Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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Commutative monoids arising from categories with finite coproducts

If $\mathcal{C}$ is a category with finite coproducts, we may associate to it a commutative monoid $\mathcal{C}/\cong$ of isomorphism-classes of objects, with addition induced by the coproduct and ...
4
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2answers
98 views

What is the use of generators of a category?

An object $X$ is a generator of a category $\mathcal{C}$ if the functor $Hom_{\mathcal{C}}(X,\_) : \mathcal{C} \rightarrow Set$ is faithful. I encountered the notion in the context of ...
6
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1answer
57 views

Realizing the monoid $\mathbb{N}/(3=1)$ from a category with finite coproducts

If $\mathcal{C}$ is a category with finite coproducts (including an initial object $0$), then the set of isomorphism-classes of $\mathcal{C}$ becomes a commutative monoid with $0 := [0]$ and $[x] + ...
2
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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 ...
0
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1answer
74 views

Is the comma category $y \downarrow X$ small? [closed]

Let $\mathcal{C}$ be a small category and $X \in \hat{\mathcal{C}}$ a presheaf. Is the comma category $y \downarrow X$ small?
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79 views

(Certain) colimit and product in category of topological spaces

Consider the diagram $$(*):\;\;\;X_0 \stackrel{i_0}\hookrightarrow X_1 \stackrel{i_1}\hookrightarrow X_2 \stackrel{i_2}\hookrightarrow \cdots $$ in category of topological spaces. Denote $I$ the ...
3
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1answer
123 views

Natural Isomorphism: how can $A \otimes B \simeq B \otimes A $ and yet $A \otimes B \neq B \otimes A $

I am reading Braided Monoidal Categories by Joyal and Street. They say cateogories with tensor product arise naturally such as the category of Abelian Groups and that of Banach Spaces. Is there any ...
5
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1answer
109 views

Category theoretic approaches to Riemann Hypothesis?

I was wondering if there has been any category theoretic advancements in the study of the Riemann Hypothesis and the theory surrounding it? This question is meant in the same vein as these ...
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2answers
142 views

A Function as a collection of Arrows

Normally you define a function to be a map on a set. But how about defining a function, in Category Theory, as a collection of arrows? Take this cateogry Objects: true, false. Arrows: ...
3
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137 views

Paper, Scissors, Rock, Category Theory! [closed]

I like to understand something about Category Theory and I think "paper scissors rock" might help. Category Data Objects: paper, scissors, rock Arrows: beats ...
6
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1answer
106 views

Are there categorifications of prime or irreducible elements (of a ring, say)?

I'm very sorry if this is a duplicate in any way or is otherwise a stupid question. I've looked around (for quite a while) but . . . no luck. There's a categorification of what it means to be an ...
3
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1answer
46 views

A question about $A\times B\times C$ and $(A\times B)\times C$.

Let $\mathcal C$ be a category with products. Find a reasonable candidate for the universal property that the product $A\times B\times C$ of three objects of $\mathcal C$ ought to satisfy, and ...
3
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1answer
84 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 ...
3
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0answers
32 views

Motivation of Strong Monics

Strong monics are defined as: A monic $m$ is strong iff every commutative square $mu=ve$, in $E$ with $e$ epi, has a diagonal i.e. there is a morphism $t$ such that $u=te$ and $v=mt$ in $E$. (where ...
3
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66 views

Weil restriction - from abstract nonsense to a practical procedure

Let $L/K$ be a finite field extension. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ ...
2
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1answer
42 views

Defining a monoidal category without elements

I am trying to generalize the notion of monoid object internal to a (not necessarily strict) monoidal category, by weakening the associativity and unitarity diagrams (see this nlab entry.) Of course ...
3
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3answers
124 views

Learning the topology needed for topos theory.

I have just started learning topos theory and I am going through Mac Lane and Moerdijk's book, "Sheaves in Geometry and Logic". I have, unfortunately, very little experience with topology. I started ...
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1answer
52 views

A question regarding $Hom(X,X)$ (or $Mor(X,X)$)

I refer to Rankeya's answer on this question. Shouldn't $Mor(X,X)$ consist of monomorphisms only, for each morphism to have an inverse?
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1answer
44 views

A question about the category Grp

This is a question about the category Grp (groups). The book "Chapter 0" by Aluffi says that the objects of the category are groups, and the morphisms homomorphisms. He then says that we need not ...
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1answer
40 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 ...
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0answers
153 views

Adjoint action of semi-direct product

Let $G$ and $H$ be Lie groups with associated Lie algebras $\mathfrak{g}:=\text{Lie}(G)$ and $\mathfrak{h}:=\text{Lie}(H)$ and adjoint actions $\text{Ad}^G:G \to \text{Aut}_\text{Lie}(\mathfrak{g})$ ...
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1answer
74 views

Is there way to formalize the idea that a category can be “cocomplete from the inside”?

Let $\mathrm{KSet}$ denote the category of all countable sets, including the finite ones. Then $\mathrm{KSet}$ is finitely complete. Furthermore, $\mathrm{KSet}$ admits all countable colimits, or, ...
0
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1answer
38 views

Left & right adjoints in the context of complete lattices.

This is a follow-up question from this question of mine. In the same paper as the one mentioned in my previous post, it's stated that In the context of complete lattices, a monotone map has a ...
3
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1answer
48 views

How do you construct the lifted topology of a groupoid cover?

If I have a particularly nice space $X$ (Hausdorf, locally path connected, semi-locally 1-connected, I think), then then there is an equivalence of categories between the cover category over $X$ and ...
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1answer
49 views

Left & right adjoints in the context of posets.

Definition 1: A function $\theta: X\to Y$ between posets is monotone if whenever $x\le y$, we have $\theta (x)\le\theta (y)$. Definition 2: For any pair $f:A\to B$ and $g:B\to A$ of monotone maps, we ...
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2answers
55 views

If $g$ and $fg$ are isomorphisms, then so is $f$.

Let's say you have a category, and you are looking at two arrows, $f$ and $g$, such that $fg$ is defined. You know that $g$ and $fg$ are isomorphisms, meaning there are arrows that are both their ...
2
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1answer
56 views

Coarse moduli space and rational points

Let $K$ be a field (not necessarily algraically closed). Let $\mathcal{F}$ be a contravariant functor from the category of schemes over $K$ to sets and $M$ be a coase moduli space for the functor. So, ...
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1answer
103 views

Is there any relevance between Boolean Algebras and Fields?

In some sense Boolean Algebras and Fields have same operators and constants. In both structures there are operators addition ($+$ , $\vee$), multiplication ($\times$ , $\wedge$), inverse with respect ...
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3answers
177 views

Abstract nonsense proof

What is a simple example of an "abstract nonsense" proof in category theory. For a theorem you are proving, it doesn't matter if the category or regular proof came first, it is just that the category ...
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vote
1answer
42 views

Rig categories concept which is equivalent to monoid concept in monoidal categories

In monoidal categories, there is a notion of monoid. Is there an "equivalent" concept in rig categories (i.e., categories with two monoidal structures which are related like + and * in a rig)?
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60 views

When are two morphisms the same?

Let $\mathcal{C}$ be a category and let $f,g:A\to B$ be two morphisms in $\mathcal{C}$. If for all $a\in A$ we have $f(a)=g(a)$, do we necessarily have $f=g$? Of course such a situation can make sense ...
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How to prove that $\mathcal{Y}_X$ is a sheaf by using epimorphic families

I'm trying to prove this: Let $(\mathcal{C}, J)$ be a site and suppose that for every $X\in ob(\mathcal{C})$ and $R\in J(X)$ the family $\{ \bar{f}_Y: \mathcal{Y}_Y\rightarrow \mathcal{Y}_X \}_{Y\in ...
4
votes
1answer
107 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
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1answer
51 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
59 views

Uniqueness of the Comparison Functor

Suppose $F:C\rightarrow D$ and that $F\dashv U$ is an adjunction and $C^{T}$ the Eilenberg–Moore category for the monad $T=U◦F$, with the corresponding functors $F^{T}:C\rightarrow C^{T}$ and ...
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1answer
64 views

Typed Category Theory?

This small book (or long paper) describes objects and morphisims in a cateogry as having types. He also talks about pre-categories as categories with typeless objects and morphisims. While I am ...
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Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
3
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1answer
39 views

Quasi-isomorphisms not localizing in Kom(A)

I've been reading up on the construction of derived categories. I understand why we prefer localizing with respect to a localizing class of morphisms (to get a nice representation of morphisms as ...
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 ...
2
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1answer
27 views

Is the class of principal $G$-bundles over $M$ a set?

Let $G$ be a Lie group and $M$ a manifold. Question: Is the class of principal $G$-bundles over $M$ a set? This question came up when I was thinking about the classifying stack of $G$. It is the ...
3
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1answer
39 views

Definition of subobject

In the definition of subobject there is an equivalence relation defined on monomorphisms into a fixed codomain. My question is that how do we know that the collection of monomorphisms into a fixed ...
5
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3answers
90 views

Types, Sets and Categories

I am learning Category Theory and at first I just pictured a category like a class in object oriented programming: type definition + methods (morphisms). However the author I am reading uses maps ...
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1answer
71 views

Center of the categories $\mathbf{Grp}$ and $\mathbf{Ab}$.

This is Exercise II.5.8 from Mac Lane, Categories for the Working Mathematician. For the identity functor $I_C$ of any category, the natural transformations $\alpha:I_C\dot{\to}I_C$ form a ...
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Algeraic theory and definition of multiplication: tensor product v.s. Cartesian product

In a monoidal category, one can define multiplication on a object $M$ as a morphism \begin{equation} M\otimes M\longrightarrow M, \end{equation} or a morphism \begin{equation} M\times ...
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41 views

What is the opposite of category of schemes

The category of schemes sits between $Aff=CRing^\text{op}$ and its free cocompletion $\hom(Aff^\text{op},Sets)$, as the locally representable functors. Dualizing, we have $CRing\subseteq ...
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1answer
29 views

Coproduct of $(0,1)$-Algebras

I am trying to find the coproduct of $(\mathbb {Z},0,+1) $ with itself in the category of $(0,1) $-Algebras. Finding $\mathbb {N}\sqcup\mathbb {N} $ was easy, since $\mathbb{N} $ is initial. But I ...
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1answer
55 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} & & ...
3
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0answers
56 views

Topological characterisations of freeness and separability?

According to wikipedia, a spectral space is defined to be homeomorphic to the spectrum of some commutative ring. They form a category $Spec$ where we take morphisms to be those whose preimage of open ...
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47 views

Closure operators and complete lattices.

A closure operator on a set $A$ is a function $C: \mathcal{P}(A) \to \mathcal{P}(A)$ satisfying following axioms: $X ⊆ Y \implies C(X) ⊆ C(Y)$ $X ⊆ C(X)$ It may also satisfy some additional ...
2
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1answer
44 views

Algebraic Compact manifold originates from a proper scheme?

If $M$ is a compact complex manifold, which is the analytification of some scheme $X$ of finite type over $\operatorname{Spec}(\mathbb{C})$, then must $X$ be proper over ...