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2
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0answers
63 views

On direct sum and direct product of groups

I understood the definitions of direct sums and direct products of groups (rings, modules, etc). The definitions of them can be given in terms of universal properties - certain commutative diagram in ...
4
votes
1answer
46 views

Are Free Groups the “Smallest Group” Containing their Generators

I apologize if this is a duplicate; I was not sure how to search for this. When I say "the smallest group" I mean unique up to isomorphism of course. Specifically, is "the smallest group containing ...
0
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0answers
8 views

How to prove pairwise independence of a family of hash functions?

I want to prove pairwise independence of a family of hash functions, but I don't know where to start. Given the family of hash functions: H with h(x) = a * x + b (mod M). ( Say h: U -> V, then: M ...
2
votes
1answer
71 views

“Universal property” of cross product

Let $V$ be a three dimensional euclidian vector space which is oriented. Because of the orientation, we can define the cross product $\times: V^2 \rightarrow V$ uniquely by: $<v\times w,u> = \...
1
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1answer
70 views

Universal property, localization of rings and modules, and initial element in a category

Sorry for the confusing title. I just started learning category theory and am very confused about the concept "universal property". I am not even sure whether my "proof" is a proof or is just a ...
1
vote
1answer
42 views

Citations for the proof of universality of graph classes

In Automorphisms of graphs, Peter J. Cameron mentioned following classes of graphs which are universal structures. graphs of valency k for any fixed k > 2; bipartite graphs; strongly regular graphs; ...
1
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0answers
41 views

How do I get from the universal product of the tensor product to other definitions.

I was wondering how you can "derive" the common (or "classical") definition of the tensor product before the universal property was established (I think there is no need to repeat it here), i.e. a ...
1
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0answers
50 views

Localization and the universal property

I've been trying to get to grips with the universal property and was looking at the localization of a commutative ring $A$ at some multiplicative set $S$, denoted $S^{-1}A$, to get an idea of what the ...
1
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1answer
35 views

On Schauder basic systems in universal enveloped algebra of system of countable family of bounded selfadjoint operators

Let $A = C^*(1,T_1,T_2, ... | T_i^* = T_i, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of selfadjoint operators. I want to know as more as possible about that algebra,...
1
vote
1answer
114 views

Anti-involution on universal enveloping algebra of a Lie algebra.

Let $\mathfrak{g}$ finite dimentional semisimple Lie algebra and $\sigma$ the usual chevalley anti-involution that fixes the Cartan subalgebra $\mathfrak{h}$ sends the weight space $\mathfrak{g}_\...
0
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0answers
32 views

Proving “up to unique isomorphism”: Universal property of cokernel

This is a homework question. I have already searched the web, but the proofs I have found are very generalized using category theory. Problem is the following: Suppose you have homomorphism $f: M \...
1
vote
1answer
39 views

If monoid satisifes universal mapping property over $X$, then $X$ generates the monoid

A monoid $M$ satisfies the universal mapping property (UMP) over $X$, if $X \subseteq M$ and for every map $\varphi : X \to N$, where $N$ is another monoid, there exists a unique homomorphism $\varphi ...
1
vote
1answer
49 views

Proof of the universal property of the quotient topology

In this question: universal property in quotient topology I saw the following theorem: Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. Let $\pi: X\to X/{\sim}$ be the ...
1
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0answers
21 views

Homomorphisms between two universal free groups [duplicate]

I'm trying to prove that, for positive integers, $m$ and $n$, there is a homomorphism from $F_n$ onto $F_m$ if and only if $m$ is less than or equal to $n$ (where $F_n$ is the universal free group ...
4
votes
1answer
87 views

Universal property of topology of uniform convergence

What kind of universal property does the strong dual topology on $X'$ have, for $X$ being a locally convex space. Is it possible to define $X'$ as the projective limit of the normed spaces $\mathcal{L}...
0
votes
3answers
47 views

universal property of product: must any map satisfying it be a morphism

I am thinking about the universal property of products: Let $X$ and $Y$ be objects of a category $D$. The product of $X$ and $Y$ is an object $X \times Y$ together with two morphisms $\pi_1 : X \...
2
votes
1answer
38 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) \not\...
4
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0answers
55 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
62 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
1answer
19 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
52 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 ...
0
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1answer
46 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
53 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
80 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
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1answer
46 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
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0answers
42 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 ...
5
votes
1answer
129 views

universal property in quotient topology

The following is a theorem in topology: Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. Let $\pi: X\to X/\sim$ be the canonical projection. If $g : X → Z$ is a continuous ...
0
votes
1answer
30 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
42 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
31 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 $B$-...
1
vote
1answer
41 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
43 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
103 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
50 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
104 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
59 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
58 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
79 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
38 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 quoted ...
2
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1answer
81 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
82 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
73 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
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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 A\...
6
votes
1answer
118 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
2answers
149 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, ...
7
votes
1answer
170 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 $F(\{...
1
vote
2answers
69 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
90 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
221 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 adjoint?...
2
votes
0answers
40 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 $...