9
votes
5answers
323 views

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 ...
2
votes
1answer
46 views

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 ...
3
votes
3answers
95 views

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 ...
6
votes
0answers
109 views

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 ...
3
votes
0answers
25 views

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 ...
1
vote
1answer
108 views

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 ...
1
vote
1answer
22 views

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 ...
0
votes
0answers
65 views

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 ...
5
votes
1answer
39 views

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 ...
7
votes
1answer
65 views

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 ...
4
votes
0answers
58 views

Existence of a coproduct and representability of a functor

I've found this claim: Let $\mathcal{C}$ be a category; the family $\lbrace C_i \rbrace_{i \in I }$ has a coproduct in $\mathcal{C}$ if and only if the functor $$F : \mathcal{C} \to Set$$ $$A ...
2
votes
1answer
35 views

Free objects are generated by the image of the canonical injection

Let $\mathcal{C}$ be a concrete category and $X$ be a set, let $F_X$ be free on $X$ with canonical injection $i:X\to F_X$. Is it always true that $i(X)$ generates $F_X$? It looks like an easy ...
2
votes
1answer
36 views

Is there a phrase to describe those objects of $\mathbf{C}$ that can be expressed as quotients of the algebra freely generated by $X$?

Let $\mathbf{C}$ denote the category of models of an algebraic theory in $\mathbf{Set}.$ Now suppose $X$ is an object of $\mathbf{Set}$. Is there a traditional phrase used to describe those objects of ...
0
votes
1answer
33 views

In a semi-simple module, any submodule is a direct factor?

I need help understanding the following : In a semi-simple module, any submodule is a direct factor (this is sometimes taken as the definition of semi-simple) (i) How is this equivalent to the ...
4
votes
1answer
93 views

Category theory for graph theory research

I am doing research in algebraic graph theory, focusing on the relation between graphs and groups (especially the representing groups as graphs) for my Ph.D. In particular, one of the ideas is to ...
1
vote
2answers
53 views

Question on Category Theory injective morphism

I have a basic understanding in Category Theory but haven't had any exposure to Modules but this is a question from last year's paper. Show that a morphism $u:M \to L$ in a category $\mathscr C$ of ...
6
votes
1answer
103 views

Adjoint Functor Theorem

The Freyd's Adjoint Theorem states that given a complete locally small category $\mathcal{C}$, a continuous functor $G: \mathcal{C} \to \mathcal{D}$ has a left adjoint if and only if it satisfies a ...
1
vote
2answers
92 views

Soft sheaves adapted to $f_!$

I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to ...
1
vote
0answers
56 views

Characterizing the real numbers as a dense complete monoidal poset

Let $P$ be a poset with a monoidal structure respecting the poset structure. This means there is an operation $P \times P \to P$ such that $a \leq b \implies ac \leq bc$. As usual, call a poset ...
1
vote
2answers
32 views

Uniqueness of Initial Object, or why must a morphism from an object to itself be the identity?

The definition of an initial object in a category $\mathscr{C}$ is defined as an object that only has one map going to each object in $\mathscr{C}$. A basic result supposedly says that any two initial ...
1
vote
1answer
63 views

Pullback in morphism of exact sequences

Suppose we have the following morphism of short exact sequences in $R$-Mod: $$\begin{matrix}0\to&L&\stackrel{f'} \to& M'&\stackrel{g'}\to &N' & \to 0\\ ...
2
votes
1answer
21 views

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 ...
2
votes
1answer
56 views

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 ...
3
votes
1answer
81 views

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 ...
1
vote
1answer
39 views

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 ...
4
votes
1answer
95 views

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 ...
2
votes
1answer
50 views

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 & ...
2
votes
1answer
57 views

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 ...
1
vote
1answer
50 views

Pushout of an injective map is injective

This is an exercise from Rotman , Introduction to homological algebra. Given a pushout diagram in $R$-Mod $$\begin{array} AA & \stackrel{g}{\longrightarrow} & C \\ \downarrow{f} & & ...
0
votes
1answer
34 views

Applying an additive covariant functor to a sequence

Let $F$ be an additive covariant functor and $0→A_1→A_1⊕A_2→A_2→0$ be the usual exact sequence in the category of $R$-modules with the first map injection on the first, and the second map the ...
2
votes
1answer
72 views

Newbie into categorical proofs

Let $$ F:Mod_A \to Mod_B $$ an aditive , exact and covariant functor and $$ M ∈ Mod_A $$ and $$M_1 , M_2 $$ submodules of M . Show that $$ F( M_1\cap M_2)=F(M_1)\cap F(M_2 ) $$ $$ F( M_1+ ...
5
votes
1answer
92 views

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 ...
1
vote
1answer
34 views

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 ...
2
votes
1answer
50 views

Exercise from Rotman, direct limit of quotient modules

Suppose $$0 \to U \to V \to V/U \to 0$$ is an exact sequence of left $R$-modules. Let $\lbrace U_i, \alpha^{i}_j\rbrace$ be a direct sequence of submodules of $U$, where $$\alpha^{i}_j : U_i \to U_j ...
2
votes
1answer
55 views

Exercise from Rotman: formal power series ring as inverse limit

Let $A$ be a commutative ring with unit, $J = (x)$ an ideal of $A[x]$. Thus we can consider the inverse system defined as $$\psi_{n,m}: A[x]/J^m \to A[x]/J^n$$ $$g(x) + J^m \to g(x) + J^n$$ $$\forall ...
2
votes
1answer
45 views

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 ...
3
votes
0answers
49 views

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 ...
0
votes
0answers
34 views

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 ...
4
votes
0answers
58 views

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 ...
1
vote
0answers
72 views

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 ...
5
votes
3answers
158 views

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 ...
5
votes
0answers
63 views

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, ...
3
votes
1answer
52 views

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 ...
6
votes
0answers
69 views

Galois Group of Composite Field vs. Second Isomorphism Theorem

$\DeclareMathOperator{\Gal}{Gal}$ 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) ...
1
vote
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) ...
4
votes
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. ...
2
votes
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 ...
2
votes
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 ...
2
votes
3answers
239 views

Definition of groups in a more abstract way

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 ...
11
votes
3answers
186 views

How to recognize adjointness?

Reading math has gradually expanded my understanding of the word "symmetry", so that now I can recognize symmetries that I would have not noticed before, and without having them pointed out to me. I ...