Questions tagged [universal-property]

For questions about universal properties of various mathematical constructions.

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Can we approximate Riemann zeta function in the half strip using its universality principe?

Universality of Riemann zeta function , Which related to the approximation of every Holomorphic function $f(z)$ by Riemann zeta function in the strip , it states for lower density: Corollary: Let $...
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1answer
42 views

Universal morphism in the first isomorphism theorem for groups.

I almost don't know anything about categories but I was reading about universal morphisms and I wanted to see this in the context of the first isomorphism theorem for groups. What would the functor $F$...
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Question from Mac Lane and Birkhoff (Chapter II Section 5) — Universal property(?) of direct product of cyclic groups

This is problem 10 from section 5 of chapter II from "Algebra" by Mac Lane and Birkhoff. It says: Let $E$ be the direct product of two cyclic groups $G$ and $H$ with generators $b$ and $c$ ...
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How to understand the effect of adjoint functors?

I have a good grasp of all different definitions/interpretations of adjoint functors, but still do not know have to interpret the left or right adjoint of a give functor, when it exist. It would be a ...
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25 views

Universal property of 2-pullback

According to the n-lab article on 2-pullbacks, A 2-pullback in a 2-category is a square $$ \begin{array}{ccc} P & \xrightarrow{p} & A \\\ q\downarrow & & \downarrow{f}\\\ B &...
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$f:X\to Y$ extends to a map $Z\to Y$ iff $f_*[g] = 0$

Let $f \colon(X,x_0)\to (Y,y_0)$ and $g \colon(S^n,s_0)\to (X,x_0)$ be base point preserving maps. Let $Z$ be the space that arises from $X$ by attaching an $(n+1)$-disk via $g$. I want to prove ...
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42 views

Prove universal morphism is unique up to unique isomorphism.

I'm following along Wikipedia page(https://en.wikipedia.org/wiki/Universal_property) on universal property, and this seems it should be trivial, but I couldn't finish the proof. The definition I am ...
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1answer
97 views

Understanding Universal Property and Universal Element (from Category Theory in Context, Riehl)

I'm trying to understand universal Property and universal element as stated in Category Theory in Context. The book states So from what I understand, since $C(c, -)$ is representable, there exists ...
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55 views

Higher homotopy group $\pi_n(X,x_0)$ is the set of homotopy classes of maps $(S^n,p) \to (X,x_0)$

I recently learned about the higher homotopy group $\pi_n(X,x_0)$ which is defined as the set of mappings $$f:[0,1]^n \to X,\ f(\partial [0,1]^n) = x_0$$ up to homotopy. Given $S^n \simeq [0,1]^n/\...
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Making a “definition” by way of a universal property

This is somewhat a question out writing styles. In a paper, I want to introduce and "define" an object $X$ by just stating the universal properties which is satisfies. Logically, this is not quite ...
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59 views

If $S\to T$ is a closed immersion, then $X\times_S Y\simeq X\times_T Y$.

Let $X,Y$ be $S$-schemes and $S\to T$ a closed immersion of schemes. Prove that we have a natural $T$-isomorphism $X\times_S Y\simeq X\times_T Y$. Let $f:X\to S$ and $g:Y\to S$ the structural ...
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39 views

A module homomorphism is injective iff the kernel is isomorphic to the trivial module

Consider a $R$-module homomorphism $\varphi\colon M\to N$. It is well-known that $\ker\varphi=\{0\}$ iff $\varphi$ is injective (equivalently is monic). However, suppose we only have $\ker\varphi\...
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63 views

does an ordinary map have a universal property?

I use Lang's Algebra book to define universal objects: Let $\mathcal{C}$ be a category. An object $P$ of $\mathcal{C}$ is called universal attracting if there exists a unique morphism of each ...
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41 views

Why does the surjectivity of the canonical projection $\pi:G\to G/N$ imply uniqueness of $\tilde \varphi: G/N \to H$

Let's look at the universal property of quotient groups: Let $\varphi:G \to H$ be a homomorphism, $N$ a normal subgroup of $G$ and $\pi:G \to G/N$ the canonical projection. If $N \le \ker \...
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55 views

Doubt on hypothesis of universal property of direct sum

I have seen that the following theorem, known as universal property of the direct sum, is always stated for direct sums of abelian groups: (Universal mapping property of external direct sum): let ...
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How can I prove that an different compactification to the Stone-Cech compactification doen't have it universal property?

Let $S(X)=\prod_{U \in \tau_{X}} \{ 0,1 \}_{U} $ with $(X,\tau_{X})$ $T_0$ topological space and $\{ 0,1 \}$ the Sierpinski's space. Cosiderer the map $e$ from $X$ to $S(X)$. $e:X\rightarrow S(X)$ ...
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25 views

If a presheaf $\mathcal{F}$ coincides locally with a sheaf $\mathcal{G}$, then $\mathcal{F}^{sh}\simeq \mathcal{G}$?

I'm trying to prove whether or not this is true: Let $\mathcal{F}$ be a presheaf and $\mathcal{G}$ a sheaf (of rings, let's say) on a space $X$. If $\{U_i\}_i$ is a basis for the topology of $X$ ...
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1answer
81 views

Is the standard definition of tensor product an effective way to introduce it?

I was taught the tensor product via its universal property: the only object that satisfies ... up to isomorphism. Later, I literally discovered one could actually write down the elements of (some) ...
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121 views

Homomorphisms from $\prod_{i\in\mathbb Z}\mathbb Z $ to $\oplus_{i\in\mathbb Z}\mathbb Z$ that fixes $\oplus_{i\in\mathbb Z}\mathbb Z$

I'm trying to verify that $\prod_{i\in\mathbb Z}\mathbb Z $(the direct product of countably many $\mathbb Z$) is not a coproduct in the category of abelian groups. We know that the coproduct object is ...
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Can I generalize this property to hold for unimodal distributions

I was able to prove this property: $-G(x)x + B = \int_x^B G(k) dk$ has at most one solution $x\in [A,B]$ for all uniform distributions $g(x)$ (with c.d.f. $G(x)$), with support [A,B] and $B> A\geq ...
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Universal properties of kernel and equalizer

The following diagram of $R$-modules is commutative. $\require{AMScd}$ \begin{CD} M @>f>> P\\ @VVgV @VV\varphi V\\ Q @>\psi>> N \end{CD} Prove: $M$ is the pullback ...
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Direct product of groups has inclusions, but it's still not the free product (coproduct)?

Conceptually, I get the difference between products and coproducts: the first has projections, the second has inclusions. There are all sorts of circumstances in which you can be convinced that these ...
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Proving the inclusion map of an integral domain into its quotient field is an epimorphism

As a well-known example of a ring homomorphism which is monic and epi, but not a ring isomorphism, serves the inclusion map $\iota:\Bbb Z\hookrightarrow\Bbb Q$. While the monocity follows immediately ...
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Proof that the exterior power fulfils an universal property using tensor product

before I ask my question, I briefly review two definition which I use: (1) Tensor product of vector spaces: Let $V_{1},\dots,V_{n}$ be vector spaces over the same field $\mathbb{F}$. Then the ''...
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free group quotient by its commutator is a free abelian group via universal property

Let $F=F(A)$ be a free group, and let $f:A→G$ be a set-function from the set $A$ to an abelian group $G$. Show that $f$ induces a unique homomorphism $F/[F,F]→G$, where $[F,F]$ is the commutator ...
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55 views

Are the categories $U(\mathfrak{g})$-$\mathrm{mod}$ and $\mathfrak{g}$-$\mathrm{rep}$ equivalent?

Let $\mathfrak{g}$ be a Lie algebra. Denote by $U(\mathfrak{g})$ its universal enveloping algebra. Let $U(\mathfrak{g})$-$\mathrm{mod}$ be the category of modules over $U(\mathfrak{g})$, and $\...
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A property of pullbacks I cannot prove!

Let $\require{AMScd}$ \begin{CD} P @>{g'}>> B\\ @Vf'VV @VVfV\\ A @>>g> C \end{CD} be a pullback. $f'$ is iso iff there exists $h:A\to B$ such that $\require{AMScd}$ $f\...
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51 views

Are “free products” just coproducts in categories admitting presentations of objects (generators and relators)?

For groups, the "free product" can be taken "generator-wise" and "relator-wise" as done here: https://ncatlab.org/nlab/show/free+product+of+groups It is also the case that the "free product" is the ...
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1answer
72 views

Polynomial Rings “Most Efficient” Ring?

I'm currently reading though Aluffi's Algebra 0, and within III.2.2 Aluffi shows how $\mathbb{Z}[x_1,\dots,x_n]$ satisfies a universal property like which free groups satisfy for groups. Specifically,...
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1answer
49 views

Free vector spaces and construction of Tensor Product

The whole (intuitive) idea of the necessity of tensor product vector space is almost understood, I mean: given our prior experience with the concept of multiplication (basic elementary one like in ...
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Describing the universal property of the evaluation map

Emily Riehl's "Category Theory in Context", ${\rm Exercise}~2.3.{\rm iii}.$ The set $B^A$ of functions from a set $A$ to a set $B$ represents the contravariant functor ${\rm Set}(-\times A,B):{\rm ...
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100 views

Question on coproducts and products

Given a family of modules $\{A_i\}_{i \in I}$, I always understood that the main difference between an element of the product $\Pi A_i$ and the direct sum $\oplus A_i$ to be that if you take an ...
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26 views

There is a unique map $P \to \text{Hom}(A, \prod_i C_i)$ whenever $P \xrightarrow{p_i} \text{Hom}(A, C_i)$?

I'm trying to prove directly that $\text{Hom}(A, \prod_i C_i) \simeq \prod_i \text{Hom}(A, C_i)$ whenever $\prod_{i\in I} C_i$ is a product of a family of objects $C_i$ in a category $B$, where $A$ is ...
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90 views

Kernel of group epimorphism with finitely presented is the normal closure of a finite subset .

Suppose $F : G \rightarrow K $ is a group epimorphism with $G$ finitely generated and $K$ finitely presented. Then, show that $H := \ker F$ is the normal closure of a finite subset in $H$. My attempt ...
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188 views

How does Universal Mapping Property encode “no-junk” and “no-noise” in free monoid?

I am going trough the "Category Theory" book by Steve Awodey. In the "1.7 Free categories" chapter the author introduces the following algebraic definition of free monoid: A ...
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1answer
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Is the image a universal object?

Given a function $f:X\to Y$ in category $\mathcal{C}$, one can construct the image as a factorisation $f=(e:I\hookrightarrow Y)\circ(g:X\to I)$ that is universal (initial) among all such ...
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1answer
30 views

Product projections are a universal arrow?

From Mac Lane's Categories for the Working Mathematician: Let $S: D \rightarrow C$. A universal arrow arrow from $S$ to $c$ is a pair $\langle r,v \rangle$ consisting of an object $r \in D$ and an ...
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81 views

Cokernel within the Category of Groups

This question is in regards to Aluffi's Algebra: Chapter $0$, II.$8.22$ $\textbf{8.22: }$Let $\varphi: G \rightarrow G'$ be a group homomorphism, and let $N$ be the smallest normal subgroup ...
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1answer
137 views

Trying to understand the definition of a universal property

Here's the definition of a universal property in Wikipedia: (where $U:D\to C$ is a functor and $X$ is an object in $C$) A terminal morphism from $U$ to $X$ is a final object in the category $(...
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68 views

Categories with $C \times 0 \cong 0$ for all objects $C$

In $Sets$, the initial object is $0 = \emptyset$. We have $C \times \emptyset = \emptyset$ for any set $C$. As there are no maps to the empty set, we don't get much of the universal property of the ...
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Coproduct of abelian groups

In the comments to this answer, Martin Brandenburg told me that it is possible to derive the element structure of the coproduct $A\oplus B$ of two abelian groups $A$ and $B$, by knowing that such a ...
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When do equalizers preserve epis?

The question cannot be completely wrong, as it does not completely make sense, whence let me detail a bit. I'm in a monoidal category $(\mathcal{C},\otimes,\mathbb{I},\alpha,\lambda,\rho)$ with ...
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1answer
73 views

The variety generated by a free algebra

I am working on the following problem: Take a free algebra $\mathcal{A}$ (on some signature $\mathcal{L}$) with set of generators $G$ and consider every other $\mathcal{L}$-algebra, say $\mathcal{B}$,...
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62 views

The monoid of fractions associated with the submonoid of cancellable elements of a commutative monoid E

Let $E$ be a commutative monoid, $\Sigma$ the submonoid of cancellable elements of $E$, $E_{\Sigma}$ the monoid of fractions of $E$ associated with $\Sigma$ and $\varepsilon$ the canonical ...
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106 views

Universal properties of $\mathbb Z$

What is the universal property of ℤ as a group? What is the universal property of ℤ as a ring? Explain why the answers to each of these questions are different. I have been able to show that this ...
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1answer
62 views

Viewing $\mathbb{Q}_p/\mathbb{Z}_p$ as a direct limit

I have been told that taking the direct limit of the system $\mathbb{Z}/p^n\mathbb{Z}\rightarrow\mathbb{Z}/p^{n+1}\mathbb{Z}$ with respect to multiplication by $p$ maps gives us $\mathbb{Q}_p/\mathbb{...
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31 views

Universal property for rings of fractions

Let $R$ denote a (commutative and unital) ring and $S\subset R$ a multiplicative subset. Let $S^{-1}R$ denote the ring of fractions. I now want to consider the canonical projection $\pi: R\rightarrow ...
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29 views

Exponential decay sampled geometrically

Question: I'm trying to relate an exponential lifetime model where I know the average lifetime (1000hr) and a geometric sampling distribution where I would "check" the decay model, for example to see ...
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1answer
51 views

Smallest lattice containing a poset

Given a poset $P$(we can assume $P$ is finite if necessary), how can we construct the smallest lattice containing $P$? (Does this exist?) To make the question precise, I am looking for a lattice $L$ ...
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56 views

Bijection between tensors and multilinear maps for infinite dimensional case

When $V$ is a finite dimensional $k$-vector space, I understand that there is the natural bijection between $(p, q)$-tensor product $\bigotimes_{p}V\otimes \bigotimes_{q}V^*$ and the space of ...

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