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Questions tagged [morphism]

In category theory, a morphism is a structure-preserving map, such as continuous mappings on topological spaces, measurable functions, and linear maps.

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How to say these two distinct functions have the same structure?

Yesterday, I posted this question, which remains unanswered. In this related question, I ask a different yet more precise question that may help me solve the other question. Let $N=\{1,2\}$ be a two-...
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algebras morphism [closed]

I have this exercise: "let (A, µ, e) be an algebra (e.g., over a field $\mathbb{K}$). Suppose that there exists a second algebra structure on A, (A, µ', e'), such that $\mu' : A ⊗_\mathbb{K} A → ...
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Affine Map as a Morphism of Affine Vector Spaces

I've recently took interest in morphism and category theory and I'm amazed how it offers a very general notion. However, I'm struggling to apply this for the affine vector spaces. I've seen that a ...
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Are the arrows in commutative diagram always morphisms?

During my learning algebra, commuting diagrams are frequently used in many books, but I found sometimes that the authors do not specify whether the arrows are homomorphisms or just maps. For instance, ...
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Morphisms as Homomorphisms

It is usually said that when we consider the category of groups, the morphisms are homomorphisms. The category of groups can also be considered as a category of sets and morphisms as usual functions, ...
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Morphisms in Monoids

As I am trying to learn Category Theory (CT) which is very relevant to my line of research, I am coming across the idea that, in CT, we don't have an "internal view" of say a group as made ...
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Is a surjective endomorphism of a finitely generated module over a *non-commutative* ring with unity necessarily an isomorphism?

As is well known, any surjective endomorphism of a finitely generated module $M$ over a commutative ring with unity $R$ must be an isomorphism. What about the non-commutative case? In other words, is ...
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On Wikipedia's definition of zero morphisms

Wikipedia defines $f : X \to Y$ to be a zero morphism if $(1)$ $gf = hf$ for any object $Z$ and $g, h:Y \to Z$, and $(2)$ $fg = fh$ for any object $W$ and any $g, h : W \to X$. It then defines a ...
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Carpi's sufficient condition for the abelian nth power freeness of a morphism in 1993

I'm working on Carpi's article on Abelian k-power free mentioned below: "ARTURO CARPI, ON ABELIAN POWER-FREE MORPHISMS, International Journal of Algebra and Computation Vol. 03, No. 02, pp. 151-...
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rational map by F. Mangolte

I'm reading Real Algebraic Varieties by F. Mangolte. Definition 1.3.22 (in the book) If $X$ and $Y$ are algebraic varieties over a base field $K$ a rational map $\phi:X\dashrightarrow Y$ is an ...
isz's user avatar
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A Natural Transformation Property in Single-Sorted Categories

Background: As a learning exercise, I am reproving some basic results in category theory within the single-sorted categories framework. Natural transformations in this framework have a slightly ...
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Prove that a set of homomorphism is Noetherian

I am currently working on the following questions ($M$ is a finite $A$-module and $N$ a Noetherian $A$-module): Prove that for all $l$ in $N$, A-module $N^l$ is Noetherian (this part was ok I proved ...
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What does $Hom(h, B)$ mean in the contravariant functor?

The Wikipedia stated that Let C be a locally small category (i.e. a category for which hom-classes are actually sets and not proper classes). For all objects A and B in C we define two functors to ...
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If $\phi:V_1\rightarrow V_2$ is a morphism of varieties then $V_1\cong \phi(V_1)$

I am reading Silverman's The Arithmetic of Elliptic Curves. I am wondering if with the definition of morphism he gives, we can conclude that if $\phi:V_1\rightarrow V_2$ is a morphism of projective ...
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What does it mean for a map to factor through another map?

In Darij Grinberg's "Hopf algebras in combinatorics", there is a statement about existence of quotient coalgebras: "Indeed, $J ⊗ C + C ⊗ J$ is contained in the kernel of the canonical ...
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A Full and Faithful Functor Transforms Surjection to Injection.

In general, I do not think the statement in the title is true. But from Galois Theories, by Francis Borceus and George Janelidze, they claimed a similar fact without proof. I shall give sufficient ...
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Silverman's AEC Exercise I.1.7

I am attempting Silverman's AEC exercise I.1.7 part (c). Instead of using intrinsic definitions or results, I am trying working with the definitions stated in the book, i.e. the two definitions in ...
Mystery girl's user avatar
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Subobject in the category of topological spaces

Given an object $X$, we can define an equivalence relation on the monomorphisms with range $X$: $u:S\to X,v:T\to X$ are equivalent iff exists an isomorphism $\phi:S\to T$ such that $u=v\circ \phi$. By ...
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Monoid morphisms between naturals with multiplication and naturals with addition

We define $\mathbf{N} = \{0,1,2,...\}$ and $ \mathbf{N}^* = \{1,2,...\}$, each a monoid with addition and multiplication respectively. I am looking for monoid morphisms between these two monoids. For ...
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Orders of isogenous curves over the algebraic closure

Let $E/k$ and $E'/k$ be isogenous over $k$. We know: $E$ and $E'$ are isogenous if and only if $\# E(k) = \# E'(k)$ [Ex V.5.4, Sil86]. Any non-constant morphism of curves is surjective [Thm. II.2.3, ...
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prove or disprove that there is an injective morphism of G-representations iff there is a surjective morphism of G-representations

Prove or disprove that given two G-representations V,W over $\mathbb{c},$ there is an injective morphism of G-representations $\theta : W\to V$ iff there is a surjective morphism of G-representations $...
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Prove that if $p$ and $q$ are projections with the same kernel, then $p\circ q=p$ and $q\circ p=q$

Let $K$ be a field, and let $E$ be $K$-vector space. Let $p$ and $q$ be two endomorphisms of $E$. Prove the following proposition: ($p$ and $q$ are projections, and $\ker{p}$ = $\ker{q}$) $\...
virtualcode's user avatar
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If $f:X\rightarrow Y$ is a $k-$morphism of projective varieties and $X(k)\neq\varnothing$, then $Y(k)\neq\varnothing$

Suppose $X\subset \mathbb{P}^m,Y\subset\mathbb{P}^n$ are projective varieties defined over a field $k$, and that $f:X\rightarrow Y$ is a $k-$morphism, i.e. $f$ is a morphism which induces a $k-$...
Aaron Andersen's user avatar
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Closure of morphisms with respect to composition.

Let $C$ denote some arbitrary category, and denote $X,Y,Z\in\text{Obj}(C)$. If $\mu_{XY}\in\text{Hom}_C(X,Y)$ and $\mu_{YX}\in\text{Hom}_C(Y,Z)$ are well-defined (total) functions, then can we ...
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An exercise about structure sheaf of product of two algebraic varieties

I am interested in the Exercise 5.5.8 of this lecture notes. I have my own solution for this exercise but I need someone here to verify if there are any flaws in my arguments. To recall, this exercise ...
Mystery girl's user avatar
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Restrictions of a morphism that is piecewise smooth

My lecture notes of classical algebraic geometry on complex field has presented a following result. Theorem. Let $X$ and $Y$ be (quasi-projective irreducible) varieties, and $f \colon X \to Y$ a ...
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Extension of a morphism of affine varieties $f: X \to Y$ to $\overline{f}: \mathbb{A}^m \to \mathbb{A}^n$

I am attempting Exercise 5.5.7 from this lecture notes. Let $X \subset \mathbb{A}^m$ and $Y \subset \mathbb{A}^n$ be closed subsets and a morphism of varieties $f: X \to Y$, extend $f$ to a morphism ...
Mystery girl's user avatar
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Restriction of dominant morphism on open subset is dominant

Suppose $X$ and $Y$ are two varieties and $f: X \to Y$ is a dominant morphism (i.e. $\overline{f(X)} = Y$) between them. Prove that for any nonempty open subset $U \subset X$, the restriction $f|_U : ...
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Why the need to show well definedness for the following case of restriction maps?

The following is based on an exercise from the book $linear algebra and geometry" by Leung. $\color{Green}{Background:}$ Show that if the diagram of linear spaces and linear transformations $$\...
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Are both $g', g$ assumed to be surjective in the commutative diagram for $3 \times 3$ lemma?

The following is taken from Module Theory An Approach to Linear Algebra} by T.S. Blyth $\style{font-family:inherit;}{\color{Green}{\textbf{Background:}}}$ Theorem 3.4: Consider the diagram of $R$-...
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How does the following commutative diagram within the following proof match the definition for universal property of the kernel and also what is $g?$

$\color{Green}{Background:}$ $\textbf{Definition:}$ (Universal property of the kernel) Let $R$ be a commutative ring and $f:A\to B$ a morphism of $R-$modules. Recall that the $\textit{kernel}$ of $f$ ...
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How is defined a domain (/codomain) in Category Theory? As a function, it is a morphism hence has a domain (/codomain). This, infinitely?

I'm basing myself on the Wikipedia article on Categories. A category $C$ consists on: a class of objects $Ob(C)$ a class of morphism $Mor(C)$ a class function going from $Mor(C)$ to $Ob(C)$ called ...
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Inverse morphism of group object is iso

Let $C$ be a category with finite products (and a terminal object). We can then define group objects as tuples $(G, m, e, inv)$ by requiring that the usual diagrams commute. Is it true that the ...
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Clarifications need for unclear notations in passage from Cohn's $\textit{Further Algebra with Applications}$ text

The following is taken from pg 37-38 of: Further Algebra with Applications by: P Cohn. It is also a continuation of this post $\color{Green}{Background:}$ Dually the cokernel of $\alpha:X\to Y$ is an ...
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Clarifications need for confusing passage about kernel in Cohn's $\textit{Further Algebra with Applications}$ text

The following is taken from pg 37 of: Further Algebra with Applications by: P Cohn. $\color{Green}{Background:}$ Let $A$ be an additive category; given a map $\alpha:X\to Y,$ we shall define the ...
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Clarifications need for passages explaining about cokernel, kernel, image and co-image in Cohn's $\textit{Further Algebra with Applications}$ text

The following is taken from pg 37-38 of: Further Algebra with Applications by: P Cohn. It is also a continuation of Meaning of: "$M'$ is the kernel of the canonical surjective morphism" and &...
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how to understand the meaning of kernel, cokernel, coimage, etc of another morphism.

The following retyping of four slides are taken from: Richard Crew's Homological Algebra Lecture 2 and is a continuation of this post Meaning of: "$M'$ is the kernel of the canonical surjective ...
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Meaning of: "$M'$ is the kernel of the canonical surjective morphism" and "$\text{Im} f$ is the kernel of $\text{Coker }f$?"

The following is taken from: $\textit{Groups, Rings, Modules}$ by: Auslander and Buchsbaum $\color{Green}{Background:}$ $\textbf{Proposition:}$ Let $M$ be an $R-$module. Supppose $f:M\to N$ is a ...
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Given $\varphi:A\to B$, how do I describe the following six maps in terms of elements?

$\color{Green}{Background:}$ In the context of the topic of exact sequences, I often see the following: suppose I have a homomorphic map between groups, rings, modules, etc or a linear transformations ...
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Meaning of: any exact sequence'can be recovered by "composing" the short exact sequences...

The following is taken from: $\textit{Abstract Algebra}$ by: P. A. Grillet $\color{Green}{Background:}$ $\textbf{Exercise:}$ Explain how any exact sequence $A\xrightarrow{\varphi}B\xrightarrow{\psi}C$...
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Commutative square inducing a linear transformation $r:\text{Coker }b\to \text{Coker }c.$

The following is based on an exercise from the book $linear algebra and geometry" by Leung. $\color{Green}{Background:}$ Given the linear transformation $c:X\to Y$ and the commutative square ...
Seth's user avatar
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Defining linear transformation: $r:\text{Coker }b \to \text{Coker }c.$ given a commutative sqaure diagram.

$\color{Green}{Background:}$ Given the linear transformation $c:X\to Y$ and the commutative square diagram below also with linear transformations $b,c,d.$ $$\begin{array}{ccccccccc} X' & \...
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Defining $\text{Ker }f\xrightarrow{j}A,\text{Im }f\xrightarrow{k} B,p:\text{Ker }c\to \text{Ker }f,q:\text{Im }c\to \text{Im }f,$ using elements?

$\color{Green}{Background:}$ Suppose I have a linear transformation or homomorphism map $f:A\to B,$ where $A,B$ can be groups, rings or modules. If I have the following two sequences of maps: $\text{...
Seth's user avatar
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How to define linear transformations: $\text{Ker }b \to \text{Ker } c,$ $\text{Im }b \to \text{Im } c,$ and $\text{Coker }b \to \text{Coker } c?$

The following is taken from linear algebra and geometry by Leung $\color{Green}{Background:}$ The $\textit{cokernel}$ of a linear transformation $c:X\to Y$ is defined as $$\text{Coker }c=Y/\text{Im }c$...
Seth's user avatar
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6 votes
2 answers
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Can two fields which are extensions of one another be non isomorphic

Let $\mathbb L$ and $\mathbb K$ be two fields. We say that $\mathbb L$ is an extension of $\mathbb K$ if there exists a ring homomorphism $\varphi$ from $\mathbb K$ to $\mathbb L$. In that case, $\...
Will's user avatar
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Minimally sufficient condition for composability of quiver morphisms?

Given three quivers $(X_0, X_1, \sigma, \tau)$, $(Y_0, Y_1, \phi, \psi)$, and $(Z_0, Z_1, \chi, \omega)$, and two morphisms $F := (F_0 : X_0 \to Y_0, F_1 : X_1 \to Y_1) : (X_0, X_1, \sigma, \tau) \to (...
Jos van Nieuwman's user avatar
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1 answer
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Terminology: (Co)equalizer of a family of morphisms?

Let $A,B\in \mathsf{C}$ be two objects in a category $\mathsf{C}$. Let $(f_i\colon A\rightarrow B)_{i\in I}$ be a family of parallel morphisms in $\mathsf{C}$. Consider the corresponding diagram in $\...
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What are colored morphisms/arrows intended to mean in these diagrams?

I've been reading more category theory as a prerequisite to understanding some more complicated theorems, and for the first time I'm running into arrows that are distinctly colored. Examples include ...
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Is Inverse of bijective morphism also morphism?

Let $V_1,V_2$ be affine varieties. Let $f$ be a morphism(https://en.wikipedia.org/wiki/Morphism_of_algebraic_varieties) between $V_1,V_2$. If $f$ is bijective, is inver of $f$ also morphism ? If not, ...
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Understanding rational maps from Shafarevich

Can someone please explain it to me what does the highlighted text mean? I am trying to learn Algebraic Geometry from Fulton and Shafarevich but I am stuck now. This says that to check whether an $m-$...
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