# Tagged Questions

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### Product in the category of pointed sets..

I have the category $C$, where: objects are nonempty sets with one fixed element $Obj = \{(A,a)$, where $A$-nonemty sets, $a\in A\}$, morphisms are $Mor=\{ f:(A,a)\rightarrow (B,b)$; where $f$ - is ...
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### Category with only one object.

Suppose we have the category with only one object - the group G. Why can we think of the morphisms in this category as of the elements of the group G? I would be very grateful for explanation.
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### What are the algebraic and topological properties of a tree (graph theory), and what are any possible connections?

I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup ...
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### Kernels of induced maps in cohomology

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ ...
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### Free object in category of groups.

Suppose $X$ is a set and $F$ is a free object on $X$ (with $i:X\rightarrow F$) in the category of groups. Prove that $i(X)$ is a set of generator for the group $F$. I have the following hint: If ...
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### The cohomology ring of the nerve of a category associated to a vector space

Let $n\ge2$, and let $V$ be an $n$-dimensional vector space over a field $k$. Consider the category $\mathcal{C}$ whose objects are nonzero, proper subspaces of $V$, and whose morphisms are ...
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### Every Abelian group is canonically a $\mathbb{Z}$-module. Is this just a coincidence?

Every Abelian group is canonically a $\mathbb{Z}$-module, where $\mathbb{Z}$ is the initial monoid in the monoidal category of Abelian groups. And every Abelian monoid is canonically an ...
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### Generalizing a statement about direct limits in the category of $A$-modules to other categories

The original problem is from Atiyah and Macdonald's Introduction to Commutative Algebra, Ex 2.15: Suppose $(M_\alpha,\mu_{\beta\gamma})$ is a direct system of modules over a unital commutative ...
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### Presentation of an object in an Eilenberg Moore category by generators and relations

Let $\cal C$ be a category and let $T \colon \cal C \longrightarrow C$ be a monad. An $T$-algebra $A$ is presented by generators $G \in \cal C$ and relations $R \in \cal C$ if it is the coequaliser of ...
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### Universal property of polynomial ring in $\mathbf{CRING}$

I know that the polynomial ring $A[x]$ is the free $A$-algebra on $\{x\}$; this is its universal property in the category of $A$-algebras. Is there also a universal property for $A[x]$ considered as a ...
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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 ...
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 ...
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 ...