Questions tagged [universal-property]
For questions about universal properties of various mathematical constructions.
331
questions
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42 views
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 $...
1
vote
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 $F$...
1
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0answers
32 views
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 $c$ ...
6
votes
0answers
132 views
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 ...
0
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0answers
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
$$ \begin{array}{ccc}
P & \xrightarrow{p} & A \\\
q\downarrow & & \downarrow{f}\\\
B &...
0
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0answers
41 views
$f:X\to Y$ extends to a map $Z\to Y$ iff $f_*[g] = 0$
Let $f \colon(X,x_0)\to (Y,y_0)$ and $g \colon(S^n,s_0)\to (X,x_0)$ be base
point preserving maps.
Let $Z$ be the space that arises from $X$ by attaching an $(n+1)$-disk via $g$.
I want to prove ...
0
votes
1answer
42 views
Prove universal morphism is unique up to unique isomorphism.
I'm following along Wikipedia page(https://en.wikipedia.org/wiki/Universal_property) on universal property, and this seems it should be trivial, but I couldn't finish the proof.
The definition I am ...
1
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1answer
97 views
Understanding Universal Property and Universal Element (from Category Theory in Context, Riehl)
I'm trying to understand universal Property and universal element as stated in Category Theory in Context.
The book states
So from what I understand, since $C(c, -)$ is representable, there exists ...
0
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1answer
55 views
Higher homotopy group $\pi_n(X,x_0)$ is the set of homotopy classes of maps $(S^n,p) \to (X,x_0)$
I recently learned about the higher homotopy group $\pi_n(X,x_0)$ which is defined as the set of mappings $$f:[0,1]^n \to X,\ f(\partial [0,1]^n) = x_0$$ up to homotopy.
Given $S^n \simeq [0,1]^n/\...
0
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0answers
20 views
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 ...
1
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1answer
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 ...
1
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1answer
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\...
4
votes
1answer
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
...
0
votes
2answers
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 \...
1
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2answers
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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0answers
30 views
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 $S(X)$. $e:X\rightarrow S(X)$ ...
0
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1answer
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 $X$ ...
2
votes
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) ...
4
votes
2answers
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 ...
0
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1answer
16 views
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 ...
2
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1answer
89 views
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 ...
7
votes
2answers
178 views
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 ...
3
votes
1answer
41 views
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 ...
0
votes
0answers
27 views
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 ''...
3
votes
0answers
47 views
free group quotient by its commutator is a free abelian group via universal property
Let $F=F(A)$ be a free group, and let $f:AāG$ be a set-function from the set $A$ to an abelian group $G$. Show that $f$ induces a unique homomorphism $F/[F,F]āG$, where $[F,F]$ is the commutator ...
0
votes
0answers
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 $\...
2
votes
2answers
88 views
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\...
1
vote
1answer
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 ...
1
vote
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,...
2
votes
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 ...
2
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0answers
72 views
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 ...
3
votes
2answers
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 ...
0
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1answer
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 ...
1
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0answers
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
...
3
votes
3answers
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 ...
1
vote
1answer
34 views
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 ...
1
vote
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 ...
3
votes
1answer
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 ...
1
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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 $(...
4
votes
1answer
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 ...
3
votes
0answers
52 views
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 ...
2
votes
0answers
22 views
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 ...
3
votes
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 $\mathcal{B}$,...
4
votes
2answers
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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1answer
106 views
Universal properties of $\mathbb Z$
What is the universal property of ā¤ as a group? What is the universal property of ā¤ as a ring? Explain why the answers
to each of these questions are different.
I have been able to show that this ...
1
vote
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{...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
4
votes
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 $L$ ...
2
votes
0answers
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 ...
