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.
0
votes
1answer
34 views
${(F^*)}^{-1}(m)$ 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 ...
1
vote
1answer
18 views
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 ...
1
vote
1answer
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 ...
1
vote
2answers
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 ...
0
votes
1answer
83 views
$\lambda$-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 ...
0
votes
0answers
43 views
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:
$$
\text{Hom}_{C^{\wedge}}(h_C(X), A) \to \text{Hom}_{\text{...
0
votes
1answer
31 views
Given a $G$-Set isomorphism from $G/H$ to $G/S$, can I make a morphism from $G/H$ to $G/K$ if $S<K$?
Let $G$ be a group and let $H,K$ be subgroups. I have an $S<K$ such that $S=g^{-1}Hg$. I'm asked to show that there is a morphism from $G/H$ to $G/K$.
I have a theorem that says that since $H$ and $...
1
vote
1answer
49 views
Coproduct of Group Homomorphisms
I've been trying to work through the following problem:
Let $H$, $G$, and $G'$ be groups, and let $f:H\to G$ and $g:H\to G'$ be two homomorphisms. Define the notion of coproduct of these two ...
0
votes
0answers
37 views
Bijective comorphism of a polynomial map
Let $f:G\rightarrow H$ be a polynomial map, between algebraic groups $G,H$ (ie defined by polynomial equations), such that the comorpism $f^*:K(H)\rightarrow K(G),\ g\mapsto g\circ f$ is bijective.
...
3
votes
1answer
65 views
Every monic is a kernel
This is part of Weibel's Exercise 1.2.2, where I have to show that in the category R-Mod, every monic is a kernel.
A monic morphism is defined to be a map $i \colon A \to B$ such that if $g \colon A ...
1
vote
2answers
63 views
Kernel of a surjective module morphism $f : R^2 \rightarrow R$ is a free submodule
I have to prove this :
Let $R$ be a commutative ring and : $f : R^2 \rightarrow R$ a morphism of
$R$-module. I want to show : $\ker f$ is a free module.
My first try is to say : $R^2/\ker(f) \...
2
votes
0answers
137 views
Understanding De/Suspension $\Sigma^{-1}(\Sigma{X})\neq X$
The question is about understanding suspension and desuspension, see also a previous question.
Question: How do we define desuspension exactly? (Please see the comments below, people complain about ...
1
vote
1answer
51 views
Understand the suspension (topology) and some Lie group examples
Let $\Sigma$ denotes a suspension
$$\Sigma X =S^1 \wedge X \equiv (S^1\times X)/(X\vee S^1)$$
where $\wedge$ and $\vee$ are the smash product and the wedge sum (one point union) of pointed ...
2
votes
0answers
42 views
Lie group homomorphism to $S^1$
I wish to know the general strategy to deal with a compact Lie group $G$ and its Lie group homomorphism $G$ to $S^1$ as:
$$
\text{Hom}(G, S^1).
$$
Here $S^1=U(1)$.
For example, how could we determine:...
-1
votes
1answer
33 views
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 ...
1
vote
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 ...
3
votes
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 ...
2
votes
0answers
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. ...
3
votes
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 ...
1
vote
0answers
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$ ...
1
vote
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 ...
2
votes
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
...
0
votes
0answers
35 views
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 ...
2
votes
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 ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
1
vote
0answers
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 $\...
0
votes
0answers
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 ...
0
votes
0answers
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$ ...
0
votes
0answers
15 views
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 ...
11
votes
2answers
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 ...
0
votes
1answer
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)...
0
votes
1answer
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 ...
1
vote
1answer
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. [...
0
votes
1answer
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 ...
0
votes
1answer
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 $...
1
vote
1answer
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 ...
7
votes
1answer
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 ...
1
vote
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 ...
1
vote
0answers
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_{\...
1
vote
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. ...
1
vote
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?
0
votes
1answer
50 views
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 ...
0
votes
0answers
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}$. ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
2
votes
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 ...
1
vote
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 ...
2
votes
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, ...
9
votes
1answer
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\...
