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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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${(F^*)}^{-1}(m)$ is maximal ideal where $m$ is maximal

Our main objective is to interpret $F:V \to W$ a morphism as a map $F:maxSpec \Bbb C[V] \to maxSpec \Bbb C[W]$, $V,W$ are algebraic varieties. Now from $F:V \to W$ using Hilbert Nullstelensatz we ...
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image of the neutral in group homomorphism

It it always true that the image of the neutral element in a group homomorphism $f$ is the neutral element of the codomain group regardless whether $f$ is injective/surjective? The answer is most ...
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33 views

Is there a general formalized method to define the homomorphism (structure preserving map) of a structure?

This is a follow-up to this question. There I asked for an intuition of what "structure preserving" means. My question here is, is there a universally applicable method (given the objects that define ...
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102 views

What is the colon ':' called in category theory morphism definitions?

Category theoretic morphism definition syntax, e.g.: $\varphi : G \to H$ uses the colon character rather than, for instance, the member-of sign e.g.: $\varphi ∈ G \to H$ What is the descriptive ...
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$\lambda$-pure morphisms in $\lambda$-accessible categories are monos, unclear proof

This is Proposition 2.29 from the book Locally Presentable and Accessible Categories by Jiří Adámek and Jiří Rosický. Above is a proof that $\lambda$-pure morphisms in $\lambda$-accessible categories ...
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In the proof of the Yoneda Lemma in “Categories & Sheaves” by Kashiwara & Schapira.

Our goal is to show that $\text{Hom}_{C^{\wedge}}(h_C(X), A) \simeq A(X)$. We first want to show a map from left to right. The book says: $$ \text{Hom}_{C^{\wedge}}(h_C(X), A) \to \text{Hom}_{\text{...
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Given a $G$-Set isomorphism from $G/H$ to $G/S$, can I make a morphism from $G/H$ to $G/K$ if $S<K$?

Let $G$ be a group and let $H,K$ be subgroups. I have an $S<K$ such that $S=g^{-1}Hg$. I'm asked to show that there is a morphism from $G/H$ to $G/K$. I have a theorem that says that since $H$ and $...
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49 views

Coproduct of Group Homomorphisms

I've been trying to work through the following problem: Let $H$, $G$, and $G'$ be groups, and let $f:H\to G$ and $g:H\to G'$ be two homomorphisms. Define the notion of coproduct of these two ...
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37 views

Bijective comorphism of a polynomial map

Let $f:G\rightarrow H$ be a polynomial map, between algebraic groups $G,H$ (ie defined by polynomial equations), such that the comorpism $f^*:K(H)\rightarrow K(G),\ g\mapsto g\circ f$ is bijective. ...
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65 views

Every monic is a kernel

This is part of Weibel's Exercise 1.2.2, where I have to show that in the category R-Mod, every monic is a kernel. A monic morphism is defined to be a map $i \colon A \to B$ such that if $g \colon A ...
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63 views

Kernel of a surjective module morphism $f : R^2 \rightarrow R$ is a free submodule

I have to prove this : Let $R$ be a commutative ring and : $f : R^2 \rightarrow R$ a morphism of $R$-module. I want to show : $\ker f$ is a free module. My first try is to say : $R^2/\ker(f) \...
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Understanding De/Suspension $\Sigma^{-1}(\Sigma{X})\neq X$

The question is about understanding suspension and desuspension, see also a previous question. Question: How do we define desuspension exactly? (Please see the comments below, people complain about ...
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1answer
51 views

Understand the suspension (topology) and some Lie group examples

Let $\Sigma$ denotes a suspension $$\Sigma X =S^1 \wedge X \equiv (S^1\times X)/(X\vee S^1)$$ where $\wedge$ and $\vee$ are the smash product and the wedge sum (one point union) of pointed ...
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Lie group homomorphism to $S^1$

I wish to know the general strategy to deal with a compact Lie group $G$ and its Lie group homomorphism $G$ to $S^1$ as: $$ \text{Hom}(G, S^1). $$ Here $S^1=U(1)$. For example, how could we determine:...
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Vertical and top arrows??

Here in her short note in the middle of the page 1, Emily Riehl says ...with the top two vertical arrows ... I just wonder how vertical arrows can be top or bottom. I would say that vertical arrows ...
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1answer
88 views

Proof of Theorem of Dimension of Fibres

I am following a lecture series on YouTube, and in the series the lecturer skipped the proof of the Theorem on the Dimension of Fibres. I tried to follow the style of the professor's proof but I ...
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1answer
70 views

Is there such thing as a morphism of natural maps and what does it look like?

I'm writing software to be general, so right now I'm writing a NaturalMap class which will be a graphical arrow that goes between any two ...
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46 views

Strategy to prove the isogeny with arbitrary kernel is unique

I encountered a theorem saying that, Given an elliptic curve $E_1$ and arbitrary subgroup $H$, there only exist an isogeny $\phi:E_1\rightarrow E_2$ with $\ker\phi=H$ up to isomorphism of its image. ...
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2answers
118 views

Why are 'finite morphisms' important in algebraic geometry? And what does a module of finite type mean?

Linear transformations, Group, ring, $k$-algebra morphisms and many other types of morphisms that appear throughout mathematics are more or less obvious in the sense that we can clearly see why they ...
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64 views

Injective and dominant morphism bewtween quasi-projective varieties implies same dimension?

Is the following assertion true? "Let $X$ and $Y$ be two quasi-projective varieties and $f:X \longrightarrow Y$ a morphism which is injective and dominant. Then, the dimensions of $X$ and $Y$ ...
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2answers
71 views

Definition morphism of Hopf algebra

this is my first post, so let me know if you need some more information. I am currently studying Hopf Algebras and and exercise I have tells me to show that $f:H \rightarrow H'$ is a morphism of ...
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1answer
59 views

Isomorphism between two exact sequences

I am trying to understand a proof and came across the following diagram (for any abelian category): $$$$ $\hskip2in$ The proof now says: Since the rows of the diagram are exact, there are unique ...
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Is the biunivocal transformation or bijective function an $automorphism$ if morphism that I choose is a homeomorphism?

Question seems stupid but I see this is related with Continuous Transformation of Continuous Space read this pdf ((a topological examples are homeomorphisms) and about Orientation (vector space) A ...
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1answer
74 views

Functors between morphism categories

Let ${\cal{C}}$ and ${\cal{D}}$ be categories, and let $\textsf{Mor}({\cal{C}})$, $\textsf{Mor}({\cal{D}})$ be their morphism categories (objects are morphisms, morphisms are pairs of morphisms making ...
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62 views

Existence of morphisms in a free completion under directed colimits,$\lambda$-accessible category

Let $\cal K$ be a $\lambda$-accessible category with directed colimits and $\cal C$ be its representative full subcategory consiting of $\lambda$-presentable objects. Let $\cal L$ be free completion ...
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32 views

Monomorphisms, unclear basic property, Functor

Suppose that for morphisms in a category it holds that $f\circ u=v\circ f'$ and $g\circ u=v\circ g'$ and that for a functor $F$, $Fv$ is a monomorphism. Suppose that $Ff'$ and $Fg'$ are distinct. WHY ...
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28 views

Morphism, functional equations and bijection

I've got a question about something. We consider $\mathbb{T}=\{ z \in \mathbb{C} \; | \; |z|=1 \}$, and : $car(\mathbb{R}) = \{ f : \mathbb{R} \rightarrow \mathbb{T} \; | \; f$ continuous and $\...
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22 views

Formula of morphism $\pi $ to general element $K[x]/(m)$

I need to write formula, how morphism $\pi $ is influencing general element of $K[x]/(m)$. We have field K and polynomial $m \in K[x]$ which is irreducible over K. F is field, which is isomorphic ...
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19 views

homomorphism realised by an element of a ring

I have an endomorphism $\phi:A\rightarrow A$ of a left $R$-module $A$. What does it mean that $\phi$ can be realised by en element of $R$? I suppose it means that $\phi(a) = ra$ for some $r\in R$ ...
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Does preservation of addition and multiplication between star continuous Kleene algebras imply preservation of star?

Suppose I had a mapping $h : K_1 \rightarrow K_2$ for star continuous Kleene algebras $K_1,K_2$, and I was able to show that $h$ preserves addition and multiplication. Does it then automatically ...
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281 views

Are continuous maps “weaker” than other morphisms?

The property of continuity (and hence smoothness) seems weaker than the properties of other morphisms, in the sense that a homeomorphism is a "continuous bijection whose inverse is continuous". In ...
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167 views

Is a bijective morphism between affine varieties an isomorphism?

I'm trying to prove the following: Let $X,Y$ affine varieties (both irreducible) and a morphism $f:X\to Y$. The pullback $f^*:A(Y)\to A(X)$ is surjective $\Leftrightarrow$ $f$ is injective and $f(X)...
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153 views

Restriction of a morphism is a morphism

Let $X,Y$ be quasi-projective varieties and $f:X\to Y$ a morphism. If $X',Y'$ are closed subvarieties of $X,Y$ respectively such that $f(X')\subset Y'$, prove $f|_{X'}:X'\to Y'$ is a morphism. I ...
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56 views

Describe the corresponding $k$-algebra homomorphism $\tilde{\varphi}:k[V]\to k[\mathbb{A}^1]$.

Let $V=\mathcal{Z}(xz-y^2,yz-x^3,z^2-x^2y)\subseteq\mathbb{A}^3$. First, I am asked to prove that the map $\varphi:\mathbb{A}^1\to V$ defined by $\varphi(t)=(t^3,t^4,t^5)$ is a surjective morphism. [...
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28 views

Finitely Generated Free Group to Finitely Generated Free Monoid

Let $F_n$ be the free group on $n$ generators $u_1,...,u_n$ and $M_n$ the free monoid on $n$ generators $v_1,...,v_n$. Would $u_i \to v_i$ and $u_i^{-1} \mapsto v_i$ extend to a well-defined map that ...
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75 views

Functor preserves restriction and corestriction

I am working with modules, but I guess this question is valid with any abelian category. Let $R$ be a ring, and $F$ a functor. Let $A, M$ be $R$-modules and $f : A \longrightarrow M$ a morphism. Let $...
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38 views

What do you call the relationship between two functions that perform essentially the same operation over isomorphic objects?

This question applies to any category, but I'm going to use vector spaces as an easy example. Suppose we let $V$ be the space of column vectors in $\mathbb R^3$, and consider the dual space $V^*$ as ...
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88 views

“Perfecting” an endomorphism in a category

Let $\mathcal{C}$ be a complete category and suppose $f: X \to X$ is an endomorphism in $\mathcal{C}$. Associated to $f$ is an inverse system, $$X_\bullet: \dots \to X \to X \to X \to X,$$ where every ...
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1answer
183 views

Monomorphism, Split Monomorphism and “Injective”

I know it's bad habit studying wikipedia as a proper source. But I am particularly troubled by the statement that: "In concrete categories, a function that has a left inverse is injective. Thus in ...
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55 views

Locally finite morphism implies globaly finite

I'm trying to understand the following theorem from Shafarevich's Basic Algebraic Geometry I: I understand why we can assume the system $D(g_{\alpha})$ to be finite, so that $(g_{\alpha_1},...,g_{\...
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2answers
221 views

Regular functions on a variety

In Hartshorne's Algebraic Geometry, remark 3.1.1., it is said that: if $f,g$ are regular functions on a variety $X$, and if $f=g$ on some nonempty open subset $U\subset X$, then $f=g$ everywhere. ...
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1answer
51 views

Are set-theoretically defined functions and category-theoretic morphisms equivalent notions?

What is the relationship, if any, between set-theoretically defined functions and morphisms (or the arrow thingies) from category theory?
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Help with avoiding the usage of syntactic abbreviations to sidestep the possibility of paradoxes.

On occasion I've informally written statements like: "For any set $X$ we let $\mathcal{L}(X)=X\times X\cup \{X,(\emptyset,X)\}$" Now this has never really been a problem because most of the time, I ...
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58 views

Usage of the Frobenius endomorphism

I just read something [1] about the usage of the Frobenius endomorphism but I can't get it clearly in usage. Say we are operating on a field with prime characteristics $p$, say $G_T=\mathbb F_{p^k}$. ...
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73 views

Differences between linear functions, multi-linear functions and morphisms.

I'm trying to understand some fundamental concepts of abstract algebra and, since i'm not a mathematician but a math enthusiast, i'm having some difficulties establishing connections. A morphism ...
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31 views

proposition im(f) f span E in category theory

I'm not sure if the title of the question is right, so there it goes an explanation. Some algberaic structures such as tensor products or Clifford algebras for example, can be defined via a universal ...
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3answers
874 views

What is the difference between the identity functor and the identity morphism?

The only difference that I can think of is that - because the identity functor is also an endofunctor - its only extra ability is being able to map morphisms to themselves and not just category ...
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1answer
24 views

What's the term for my function's aspect of equivalence structure preservation?

$A, B$ are two sets, $\equiv$ is an binary equivalence relation on $A$, and $f : A \rightarrow B$ is a function. If $x \not \equiv y \Rightarrow f(x) \neq f(y)$, what does that make $f$? This answer ...
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1answer
584 views

Definition of Epimorphism in Category Theory

Given a morphism $f : A \to B$, $f$ is en epimorphism iff $\forall g_1, g_2 : B \to C. f \circ g_1 = f \circ g_2 \implies g_1 = g_2$ In words, a morphism $f : A \to B$ is epic if it has the property, ...
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615 views

Does this theorem/collection of theorems have a name? Or is it just seen as obvious?

After some thought I realized that given any sets $X$ and $Y$ with equal cardinality and an arbitrary bijection $f:X\to Y$ we get that the set of all bijections from $X$ to $Y$ is given by: $$\{f\...