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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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.
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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
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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?$
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$\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
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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.
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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
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$\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?
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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
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$\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.
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Given the linear transformation $c:X\to Y$ and the commutative square ...
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Defining linear transformation: $r:\text{Coker }b \to \text{Coker }c.$ given a commutative sqaure diagram.
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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?
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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{...
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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
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The $\textit{cokernel}$ of a linear transformation $c:X\to Y$ is defined as
$$\text{Coker }c=Y/\text{Im }c$...
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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, $\...
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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 (...
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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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Monomorphism under a functor
Let $\mathcal{C}$ be a category. Let $X,Y$ be object in $\mathcal{C}$ and $f \in Hom_{\mathcal{C}}(X,Y)$. I want to know the condition on $F$ for which $F(f)$ is a monomorphism.
It's clear that if $F$ ...
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Objects of categories with no morphisms from them other than endomorphisms
Let $\mathcal{C}$ be a category. My question is the following. Is there a name for an object $\zeta \in \mathrm{ob}(\mathcal{C})$ such that, for every object $X \neq \zeta$, there exists no morphism $...
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Show every hyperelliptic curve admits a morphism of degree 2
I am given an alternative definition for a hyperelliptic curve as follows. A hyperelliptic curve is a smooth projective curve $C$ over an algebraically closed field $\bar{F}$ with a divisor $D\in \...
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$X\times Y$ and the product topology [duplicate]
I'm studing Andreas Gathmann's notes on algebraic geometry (pdf here: https://agag-gathmann.math.rptu.de/de/alggeom.php). In chapter 4 (about Morphisms) he was using the universal property of products ...
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Smooth morphisms of schemes
I am trying to understand smooth morphisms of schemes as in https://stacks.math.columbia.edu/tag/01V4
All the definition seems to only take in considerations the source of the morphism, as the ...
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A monoid $M$ is $\omega$-presentable in the category $M$-$\mathbf{Set}$
I feel that this is true but I'm unable to prove it formally: a monoid $M$ is $\omega$-presentable in the category $M$-$\mathbf{Set}$. This is the category of $(X,\rho)$ where $\rho:M\times X\to X$ ...
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Representing object for a functor mapping a category to composable morphisms
If I have a functor $D_n : \mathsf{Cat} \to \mathsf{Set}$ that maps a category into the set of all $n$-tuples of composable morphisms,
$D_n(C) = A_1 \to A_2 \to A_3 \to \dots \to A_n$, what would its ...
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Problems with Vignéras' proof of Skolem-Noether th. in "Arithmétiques des Algèbres de Quaternions"
I have some problems in understanding some details in the proof of Skolem-Noether Theorem from Vignéras' book. The statement is
"Let L, L' be two commutative $k$-algebras inside a quaternion ...
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Is this the correct formulation for the universal property and pushout diagram for cokernel?
The following question is taken from $\textit{Handbook of Mathematics}$ by Thierry Vialar,page 608
I consulted the Vialar's handbook and hoping to see what the dual formulation of cokernel in terms of ...
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Some questions about the functor of points perspective
If I were to define schemes as functor of points from rings to sets, what would be the morphisms? My guess is that it is not the set of all natural transformations between the functors, but some ...
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When is a quotient $\mathbb Z$-algebra a ring quotient
Suppose that I have two rings with identity, $S,R \in $ Ring, and a surjective $\mathbb Z$-algebra homomorphism $f: S \to R$ (hence $R$ is a quotient $\mathbb Z$-algebra of $S$).
Is it true that $R$ ...
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Question about the meaning of $0\rightarrow B$ used in category theory and elsewhere in mathematics
The following question is taken from Arrows, Structures and Functors: The Categorical Imperative by Arbib and Manes.
Definition: Let $\textbf{K}$ be a category with zero object $0$. Then the kernel $...
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A morphism of bialgebras between two Hopf algebras is necessarily a morphism of Hopf Algebras.
Here is the question I am trying to solve:
Use the previous exercise to show that a morphism of bialgebras between two Hopf algebras is necessarily a morphism of Hopf algebras.
Here is the previous ...
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Showing that a morphism is an equivalence relation
Let $\mathscr{C}$ be a category. $X,Y \in ob(\mathscr{C})$, $X$ and $Y$ are said to be equivalent if there is an equivalence $f:X \rightarrow Y$. Show this is an equivalence relation.
For the ...
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Associativity as a strong axiom (morphisms)
I am a physics student who is currently reading Szekeres's A Course in Modern Mathematical Physics. In the first chapter, he makes the tiniest bit of contact with category theory (presumably just to ...
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Questions about the specifics in the definitions for cone, and limits in Category theory
The following are taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes
$\quad$$\textbf{Definition 1}$ A $\textbf{directed graph}$ is an arbitrary class ...
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When is a pair of morphisms is a pullback of some pair of morphisms?
Given $p$ and $q$ morphisms on a category (preferably arbitrary, but it could be interesting to know on some other class of categories too), are there any useful necessary and/or sufficient conditions ...
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If for monos $u\leq v$ and $v\leq u$ then their domains are isomorphic
I'm unable to prove that if a mono $u:B\to A$ is less then mono $v:C\to A$ by $f:B\to C$ with $v\circ f=u$ and also
$v\leq u$ by $u\circ g=v$ then $B\cong C$ by using some morphisms as above, but I ...
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Is every bijective morphism an isomorphism in a small category?
In any category, an isomorphism is an arrow $f \colon X \to Y$ together with an inverse arrow $f^{-1} \colon Y \to X$ such that $f \circ f^{-1} = 1_{Y}$ and $f^{-1} \circ f = 1_{X}$.
Now, clearly in ...
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Every continuous map induces a morphism of groups
This statement was recalled by my instructor as general fact in my class of Topology but it doesn't looks trivial to me at all.
Statement: Every continuous map $f : X \to Y$ induces a morphism of ...
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The morphism induced in the derived category and in the homotopy category
During a lecture my professor made an observation, she said that it can happen the following fact but she never explained how/why: let $\mathcal{A}$ a category; there are examples of morphisms $f\...
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help with Kernel and image of an endomorphism
I am given the kernel and image of an endomorphism in $\Bbb R^3 $, which are:
$\ker f =\langle(1,1,0)\rangle$
$\text{Im}\, f = \langle(0, 1, -1), (2, 1, 2)\rangle$
and I have to find the matrix ...
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Does a would-be isomorphism between a known and suspected category object guarantee the latter object to be in the category?
In this question from a few day ago, it became evident that, given a known vector space $(V,+,\times)$, a possible vector space $(W,\oplus,\otimes)$, and a transformation $T:V\rightarrow W$ that is is ...
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diagonal morphism preserves properties that are satisfied by isomorphisms under base-change
Let $\mathcal{P}$ be a property of morphisms of schemes that is fulfilled by isomorphisms.
Now assume we have a property $\mathcal{P}$ that is invariant under base-change, that means if $f:X \to Y$ ...
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Show that $\mathbb{R}^n$ is a direct sum of lines and planes
Let $A \in M_n(\mathbb{R})$ be a matrix which is diagonalizable in
$M_n(\mathbb{C})$. We note $u_{A, \mathbb{R}}: \mathbb{R}^n
\rightarrow \mathbb{R}^n$ the endomorphism defined by the left
...
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Non existence of homogeneous polynomials defining a morphism between projective varieties
Given the quadric $Q=V(XT-YZ)\subset\mathbb{P}^3$ and the lines $L_{X,Y}=V(X,Y)\subset Q$ and $L_{Z,T}=V(Z,T)\subset Q$, we have the morphism $\Phi: Q\rightarrow\mathbb{P}^{1}$ given by:
$\Phi(X,Y,Z,T)...
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A Monomorphism in a category is an epimorphism in opposite category.
I am studying Emily Reihl's Category theory in context:
Let $f: x \rightarrow y$ be a monomorphism in a category. Then, for any parallel morphisms $h,k : w \rightrightarrows x, fh=fk \implies h=k $.
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A morphism in $Hom (a,a)$ which is not the identity morphism
So, when defining a category, one is careful enough to define the identity of a object $a\in\text{Ob}(\mathcal C) $ as a particular element of the hom-class, $\text{id}_{a}\in\text{Hom}(a,a)$. That ...
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