Tagged Questions

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Exercises in category theory for a non-working mathematican (undergrad)

I'm trying to learn category theory pretty much on my own (with some help from a professor). My main information source is the good old Categories for the working mathematician by Mac Lane. I find the ...
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Conceptual Question on different representations of Hyperplanes, Higher Standpoint, Coordinate-free

In a vector space $V$ over some field $F$ a hyperplane is the kernel of some linear transformation $T : V \to F$, i.e. the kernel of an element of the dual space (this could be taken as the definition ...
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Is it possible to describe $Q(x)$ as the extension field of $R$ freely generated by $\{x\}$?

Given an integral domain $R$, the polynomial ring $R[x]$ can be defined as the commutative $R$-algebra freely generated by $\{x\}$. Also, let $Q$ denote the field of fractions associated to $R$. Then ...
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Category Theory textbook (learning through guided discovery Dummit and Foote)

sorry for asking the same question in a slightly different angle, I want a book in Category Theory similar to Dummit and Foote's book in Abstract Algebra. I want it to have tons of examples and ...
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Localization and Direct limit [duplicate]

Let $A$ be a ring. Let $I$ be a preordered set, filtering. Let $\Sigma$ a multiplicative subset of $A$. Suppose for any given $i \in I$ a multiplicative subset $S_i$ of $A$ contained ...
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Localization and direct limit

Let $A$ be a ring. Let $I$ be a preordered set, filtering. Let $\Sigma$ a multiplicative subset of $A$. Suppose for any given $i \in I$ a multiplicative subset $S_i$ of $A$ contained ...
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Proving that finite direct and inverse limits exist in an additive category having kernels and cokernels

I have attempted to prove this but am unable to complete the proof. Below is my attempt. Let $\mathcal{A}$ be a category satisfying the conditions in the title and $\{M_i,\phi^i_j\}$ be a finite ...
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Can I assign a distinct homomorphism to every function?

I consider an algebraic structure and particular structure preserving morphisms (think groups and group homomorphism). I wonder if I can assign a distinct such morphisms to every function between any ...
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If $0$ is the zero-object $\Longrightarrow F(0)$ is the zero object when $F$ additive

Let $$F : \text{A-Mod} \to \text{A-mod}$$ be an additive functor. Then if $0$ is the zero-object $F(0)$ is the zero object. Why this is true ? The definition of additive functor that I know is ...
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Is there a name for those commutative monoids in which the divisibility order is antisymmetric?

Every commutative monoid $M$ is naturally equipped with its divisibility preorder, defined as follows. $$x \mid y \leftrightarrow \exists a(ax=y)$$ Is there a name for those commutative monoids such ...
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Proving that the transformation obtained from an adjoint pair is natural

I am reading Homological Algebra by J.J. Rotman and am unable to do this problem. Given an adjoint pair $(F,G)$ where $F : \mathcal{C} \to \mathcal{D}$ and $G : \mathcal{D} \to \mathcal{C}$ are two ...
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Every category is the free category for a given graph?

I am wondering if for any category $C$ (at least a small category), we can find a graph $G$ (at least a small graph), such that $C$ is the free category generated by the graph $G$. I think this ...
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Infinite Direct Sums Vs. Infinite Direct Products

Let $|R|=|S|=\infty$. In very many concrete categories, I know $R^S$ can be identified as the set of all functions from S to R, and the much "smaller" $R^{\oplus S}$ can be identified as the subset of ...
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Set maps given by a polynomial & Yoneda Lemma

This Exercise 4.1. from the book Algebraic Geometry I, by Gortz. Problem Let $R$ be a ring, and for every $R$-algebra $A$ let $\alpha_A:A\rightarrow A$ be a map of sets such that for every ...
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Is $L^{\infty}$ a direct limit or inverse limit of the directed system $(L^p , i_{p}^q )_{p,q \in [1 , + \infty [ }$?

Let $X$ be a finite measure space. Then, for any $1≤p<q≤+∞$ : $L^q(X,B,m)⊂L^p(X,B,m)$. I would like to know if the space $L^{\infty} ( X , B , m )$ is the direct limit or the inverse limit of ...
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Action of the functor Ext$_1(-,-)$ on extensions

Suppose we have an exact sequence of $R$-modules \begin{array}{ccccccccc} 0 & \longrightarrow & L & \overset{f}{\longrightarrow} & M & \overset{g}{\longrightarrow} & E & ...
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An equivalence of categories of presheaves.

Let $C$ and $D$ be two small categories. Consider the corresponding categories of presheaves $PSh(C)$ and $PSh(D)$. Suppose we have an equivalence of categories $F: PSh(C) \to PSh(D)$. Asking for an ...
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Which mathematical structures are particular cases of small categories?

In what follows, all categories are assumed to be small (classes of objects and morphisms are sets). Which mathematical structures $X$ can be seen as particular cases of small categories ...
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Determining Objects in a Semicategory

Suppose $S$ is a small semicategory (or semigroupoid, if that's your preferred term) and $\cdot$ is the binary operation on $S$. Implicit in this definition is the set $\operatorname{Ob}(S)$ and two ...
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Do covariant functors preserve direct sums?

Suppose $T$ is a covariant functor from the category $R$-mod to Ab (the category of abelian groups) Is is necessary that $T(B \oplus C) \cong T(B) \oplus T(C)$ ? Does the answer change if we ...
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For which categories do injections induce surjections in cohomology?

I'll ask a specific question first, but I believe my question might have a rather immediate abstraction, with which I'll finish. Let $H,G$ be finite affine group schemes over an algebraically closed ...
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Alternative proof for localization isomorphism

Let $f$ be an $A$-module morphism and $\operatorname{res}_{A_m}^A$ be the restriction of scalars functor from $A_m$-mod to $A$-mod. I'm curious if you have proven that for every maximal ideal ...
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Abelization of symmetric groups and its subgroups of bounded support

For $X$ a finite set one can show easily that the abelization of the symmetric group on $X$ is given by the group of order $2$ and that the commutator subgroup of $\operatorname{Sym}X$ is given by the ...
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Limit Creating Functor

This is exercise 5.5.1 of Maclane's book "Categories for the Working Mathematician". Let $X$ be any category. Prove that the projection $P:X^2 \to X \times X$ sending each arrow $f:x \to y$ to the ...
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Does it factor through?

Let $f:F\to G$ and $g:F\to H$ be group homomorphism between groups. If $\ker f \subset \ker g$ then does there exists $h:G\to H$ such that $hf = g$? I know the the above is true for vector spaces by ...
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When are all ring homomorphisms also algebra homomorphisms?

Let $k$ be an algebraically closed field, and let $A,B$ be two unitary $k$-algebras. In general, there are more ring homomorphisms $A\to B$ than there are $k$-algebra homomorphisms. More precisely, ...
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In a Morita context ($A, B, P, Q, f, g$), why are $P$ and $Q$ projective if $f$ is surjective?

The title probably says it all :). This is probably a very very simple question, please bear with me. Let $(A, B, P, Q, f, g)$ be a Morita context (or pre-equivalence data) as defined by Hyman Bass in ...
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$\DeclareMathOperator{\Gal}{Gal}$ In my abstract algebra class, we learned about how Galois groups interact with composite fields. Namely, if $K/F$ is Galois, and $L/F$ is any extension: $$\Gal(KL/L) ... 1answer 96 views What is the relationship between the second isomorphism theorem and the third one in group theory? The second isomorphism theorem [wiki] in group theory is as follows: Let G be a group. H \triangleleft G, K \le G. Then: HK \le G, (H \cap K) \triangleleft K, and K/(H \cap K) ... 0answers 46 views ~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~ I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ... 1answer 29 views The functor of monoids I'm studying this book on introductory level category theory and I couldn't solve this exercise: In the first part I've been thinking about the monoid homomorphisms F: S\to T and regarding of ... 1answer 79 views Do we lose everything, if the natural transformations in a monad are exactly inverse? I'm currently explaining monads$$T:{\bf C}\to{\bf C},\hspace{1cm}\eta:1_{\bf C}\to T,\hspace{1cm}\mu:T\circ T\to T, to my brain and the "only" tricky thing are really the identity relations. I ...
I'm trying to understand the definition of group objects in categories, this is an extract of Paolo Aluffi's book: QUESTIONS Can I say that $e(1)$ is the identity in our group $G$ we have just ...