Questions tagged [universal-property]
For questions about universal properties of various mathematical constructions.
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Formalizing the universal property of the tensor product
Introduction (a bit long)
I am very new to category theory and algebra, in general.
I recently saw the formalized concept of a universal morphism, here in WikipediA.
I will repeat the definition with ...
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Universal property of the (Vistoli-)sheafification
Given a presheaf $F$, in Notes on Grothendieck topologies,
fibered categories
and descent theory
there is a construction of the sheafification (Proof for theorem 2.64).
The first part is the ...
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Missing universal properties
Given a category $\mathcal{C}$, we can use copresheaves $H \in [\mathcal{C},Set]$ or presehaves $H \in [\mathcal{C}^{\mathrm{op}},Set]$ to state left or right universal properties. Existence of a ...
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Functor demonstrating universal property of quotient spaces
The universal property of quotient spaces is, as quoted from wiki:
The quotient space $X/\sim$ together with the quotient map $q:X\to X/\sim$ is characterized by the following universal property: if $...
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Two Formulations of Universality
There are two common formulations of universal properties, which I will briefly state for clarity.
Anyone familiar with the definitions of universal element and universal arrow could skip to the bold ...
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Short exact sequence of pointed spaces induces isomorphism of quotients
Let $X,Y,Z$ be pointed topological spaces. I will denote the point by $*$ and write $0=\{*\}$. Let $f:X\to Y$, $g:Y\to Z$ be continuous maps preserving the point. The sequence $X\xrightarrow{f} Y\...
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Well-definedness of tensor product of modules
Given two $R$-modules $M$ and $N$, one possible construction of the tensor product $M\otimes_R N$ is by taking the quotient $F/U$ of the free module $F=\bigoplus_{m,n}R\cdot(m,n)$ and the submodule $U$...
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Can fibrations be described via universal morphisms?
Jacobs defines Cartesian morphisms and fibrations in Definition 1.1.3 of this document.
In other places in the text he uses phrases like "the universal property of this lifting." This made ...
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Does the universal set operate as the Von Neumann universe in non well founded set theories?
I've been researching non well founded set theories (E.g. NF, NFU, etc.) and have been wondering if there are any similarities between the universal set & Von Neumann universe ? Or if there the ...
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Universal Property of The Simplex Category Δop
I tried formulating the universal property of the simplex category. There seemed to be two different options to choose for the universal property of $\Delta^{op}$. I'm having trouble making ...
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Kan extensions as weighted limits
In this Blog David Jaz Myers says that right Kan extensions can be expressed using weighted limits. I would really appreciate any help understanding how this is so. I do not understand the sense in ...
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Why $\bar{f}(x + H_1) = f(x) + H_2$?
I am trying to solve part $(c)$ of the following problem:
For $i = 1,2,$ let $G_i$ be a group and $H_i$ be a normal subgroup of $G_i.$ Let $\pi_i: G_i \to G_i/H_i$ be given by $\pi_i(a) = a H_i$ for ...
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Prove: there exist a unique group homomorphism $\bar{f}: G_1/H_1 \to G_2/H_2$
I am trying to solve the following problem:
For $i = 1,2,$ let $G_i$ be a group and $H_i$ be a normal subgroup of $G_i.$ Let $\pi_i: G_i \to G_i/H_i$ be given by $\pi_i(a) = a H_i$ for all $a \in G_i.$...
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Why do we require uniqueness in the universal property for a fraction field?
I'm a university student taking a course in abstract algebra. My professor recently introduced fraction fields, giving this definition:
Let $R$ be an integral domain. There exists a field $F$, called ...
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Finite product and coproduct are the same in abelian categories
I am working with the following definition of abelian category.
a) It has a $0$ object.
b) Every morphism has a kernel and a cokernel. Every monomorphismis a kernel and every epimorphism is a ...
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Connecting Mac Lane's discussion of free categories from graphs to free algebraic systems
I want to understand how the following observation applies to free algebraic systems. In Thm. II.7.1 of Mac Lane's Categories for the Working Mathematician (p.49 of ed. 2), the functor $P:G \...
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Prove that for all rings $R$ and all $r \in R$, there exists a unique ring homomorphism $\varphi: \Bbb Z[x] \to R$ such that $\varphi(x)=r$.
Denote by $\Bbb Z[x]$ the polynomial ring over $\Bbb Z$ in one variable. Prove that for all rings $R$ and all $r \in R$, there exists a unique ring homomorphism $\varphi: \Bbb Z[x] \to R$ such that $\...
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Proving the universal property of subspace topology
I'm asking for feedback on my current work.
Preamble: It is known that if $(f_i)_{i\in I}$ a family of functions $f_i:X\to Y_i$, where $(Y_i, \mathcal{T}_i)$ is a topological space, then the topology $...
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Universal characterization of the arrow (interval, ordinal, 1-simplex) object? (Is $2$-category theory actually needed?)
In the "category of categories" $Cat$ (see e.g. here for a rigorous definition), there is a category $A$ with two objects and exactly one non-identity morphism. (Cf. "two-valued object&...
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Exercise $0.11$ from Leinster's Basic Category Theory
This is Exercise $0.11$ from Leinster's Basic Category Theory.
Let $\theta:G \to H$ be a group homomorphism. Associated with $\theta$ is a diagram $$\ker(\theta) \stackrel{\iota}{\hookrightarrow} G \...
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Confusions regarding proof of tensor products preserve right exactness
This is a follow-up to my previous question.
Question:
How are they able to replace $A \otimes M$, $B \otimes M$ and $C \otimes M$ by $A \times M$, $B \times M$ and $C \times M$ respectively in the ...
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Confusion regarding proof of tensor products preserve cokernels
I found this proof here but I don't understand what they mean when they say that the existence of $C \to Q$ is implied by assumption? What assumption exactly? Can someone explain?
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Constructing the topological product space from categorical universal property
Let $X,Y$ be topological spaces. I want to construct the categorical product, i.e. the product space, $X\times Y$ using the universal property of products. To be clear, I don't want to construct the ...
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How does it follow that $P = Q$ in this universal property of coproducts of abelian groups?
How does it follow that $P = Q$ in this universal property of coproducts of abelian groups?
I'm reading Martin Brandenburg's explanation of why:
Given that $P$ is a coproduct of the abelian groups $...
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Does the category of topological spaces with an action of a topological monoid admit coequalizers?
Let $\mathsf{Top}$ be the category of topological spaces with continuous maps and let $T$ be a monoid in $\mathsf{Top}$. Let $f,g: M \to N$ be two parallel maps in $\mathsf{Top}_T$, the category of ...
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Generators give quotients of free products of groups?
I have some confusion here.
I have this idea that:
If $\Gamma$ is generated by (all of) the elements of subgroups $G_1,\dots,G_k$, then $\Gamma=\langle G_1,\dots, G_k\rangle$ is a quotient of the ...
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Showing that every continuous map between two topological spaces $X$ and $Y$ induces a continuous map between their cones $C(X)$ and $C(Y)$
I’d like to know how to show that every continuous map between two topological spaces $X$ and $Y$ induces a continuous map between their cones $C(X)$ and $C(Y)$. I was thinking it could be a good idea ...
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Characterization of basis in terms of universal property: axiom of choice
I wonder if the proof of the following statement requires the axiom of choice:
(Characterization of basis in terms of universal property) Let $V$ be a vector space, and let $S$ be a non-empty subset ...
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Intuition on product definition in category theory
I just have a question on product of 2 objects in category theory.
Given a category $\mathcal{C}$ we would like to define a product of 2 objects $A,B \in \mathcal{C}$. The definition says that $D \in \...
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How to compute a free algebra for a given algebra over a set?
This is a question from a "demo exam" a few years ago at out university course.
The question: How would you compute the free algebra for $\mathcal{A}$ over $X = \{x, y \}$, where $\mathcal{...
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Is Stone-Čech compactification the only one with universal property?
The Stone-Čech compactification (also called the Čech-Stone compactification) is the "biggest" compactification of a topological space. (I am working with Hausdorff compactifications only, ...
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Category theoretical proof that Abelianization commutes with products of group
Let $Grps$ and $Ab$ be the categories of groups and abelian groups respectively and lets denote $\times,\sqcup, \oplus$ the product and coproduct in Grps and the biproduct in $Ab$. If I want to prove ...
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Help understanding a brief proof to show $\mathbb{Z}[X]/(X^2 - 3) \cong \mathbb{Z}[\sqrt{3}]$
I was recently asked to prove that $\mathbb{Z}[X]/(X^2 - 3) \cong \mathbb{Z}[\sqrt{3}] := \{a + \sqrt{3}b\ | a,b \in \mathbb{Z}\}$. I couldn't do it in a limited amount of time, so I received a sketch ...
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The universal property of product holds only for the product sigma-algebra
Suppose $\{X_\alpha \mid \alpha \in J\}$ is a family of measurable spaces. Let X denote the Cartesian product $\prod_{\alpha \in J} X_\alpha$ and consider the natural projections $\{\pi_\alpha: X \to ...
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Is there a theorem that roughly says "all universal properties identify objects uniquely up to isomorphism"?
The question is in the title basically. I know that we can prove for certain specific universal properties (products, etc) that they uniquely determine the object up to isomorphism. The concept of &...
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Universal property of set intersection? [duplicate]
Is there a universal property of the intersection between two sets? In particular, can we define this intersection as the limit/colimit of some diagram? Presumably we'd have to assume some particular ...
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Can one give a category theory formulation of the following statement: "On a generic real vector space there is no canonical choice of a base"
My formulation would be:
There are no functors from the category $\mathcal{V}$ of real vector spaces( where morphisms are linear maps) into the category $\mathcal{V}_b$ of real vector spaces with ...
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Homeomorphism of Coproduct
Consider the space $C$ and maps $f_A: A \to C$ and $f_B: B \to C$. Assume that there exists a unique continuous function $G: C \to T$ such that $g_A= G \circ f_A$ and $g_B= G \circ f_B$ for any space ...
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Trying to find the terminal object (and category) for this universal property
Consider an index set $I = \{1, 2, \ldots, n\}$.
Let $V$ be a vector space and $v_I$ be a basis that is,
$\{v_1, v_2, \ldots, v_n\}$. Let $\phi: I \rightarrow V$ be a map
that maps the index set to a ...
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3
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How to show that $\mathbb{K}\otimes_\mathbb{K}V\cong V$ using just the universal property of the tensor product.
Given vector spaces over a field $\mathbb{K}$, how can one show that $\mathbb{K}\otimes_\mathbb{K}V\cong V$ using just the universal property of the tensor product? Some background: This is an ...
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Formulating a universal property for an inclusion map
Let $B$ be a subset of the codomain of $f : X \to Y$ that contains $f(X)$. Formulate the universal property satisfied by the inclusion map $i : B \to Y$ that characterizes it up to isomorphism among ...
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Proving $\bigwedge^k(U\cap V) = \left(\bigwedge^kU\right) \bigcap \left(\bigwedge^k V\right)$ via the universal mapping property
Let $V$ be a (finite dimensional) vector space, its exterior algebra of order $k$ is the vector space $\bigwedge^k V$ consisting of the formal sums of terms of the form $v_1 \wedge v_2 \wedge \dots \...
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Using universal properties to show isomorphisms?
I started self-learning category theory where I encountered the universal property of tensor products. Then there are a few problems that I have no ideas on how to get started on:
Question: Let $U,V,W$...
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Proof of the Universal Property of the Tensor Products
Suppose that $h:X\times Y\to \mathbb{F}$ is a bilinear map. Then prove that there exists a linear map $h_\otimes:X\otimes Y\to\mathbb{F}$ such that $h(x,y)=h_\otimes(x,y)$.
Logically, to every ...
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Show that $\mathbb{Z}/(m_1) \otimes_{\mathbb{Z}/(n)} \mathbb{Z}/(m_2) \cong \mathbb{Z}/(m_1,m_2)$
Let $m_1,m_2\in \mathbb{N}$ s.t. $m_1\mid n$ and $m_2\mid n$. By that,
both $\mathbb{Z}/(m_1)$ and $\mathbb{Z}/(m_2)$ are
$\mathbb{Z}/(n)$-Algebras. Show that $$\mathbb{Z}/(m_1) \otimes_{\mathbb{Z}/(n)...
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The free object of the variety constructed by an m-generated subalgebra
A modal algebra is an algebra $\langle M,\vee,\wedge, ',0,1,^\circ\rangle$ such that $\langle M,\vee,\wedge, ',0,1,\rangle$ is a Boolean algebra and the operation $^\circ$ satisfies $(x\wedge y)^\circ=...
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Q: Does stictly increasing function always cross x and y axis?
I've been trying to figure out this problem:
Let f: be strictly increasing function on it's Domain with range of all real numbers.
Does it always intercept x and <...
2
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1
answer
69
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Universal Mapping Property of Free Categories
Let $G$ be a graph. We say that a category $\textbf{C}$ has a universal mapping property of free categories on $G$ if there exists a graph homomorphism $i:G\longrightarrow |\textbf{C}|$ such that ...
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Equivalence of universal properties
In "Sheaves in Geometry and Logic" Mac Lane, Moerdijk (page 292: section VI, chapter 4), say:
In a topos $\mathcal{E}$ (with terminal object $1$), take a family of objects $\{ F_{n} \}_{n \...
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2
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About $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}$ and $\mathbb{C} \otimes_{\mathbb{C}} \mathbb{C}$
We know $\mathbb{C} \otimes_{\mathbb{C}} \mathbb{C} = \mathbb{C}$ and $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} = \mathbb{C}^2$. This contradicts my current understanding. I've spent time looking up ...
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