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Questions tagged [universal-property]

For questions about universal properties of various mathematical constructions.

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Generalising idea of projection of continuous laticces into any category

Definition: A continuous lattice $D$ is said to be a projection of a continuous lattice $D'$ if and only if there are a pair of continuous maps $$i:D\rightarrow D'$$ and $$j:D'\rightarrow D$$ ...
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No canonical correspondance between bilinear forms $b:V\times V \rightarrow \mathbb{R}$ and linear forms $\hat{b}:V\otimes V \rightarrow \mathbb{R}$?

In the wiki on bilinear forms, the universal product says that to each bilnear map $b : V\times V \rightarrow \mathbb{R}$ we can associate a linear map $\hat{b} : V\otimes V \rightarrow \mathbb{R}$, ...
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Show that if a submonoid $S$ of a commutative monoid $M$ contains an absorbing element, then the localization of $M$ by $S$ contains only one element.

Here is the full question: If $M$ is a commutative monoid, $S\subset M$ is a submonoid and there is a $z\in S$ that is absorbing in $M$ (i.e. $zm=z$ for all $m\in M$), then show that $M_S$ has only ...
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Proving disjoint union is defined up to an isomorphism using coproducts in $\mathbf{Set}$

The book on abstract algebra I'm going through has the following exercise. Let $A' \cong A'', B' \cong B''$ be two pairs of isomorphic sets. Further, let them be pairwise disjoint: $A' \cap B' = \...
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Prove the uniqueness of function $X \rightarrow A \times B$

Let A and B be sets and consider their Cartesian product $A\times B\rightarrow A$. Give surjective functions $\pi_A:A\times B\rightarrow A$ and $\pi_B:A\times B\rightarrow B$. Suppose we are given a ...
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When, if ever, is this a valid universal property for $A \times (B_1 + B_2)$? (Category Theory)

Notation: Let $\pi$'s denote canonical projections, $j$'s denote canonical injections, given a pair of maps $X \overset{f_1}{\to} Y_1$ and $X \overset{f_2}{\to} Y_2$ denote the unique map $X \overset{...
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Universal property of the direct product, proof verification

Let $I$ be a set. For every $i\in I$ let a set $X_i$ be given, and for every $j\in I$ denote $p_j:\prod_{i\in I} X_i\to X_j$, $p_j((x_i)_{i\in I}):= x_j$, the j-th projection. If $X,Y$ are sets ...
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Products, coproducts and morphisms

The universal properties of products and coproducts "amount" to the statements $$ \hom(\coprod_i X_i , Y) = \prod_i \hom(X_i, Y) \quad \text{and} \quad \hom(X,\prod_i Y_i) = \prod_i \hom(X,Y_i)$$ ...
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Universal Property…The Map is not Well-defined?

How does the following proposition from this book make sense: Proposition 2.8 Let $G$ be the group defined by the presentation $(X,R)$. For any group $H$ and map of sets $\alpha : X \to H$ sending ...
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A construction of the Hodge Dual operator

This question about showing that an alternative construction of the Hodge dual operator satisfies to the universal property through which the Hodge dual is usually defined. Let me give the ...
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A question about limit cones and isomorphism

A consider a limit cone on a diagram $D: \mathbf{I} \rightarrow \mathbf{C}$: $$ \left( L \xrightarrow{p_i} D(I) \right)_{I \in \mathbf{I}} $$ Now suppose that $L' \in \mathbf{C}$ is some object ...
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Proving the Free Abelian Group is Free Abelian…?

On page 40 of these notes is the following exercise: Prove that the group with generators $a_1,...,a_n$ and relations $[a_i,a_j]=1$, $i \neq j$, is the free abelian group on $a_1,...,a_n$. On ...
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Showing that the projections of a product are jointly monic

Suppose that we have a category $\mathcal{C}$ which "has binary products" (full definition provided here: https://en.wikipedia.org/wiki/Product_(category_theory)). We want to show that given $A, B, C \...
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Meaning of “universal” in category theory (equalizer definition)

In Awodey's Category Theory on page 62, equalizers are defined like so: In any category C, given parallel arrows $f, g : A \rightarrow B$, an equalizer of $f$ and $g$ consists of an object $E$ and ...
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Understanding the inverse limit and universal property of topological spaces

Let $\{X_i, \varphi_{ij},I\}$ be an inverse system of topological space index by a directed poset $I$. Now I would like to understand the proof for the existence of an inverse limit $(X,\varphi_i)$. ...
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Does the box topology have a universal property?

Given a set of topological spaces $\{X_\alpha\}$, there are two main topologies we can give to the Cartesian product $\Pi_\alpha X_\alpha$: the product topology and the box topology. The product ...
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Uniqueness in the Universal Property of Quotient Maps

Here is Munkres' way of phrasing the universal property of quotient maps: Let $p : X \to Y$ be a quotient map. Let $Z$ be a space and let $g : X \to Z$ be a map that is constant on each set $p^{-1}...
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Product property [closed]

What is the name of the following property of two equal products? If $ab = cd$, then $a(b-d)=(c-a)d$
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Is there a “universal family of hypersurfaces”?

Consider the category of (flat) families of degree $d$ hypersurfaces in $\mathbb P^n$, i.e. objects are $X \to Y$ whose fibers are degree $d$ hypersurfaces in $\mathbb P^n$, and morphisms are the ...
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Calculating a presentation of $\mathbb{Z}_{3}$ in detail.

Theorem: Let $G$ groups and $S\subset G$ such that $\langle S\rangle =G$. (Here $G=\left\{s_1\ldots, s_n:s_i\in S\cup S^{-1}, n\in\mathbb{N}\right\}$.) Let $\varphi:S\to G$ with $\varphi(s)=s$. By ...
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Can it be useful to think of functors as representing themselves ?

Here's a thought I had and I wonder if it can be of any use, for instance has it ever helped proving any result (however minor the result). Say you're in a situation where you have some objects in ...
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Direct sum of non-abelian groups doesn't satisfy the universal property of direct sum.

Let $C$ and $D$ be non-abelian groups. Show that $C\oplus D$ doesn't satisfy the universal property of direct sum. I think I must assume that the universal property is true and then use the free ...
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Universal Property of the Ring of Quotients.

Let $R$, $R'$ be commutative rings with multiplicative subsets $S, S'$ and the corresponding ring homomorphisms $\phi, \phi'$ into their respective ring of fractions. Prove that for any ring ...
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The peculiar properties of 14732

About a year ago, I delved into the properties of the number 14732. A number that have popped into my brain at random times for well over a decade. Through a bit of experimenting with base10 and ...
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The Universal Mapping Property of a free vector space.

Definition: Let $X$ be a non-empty set. A free vector space on $X$ is a pair $(V,i)$ consisting of a vector space $V$ and a function $i:X\to V$ satisfying the following universal mapping property. I ...
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Inequality properties between their multiplications [closed]

If $A>B>C>D$ then will $$(A/B * A/C * A/D) > (B/A * B/C * B/D) > (C/A * C/B * C/D) > (D/A * D/B * D/C)$$ be always true? If not, in what intervals will it not be true? Obviously $A,...
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Find the properties of the function f(x) which fulfills the following inequality

Let us assume that $f:\mathbb{R}^+ \rightarrow \mathbb{R}$ fulfills the following inequality: $$ \exists_{C \in \mathbb{R^+}} \quad|f'(x) \cdot x| < C. $$ In other words we can say that $f'(x)\...
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A topological space with the Universal Extension Property which is not homeomorphic to a retract of $\mathbb{R}^J$?

A topological space $Y$ has the universal extension property if for every normal space $X$, every closed subset $A$ of $X$, and every continuous function $f:A\rightarrow Y$, we can extend $f$ to a ...
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Free monoids as universal constructions (example)

I'm a beginner and I'm desperately trying to understand free monoids as universal constructions using an example. What is the relation between the following? a generator set (e.g. $\mathbb{N}$ (...
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Does universal construction apply to any arbitrary category? [closed]

Does universal construction presuppose a category equipped with morphisms that implement an order? Otherwise how does the universal construction pick the most suitable element for the given purpose? ...
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Looking for a pair of adjoint functors

I'm struggling with finding the left adjoint to a functor, and the right adjoint to another one. Here's some context. Given any ring $R$, we can associate two categories to it. The first one is a ...
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How is functor with “image” unique up to a unique isomorphism defined exactly?

In an abelian category $\mathscr A$ we encounters the notions of kernel, cokernel, chain homology, derived functors, etc. These notions are frequently referred to as functors, and yes, they actually ...
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The universal property of polynomial rings and universal identities

To verify some identity involving say two variables $x, y \in R$ for any commutative ring $R$ it suffices to verify this identity in $\mathbb{Z}[x,y]$. This is just the sort of trick that should be ...
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Being universal with respect to a property

I have always concerned universal properties under a statement of the form Universal Property of the Quotient Group: Let $G$ be a group and $H$ a subgroup of $G$. If $f: G \to G'$ is a group ...
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Soft question on universal properties

My question heads towards an intuition of universal properties. Take for example a group $G$ and a subset $M$. Then $F_M$, the free group on $M$ together with a map of sets $i:M\to U(F_M)$ where $U$ ...
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Universal Property of free vector space

Let $X$ be any set and $F$ a field. Let $V_{X}$ be set of functions $f:X \to F$ for which $f(x)$ is nonzero for finitely many $x \in X$. Note that $V_{X}$ is a vector space. Call $\sigma_{x}$ a ...
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Is there a category theoretic characterisation of the exponential map from differential geometry?

While I have only a shallow understanding, I like category theory. I find definitions and proofs in terms of category theoretic concepts to be very clean and deep, often cutting to the core of a ...
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Dummit and Foote Universal Property of Tensor Products

Here is the theorem giving me trouble: Here is theorem 6, which is invoked in the proof of theorem 6: Here is the proof of theorem 8: Here is the sentence giving me trouble so far: "Since $\...
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Functional property name?

I would like to know if this functional property has a name. Definition. We say that a functional $L$ has a property if and only if $L(f) \leq L(g)$ implies $$ L(f) \leq L\left(\frac{f+g}{2}\right) \...
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Compact open topology on $\operatorname{GL}(n, \mathbb{R})$ coincides with Euclidean topology.

There are two ways to assign $\operatorname{GL}(n,\mathbb{R})$ topologies: as subspace of $\mathbb{R}^{n^2}$, or subspace of $\operatorname{Maps}(\mathbb{R}^n, \mathbb{R}^n)$ where the latter is ...
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Why do exponential objects (in category theory) require currying?

I'm a bit confused about exponential objects in category theory. I find them pretty intuitive when I think about them as "arbitrary-arity cartesian products"; in the sense that if I had never seen ...
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Universal property of quotient group to get epimorphism

I know there are already answers to the question of the universal property of quotient groups. For example here: Universal property of quotient group. My question is now: If I have the homomorphism: $...
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Equalizer of reflexive pair

A reflexive pair is a pair of parallel morphisms $f,g:X\to Y$ having a common section, i.e. a map $s:Y\to X$ such that $f\circ s = g\circ s = id_Y$. Reflexive maps are famous because their ...
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What makes an universal mapping unique?

Maybe the title doesn't explain very well my question. I was wondering why if some object satisfies an universal property, then for any other object there is one and only one mapping between them. Let ...
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Isomorphic Free $R$-module has Universal Property

Here is corollary 7 in chapter 10.3 of Dummit and Foote book on Algebra: If $F_1$ and $F_2$ are free modules on the same set $A$, there is a unique isomorphism between $F_1$ and $F_2$ which is ...
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on the maps to moduli spaces

Let $M$ be a coarse moduli scheme for some moduli functor $\mathcal{F}$( from the category of schemes to the category of sets). Let $V$ be a scheme such that for each point of $v \in V$ we can ...
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Proving the universal mapping property for polynomial rings

I am studying from Patrick Morandi's Field and Galois Theory, and I am stuck on Problem 6 from Section 1 (on page 13). Verify the following universal mapping property for polynomial rings: ...
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Sum of two isomorphic objects and relation between the inclusions

Consider objects $A, B$ in some category $C$. Let $S\in C$ to be the sum of $A$ and $B$ with inclusions as $i_A:A\rightarrow S$ and $i_B:B\rightarrow S$. If $A\cong B$ with isomorphisms $f:A\...
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Universal Finitely Presented Group

Higman's embedding theorem states every finitely generated, recursively presented group embeds into some finitely presented group. A further result of Higman, Neumann and Neumann shows that every ...
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Defining Groups via the Universal Property

The snippets below on group presentation led to the following questions: What is the explanation/intuition of the universal property. Basically what (2) is saying. I understand free groups (though ...