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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Trying to understand the arrows-only description of a natural transformation, but cannot form the composition $\tau(g) \circ \tau(f)$
It is an elementary result that a natural transformation $\tau \colon S \to T$ defines a function (also called $\tau$) such that for each composable pair $(g, f)$ of morphisms we have $T(g) \circ \...
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Examples of non-split idempotent morphisms (category theory)
Let $\mathcal{C}$ be a category. A morphism $f:x\to x$ is idempotent if $f^2=f$, and it is split idempotent if there exists $y$ and $x\stackrel{g}{\to} y\stackrel{h}{\to }x$ s.t. $h\circ g=f$ and $g\...
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Map induced between orthogonal Grassmanians by taking a quotient of vector spaces
Let $V$ be a complex vector space endowed with a nondegenerate symmetric bilinear form. For simplicity, let us just consider the case $V$ is of odd dimension. Let $U$ be an isotropic subspace of $V$ ...
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Morphisms between projective spaces $\mathbb P^n$ to $\mathbb P^m$.
I am looking for all morphisms $f:\mathbb P^n\to \mathbb P^m$ where $\mathbb P^r$ denotes the projective $r$-space over some algebraically closed field $k$. I think a morphism $f:\mathbb P^n\to \...
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Interesting examples of homotopic homeomorphisms
This question is motivated by the discussion in May's topology notes (p. 190), wherein it is postulated that two different isomorphisms $\mathbb{R}^{\infty} \oplus \mathbb{R}^{\infty} \to \mathbb{R}^{\...
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Question about components of natural transformations
I'm reading this paper which tries to explain categories from the ground up for use in the theory of anyons (my background is physics; I have no prior exposure to category theory). One aspect about ...
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An alternative way to prove that $\Bbb A^1$ is the normalization of $V(y^2-x^3-x^2)\subset\Bbb A^2$ [duplicate]
I had some doubts to resolving this exercise. Now thanks to the answers in Normalization of the variety $V(Y^2-X^3-X^2)\subset\Bbb A^2$ I can understand it really clear, so thank you.
My problem is ...
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Common notation for a unique morphism
Let $A,B$ be objects of a category. Assume that there is a unique morphism $A \to B$. Is there a notation for this morphism that has been used in at least one publication, or maybe is even common? Has ...
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How to show $\phi:\Bbb A^2\to\Bbb A^4$ by $(t,u)\mapsto(t-tu,u(u-1)^2,tu,u^2(u-1))$ is finite and birational on to its image, but not an isomorphism?
Let $\phi:\mathbb{A^2_\mathbb{C}}\to\mathbb{A^4_\mathbb{C}}$
$(t,u) \mapsto (t-tu,u(u-1)^2,tu,u^2(u-1))$.
I just found $\phi^{-1}$ given by $(x_1,x_2,x_3,x_4) \mapsto (x_1+x_3,\frac{x_4}{x_4-x_2})$ in ...
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Are "maps of sheaves" induced by morphisms of varieties?
A friend and I were trying to make sense of the definition of a morphism of (say affine or quasi-affine) varieties, as given by Hartshorne (p. 15-16):
If $X$ and $Y$ are two varieties, a morphism $\...
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$\mathbb{A}^1$-homotopy between embeddings of affine schemes
I know that $\mathbb{A}^1$-homotopy equivalence is not an equivalence relation on $Sch/k$ as the transitivity condition most often fails. I was wondering whether there existed a transitivity $\mathbb{...
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Basic question about 2-morphisms and homotopies
Say we are working in a category with an appropriate notion of homotopy between two morphisms given by 2-morphisms. When we say a diagram like the following
"commutes up to homotopy", what ...
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There is a natural map $\text{colim}_{\beta \in \alpha}\text{Mor}(A,B_{\beta})\rightarrow \text{Mor}(A,\text{colim}_{\beta \in \alpha}B_{\beta}) $
In the very beginning of 19 chapter in stack project I found this
Let $\{B_{\beta}\}_{\beta \in \alpha}$ be an inductive system (which means the index category is a filtering.) of objects in category ...
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Reversible k-morphism?
I'm reading this nLab page right now, and I'm running into an issue trying to understand $(n, r)$ categories. nLab defines them as such:
An (n, r) category is a higher category such that, essentially:...
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Local structure at ramification points of a morphism of smooth curves
Let $f:C \longrightarrow D$ be a morphism of smooth curves, simply branched and of degree $k$. We know that if $f$ is étale at $q \in C$ and $p=f(q) \in D$, then for $t$ a regular parameter in $\...
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If $\alpha$ is a root of $x^3-3x+1$ then $f: \mathbb{Q}(\alpha)\to \mathbb{Q}(\alpha)$ with $f(1)=1$ and $f(\alpha)=\alpha^2-2$ is an automorphism
If $\alpha$ is a root of $x^3-3x+1$ then $f: \mathbb{Q}(\alpha)\to \mathbb{Q}(\alpha)$ such that $f(1)=1$ and $f(\alpha)=\alpha^2-2$ is a automorphism.
Assuming that is a homomorphism, then is fair ...
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how $\mathrm{Gal}(\overline{K}/K)$ acts on automorphisms of a $\overline{K}$-scheme
Let $K$ be a number field and $\overline{K}$ its algebraic closure. Let $(X,\mathcal{O}_{X},s)$ be a scheme over $\overline{K}$ where $s\colon X\to\mathrm{Spec}\overline{K}$ is the structural morphism....
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Underlying Foundations of Category Morphism Definition
Recently I'm studying category theory in the beginning level. This mathematical concept is really fantastic. However, the definition of morphism composition seems a little confusing:
$$\circ: \mathrm{...
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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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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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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 ...
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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 $\ker{p}$ = $\ker{q}$) $\...
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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 $k-$...
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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 ...
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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 ...
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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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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 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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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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