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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If $\mathcal{C}$ has pullbacks & coequalisers, and the pullback of a regular epi is epic, morphisms factorise as regular epic & monic

I'm trying to show that if $\mathcal{C}$ has pullbacks & coequalisers, and the pullback of a regular epimorphism is epic, morphisms factorise as a regular epic followed by a monic. My approach so ...
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Is there example of equalizer that is not injective?

we know that in general case every arrow is not function also in the $Div \mathbb{A}b$ there is example of monic that is not injective. Is there example of equalizer that is not injective ?
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Number of ring endomorphisms of $\mathbb{Z}[\zeta_n]$

Let $n \in \mathbb{N}$ and $\zeta_n = e^{2i\pi/n} $ we define the following subring of $\mathbb{C}$ by, $$\mathbb{Z}[\zeta_n] = \{ P(\zeta_n) : P \in \mathbb{Z}[x]\}$$ One can easily show that, $$\...
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In category theory, should morphisms apply to specific objects or to any objects in the category?

I am not sure to correctly understand the notion of morphism in category theory. To try to better understand, let's take a very simple example. Let's say that we have a category $\mathcal{C}$: whose ...
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Is shape morphism invariant to the cardinality of the set of components/quasicomponents?

Let $X$ and $Y$ be two topological spaces. We denote with $CX$ and $CY$ the sets of connected components and $QX$ and $QY$ the sets of quasicomponents for the corresponding spaces. It is well known ...
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Weighted limits

I have a very trivial question about the page 80 here: how this shape of $W$ $$W:2\to \mathbf{Set}$$ with $$\ast\sqcup\ast\to \ast$$ implies that the components of $$W\Rightarrow\cal{M}(m,f)$$ are ...
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Preimage of a proper morphism

We know that proper  morphisms of  varieties take  closed subsets to closed subsets. What can I say of the preimage  of a proper morphism? For example, if I know  that $f(g(x))$  is an uncountable ...
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What does it mean that "All diagrams commute in a posetal category"?

I am quite familiar with posetal categories, however, I just randomly came accross the claim that "all diagrams commute in a posetal category" on Wikipedia. I am confused, what does it even ...
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Let $G_1$ and $G_2$ be groups and $\pi_1:G_1\times G_2\rightarrow G_1$

Let $G_1$ and $G_2$ be groups and $\pi_1:G_1\times G_2\rightarrow G_1$ be the function defined by $\pi_1(a,b)=a$. Prove that $\pi_1$ is a homomorphism, find $\ker(\pi_1)$, and prove $(G_1\times G_2)/\...
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Constructing $k$-morphisms between algebraic curves

Let $k$ be a finite field containing a sixth root of unity $\xi_6$. Let $C: t^2 = s^6 - 1$ be an hyperelliptic curve over $k$. Let $E': \eta^2 = \xi^3 + 1$ be an elliptic curve over $k$. I am trying ...
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Prove the sheaf isomorphism is local

EDIT: It turns out I have read the statement wrongly. The statement in Wikipedia actually requires $\psi_{U_i}$ are isomorphisms of sheaves. With this, it's then clear how the result follows with my ...
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When does a given type of 'mathematical structure' (groups, metric spaces...) have a 'natural' choice of morphism to turn it into a category?

Consider a class of sets sharing some structure and properties e.g. groups, vector spaces, metric spaces, topological spaces, rings, fields. I am curious about under what conditions there may be a '...
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Graph with no nontrivial endomorphisms?

For context, I'm trying to determine whether there exists a full and faithful functor $F:\mathsf{Dgr}\to\mathsf{SimpGph}$ that "encodes" directed graphs as simple graphs. Right now I believe ...
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the direction of a point of a topos

I would like to understand what is the direction of a point of a topos. A point of a topos $\cal T$ in one source (2nd snippet below) is a functor preserving finite limits and all colimits $$p^*:{\cal ...
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Morphism between algebraic (smooth) curves of degree 1 is an isomorphism. Does the converse holds?

Morphism between algebraic (smooth) curves of degree 1 is an isomorphism. Does the converse holds ? In other words, isomorphism between algebraic smooth curves has always degree 1 ? Degree of morphism ...
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Initial objects in the category of bilinear maps $M \times N \to L$. where $M,N$ are fixed $R$-modules, and $L$ arbitrary $R$-module.

In our class we are looking at the definition of tensor product, and here’s a (paraphased) remark that I don’t get. Let $M,N$ be fixed $R$-modules, and $L$ an arbitrary $R$-module. At this point we ...
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What is an isomorphism between $L:aX+bY=Z$ and $\Bbb P^1_K$?

It is known that genus $0$ smooth curve over field $K$ with base point is isomorphic to $\Bbb P^1_K$ over $K$. I want to understand this with a lot of examples. For example, let $a,b\in K^\times$ and ...
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A more succinct group object diagram (all axioms in one connected diagram), questions about its properties...

Here is the definition of group object from nLab. They give 3 associated maps $* \xrightarrow{1} G$, $m: G^2 \to G$, and $-^{-1}: G \to G$ and require 3 commutative diagrams to complete the axioms ...
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Graded algebras and symbols

Take $A$ a commutative unitary $\mathbb{C}$-algebra and take $A_0\subset A_1\subset...$ a filtration on $A$. If $$GrA=\bigoplus \frac{A_n}{A_{n-1}}$$with $A_{-1}=0$, is the associated graded algebra. ...
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Why there is no notion ´bijective´ regarding morphism of schemes?

Morphism of schemes is defined as morphism between ringed spaces, and the morphism is not a map (pair of maps), so we cannnot define notion of bijectivity of morphism in the category of schemes, is my ...
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Let $F:\mathcal{A}\to\mathcal{B}$ be a fully faithful covariant functor. Then, if $F(f):F(A_1)\to F(A_2)$ is an isomorphism, so is $f:A_1\to A_2$.

I’d like to ask for checking of my attempt below. We want to find $g: A_2 \to A_1$ such that $f \circ g = 1_{A_2}$ and $g \circ f = 1_{A_1}$. So define $g: A_2 \to A_1$ to be a morphism such that $F(...
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How do I prove the following statement about the kernel?

I have the following problem: Given a finite group $G$ and $p$ the smallest prime dividing $card(G)$. Let $H$ be a subgroup s.t. $card(G\setminus H)=p$ Let $X=G\setminus H$ and consider the action $$...
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Question about morphism of algebras

I wanna show that $(K^X)^G \simeq K^Y$. Well, I will explain what is $X,Y$ and $G$ in this context. Let $X$ be a nonempty set and $G$ a group, with an action over $X$. Let $K^X$ be the algebra of ...
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What does "up to" mean?

There are several prior questions of the form "What does 'up to X' mean?" The answers generally focus on "X", which has led some commentators to ask "What does 'up to' mean?&...
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Find a finite morphism $F:X\to\mathbb{P}^k$.

I am working in the following problem from my algebraic geometry course: Let $X$ be a projective set of $\mathbb{P}^n$. Prove that there exists a finite regular morphism $F:X\to\mathbb{P}^k$, where $k=...
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$M \hookrightarrow M[t^{-1}]$

Take $M$ an $X=\mathbb{C}[[t]]$-module. first question I've to prove that: If $M$ is flat then $M$ is a submodule of $M[t^{-1}]$. Here my attempt. We have: $$M\cong M\otimes_X\mathbb{C}[[t]] \...
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Is a morphism a special case of a function?

https://en.wikipedia.org/wiki/Morphism states that In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same ...
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"prime" or "indecomposable" morphisms in small category [duplicate]

Let me start by saying I have had to teach myself all the category theory I know, if I can even call it knowing category theory haha. I am now in a class using Emily Riehl's Category Theory in Context....
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On the existence of zeroes for non-monomorphic linear operators.

In Linear Algebra by Georgi E. Shilov the author gives the following theorem: The operator $A:X\to Y$ has a left inverse$\iff$$A$ is a monomorphism. The operator $B:Y\to X$ has a right inverse$\iff$$B$...
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How to describe all homeomorphisms on $l^2$

I know homeomorphisms are something like transformations of a space. But I am struggling to make some "system" of all homeomorphisms of given space (say $l^2$) and whether we can describe ...
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4 votes
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Description Morphisms from Category theory in the language of set theory

I have a good understanding of set theory and axiomatic set theory and I am trying to now understand Category theory. I usually understand most of mathematics through set theory, like I understand ...
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Milne, Elliptic Curves, Theorem 2.1 b), Correctly Stated? - Verifying Something is an Isomorphism of Curves.

I am quite confused about an elementary verification of something that's part of the Theorem in the title. $E(a,b)$ is an elliptic curve defined by the equation $Y^2Z = X^3+aXZ^2+bZ^3$. The statement ...
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4 votes
3 answers
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Attempt to define the notion of subobjects

I attempted to generalize the notion of subsets for an arbitrary category. Given an object $X$, an object $Y$ shall be called $X$'s subobject iff there exists a monomorphism $f : Y → X$. Dually, $Y$ ...
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Is a functor that is injective on both objects and morphisms at the same time an embedding functor? [closed]

In category theory: Assuming there is a category with n objects and j morphisms between them, is it possible that a functor simply adds m new objects and k new morphisms between these, while leaving ...
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isomorphisms are exactly the morphisms that take colimit cocones to colimit cocones

I would like to understand why isomorphisms are exactly the morphisms that take colimit cocones to colimit cocones. I believe that both directions are easy but they are not immediate to me. Which ...
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Proof of Theorem 6.1(b) in Silverman's AEC

I'm learning about the construction of the dual isogeny in Silverman's Arithmetic of Elliptic Curves. In particular, I'm reading the proof for Theorem 6.1 from Chapter III. There is a (probably easy) ...
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If $f: A \to B$ is a morphism with $f(I) \subset J \trianglelefteq B$, then there is $\tilde{f}: I/I^2 \to J/J^2$.

Let $A, B$ be commutative rings, $I\trianglelefteq A$, $J\trianglelefteq B$ and a ring morphism $f: A \to B$ such that $f(I) \subset J$. I was wondering if this map $f$ induces a map $$\tilde{f}:I/I^2 ...
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Arrow category $\cal K^\to$ [closed]

I would like to understand here on the page $6$ in the definition $3.3$ how works the functor $F:\cal K^\to \to K$. They say ...
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Sketches, the realized sketch and a model

I have a problem here with this statement in Chapter 2,Details: In particular, $T$ is realized if and only if its identity functor is a model. Namely, I do not know how the identity functor arises ...
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Understanding Hartshorne's example II 3.2.2

Example 3.2.2. If $P$ is a point of a variety $V$, with local ring $\mathcal{O}_P$, then $X:=\operatorname{Spec} \mathcal{O}_P$ is an integral noetherian scheme, which is not in general of finite type ...
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why this condition implies uniqueness

Suppose that in a category if a morphism $g$ has the same domain as $p$ and $$pu=pv$$ implies $$gu=gv$$ then $$g=tp$$ for some $t$. Then $t$ is unique. Is it true; how can I (dis)prove this ? See ...
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Show that an order-isomorphism is necessarily a bijective mapping but the converse is not true.

Show that an order-isomorphism is necessarily a bijective mapping but the converse is not true. My try for the converse: let $L=M=\{0,1\}$ and $f$ is defined by $f(0)=1,f(1)=0$. This can be easily ...
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Difference between order isomorphism and order homomorphism

I read on Applied Abstract Algebra by Rudolf Lidl and Gunter Pilz that if $L$ and $M$ be two lattices and $f:L\to M$ be a mapping, then $f$ is order homomorphism if $x\le y\implies f(x)\le f(y)$. On ...
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2 votes
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Relation between epi and mono in a category

Let $f: A \rightarrow B$ be an epic morphism in category $C$. Then is it true that the morphism $h: \text{Hom}(B,-)\rightarrow \text{Hom}(A,-)$ is mono? If yes why? What if $B$ is the initial object? ...
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Are there $f: A\rightarrow B$ and $g: B\rightarrow A$ but $g\circ f$ is not identity?

I am new to category theory. I found the following definition. An arrow $f:A\rightarrow B$ is called an isomorphism, if there is an arrow $g:B\rightarrow A$ such that $$ g\circ f=\mathrm{id}_A $$ and $...
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5 votes
2 answers
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Morphism of topoi - the canonical topology

Let $T,T'$ be Grothendieck topoi and $(f^\ast,f_\ast): T' \to T$ a morphism of topoi, i.e. $f^\ast\colon T\to T'$ is left adjoint to $f_\ast\colon T\to T'$ and $f^\ast$ commutes with finite limits. In ...
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morphism between two princial bundle over the same mainfold?

Let $ \pi : P \rightarrow B$ and $\pi' ; Q \rightarrow B$ be two principal G- bundles. Why this is true: If $f : P \rightarrow Q $ is a morphism of the pricipal G-bundles P and Q ( i.e. $f(p.g)= f(p)...
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Image of a projective variety under a morphism

Let $f:X \rightarrow Y$ be a morphism of projective varieties. If $X=\cup_{i=1}^n X_i$ is the irreducible decomposition of $X$ does the following hold? $$f(X)=f\left(\bigcup_{i=1}^n X_i\right) = \...
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Pullback of a pencil on a smooth surface with irreducible fibers is injective

We consider $f\colon X \to \mathbb{P}^1$ a non degenerate morphism of a smooth surface $X$ with irreducible fibers. I want to understand why $f^*: H^0(O_{\mathbb{P}^1}(1))\to H^0(O_X(F))$ is an ...
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What morphisms must exist in a category if the morphism between two hom-sets relative to that category's objects exist?

I'm asking this and other questions in an attempt to tackle something I'm having a hard time to understand from another point of view. Let's consider a category $C$ with 4 objects, $a$, $b$, $c$, and $...
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