. 355) so that I don't have to write down the expressions of any specific elements ...
3
votes
0answers
75 views

Universal property of l^p-spaces

The category $\mathsf{Ban_1}$ of Banach spaces together with short linear maps (i.e. those of norm $\leq 1$) seems to have a natural construction which interpolates between coproduct and product: Let ...
0
votes
1answer
42 views

Arrow From an Initial Object Does Not Disturb Commutativity of the Diagram

Suppose I have a commutative diagram $D$ in the category of abelian groups. In $D$ there appear abelian groups $A$, $B$, $A\oplus B$ and some other abelian groups. Also, the natural inclusions $i:A\to ...
0
votes
0answers
33 views

Universal property of free products.

In my textbook, the Van der Waerden trick is used to prove that every word is equivalent to exactly one reduced word. For this, the author used the universal property of free products, so I tried to ...
0
votes
1answer
20 views

Existence of unique homomorphism between free groups

We know that a free group F(S) on a set S comes equipped with a function $\alpha_S : S \to F(S)$. For any function $\beta: S \to G$, for some group $G$, there exists a unique homomorphism $\phi : F(S) ...
2
votes
1answer
41 views

$L(\bigoplus_{i\in I} H_i)\cong \prod_{i\in I}L(H_i)$ as Banach spaces?

Let $\{H_i:i\in I\}$ be a system of complex Hilbert spaces, $L(\bigoplus_{i\in I} H_i)$ the set of bounded linear maps $T:\bigoplus_{i\in I} H_i\to \bigoplus_{i\in I} H_i$, equipped with the operator ...
3
votes
1answer
18 views

Associativity of extension of scalar (tensor product)

I am trying to prove the following basic property $$(M \otimes_A B) \otimes_B C \cong M \otimes_A (B \otimes_B C) \cong M \otimes_A C$$ where $M$ is $A$-module, $B$ is an $A$-algebra and $C$ is ...
1
vote
1answer
28 views

Finite graphs forms Fraïssé Class with limit Rado/random graph

I was reading Peter Cameron's explanation of Fraïssé's Theorem in his book Permutation Groups (Chapter 5). He states the Fraïssé class of finite graphs has limit being the countable random/Rado graph, ...
2
votes
1answer
35 views

Can the choice of definition of morphisms for a slice category be justified categorically?

An example of a slice category $(\mathscr{C} \downarrow c)$ derived from some fixed object $c$ of some base category $\mathscr{C}$, would be one whose objects correspond to the $\mathscr{C}$-morphisms ...
0
votes
2answers
85 views

Observability of a System in State Space form

as we all know, each system has infinite state space representations. My question is, whether the observability/controllability (and other common properties) are a properties of the SYSTYEM or ...
3
votes
1answer
42 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
85 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
47 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
63 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
36 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
77 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
68 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
67 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
100 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
103 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 ...
8
votes
1answer
117 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
145 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
62 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
71 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 ...
15
votes
2answers
207 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
38 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
68 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: ...
6
votes
2answers
264 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
26 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
59 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
66 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
107 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
165 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
43 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
19 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
54 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
47 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
52 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
114 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 ...
7
votes
1answer
177 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
286 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
89 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 $V_1, \cdots, V_m$, ...