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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Initial Algebras in a Category with ω-limits.

I am trying to prove the following: Let $C$ be a category with an initial object, and ω-limits. Suppose $F$ is an ω-continuous endofunctor on $C$. Then $F$ has an initial algebra. I know the result ...
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52 views

How are the cardinalities of the object images of adjoint functors related?

Here is a very silly question: Adjoint functors satisfy $$\mathrm{hom}_{\mathcal{C}}(FA,B) \cong \mathrm{hom}_{\mathcal{D}}(A,GB).$$ I consider numbers $a,b$ and read this as ...
2
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1answer
54 views

Proof that left adjoints preserve direct limits

I am reading Rotman's book on Homological algbra and have a slightly different proof of the statement in the title of this question. Am writing my attempt below. Could someone please advise me if I am ...
3
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1answer
63 views

Naturality in Yoneda's Lemma

Edited (Thanks to Kevin Carlson and Zhen Lin for pointing out the mistakes in my definitions.) Assuming $C$ is a locally small category, Yoneda lemma says that for any given object $e \in ...
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1answer
224 views

Are these parallel theorems from Set Theory and Linear Algebra connected through Category Theory?

From Set Theory and Linear Algebra, we have these two theorems: Given two finite sets of the same cardinality $X$ and $Y$ and a function $f:X\rightarrow Y$, the following are equivalent: $f$ is a ...
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1answer
94 views

Understand the $\operatorname{Hom}$ Functor.

Via Wikipedia I see that $\operatorname{Hom}_C(A,-): C \rightarrow \textbf{Set}$ a covariant functor which maps each object $X$ in $C$ to the set of morphisms $\operatorname{Hom}_C(A,X)$. I am trying ...
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1answer
33 views

Showing $f:a\to b$ is epic iff there exists some iso $g: f(a)\cong b$ with $g\circ f^*=f$.

This is Exercise 5.2.4 of Goldblatt's, "Topoi: A Categorial Analysis of Logic". In any topos, given $f:a\to b$, define $p, q: b\to r$ using the pullback of $f$ along itself like so ...
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2answers
52 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 ...
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1answer
63 views

Epimorphism in Category

Let A and B are two objects in a category $\mathcal{C}$ and $u \in Hom(A,B)$. We say $u $ is a monomorphism if and only if for any $X \in Obj(\mathcal{C})$ and $v \in Hom(X,A)$, the map ...
5
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0answers
98 views

Is there such a thing as 'overtification' (dual to compactification)?

The dual to the notion of compactness for spaces is overtness. This duality is not manifest in the category of spaces but rather in the quantifiers used to define these notions. Is there a process ...
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2answers
68 views

What does “coproduct of $\Bbb{Z}*\Bbb{Z}$ of $\Bbb{Z}$ by itself” mean?

Prove that the group $F(\{x,y\})$ is a coproduct of $\Bbb{Z}*\Bbb{Z}$ of $\Bbb{Z}$ by itself in the category Grp. What does "coproduct of $\Bbb{Z}*\Bbb{Z}$ of $\Bbb{Z}$ by itself" mean?
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132 views

Profinite completion of a partial order

In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious ...
2
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2answers
54 views

Epic-monic factorisation in $\mathbf{Set}$.

I'm stuck on Exercise 5.2.1 of Goldblatt's "Topoi: A Categorial Analysis of Logic": Given a function $f:A\to B$, if $h\circ g: A\twoheadrightarrow C\rightarrowtail B$ and $h'\circ g': ...
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1answer
100 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 ...
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35 views

A term for category where every loop of morphisms is an identity

"A category where composition of every loop of morphisms is an identity." Moreover, in the case I am thinking about, morphisms are bijective functions. Is there a name for this concept?
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2answers
136 views

What kind of object is the kernel of a ring homomorphism?

The category $\mathbf{Grp}$ of groups has a zero object, namely the trivial group $1$. Since $\mathbf{Grp}$ is furthermore complete, we have the notion of a kernel of a group homomorphism. The kernel ...
5
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2answers
84 views

Abstract proof that internal hom presheaf is a sheaf

Let me recall some definitions first. Let $D$ be a small category and let $J$ be a Grothendieck topology on $D$. A presheaf $F$ on $D$ is called a sheaf when for every covering sieve $\psi \in J(d)$ ...
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89 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 ...
3
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1answer
71 views

Verifying a Construction Satisfies the $\Omega$-axiom.

I'm stuck on Exercise 4.5.1 of Goldblatt's, "Topoi: A Categorial Analysis of Logic". It's in the topos $\mathbf{Bn}(I)$ of bundles over a set $I$. Goldblatt asks the reader to verify that $\tag{1}$ ...
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2answers
36 views

Prove that the product satisfies the universal property for coproducts in Ab.

I refer to this question. The question is Is there an intuitive way to understand why finite products and coproducts in Ab coincide, while the same is not true in Grp? The author of the ...
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2answers
57 views

Explain why there is a unique morphism $(\phi\times\phi):G\times G\to H\times H$.

Let $\phi:G\to H$ be a morphism of category $C$ with products. Explain why there is a unique morphism $$(\phi\times\phi):G\times G\to H\times H$$ How is $G\times G$ defined in a category, where ...
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1answer
33 views

Adjoint situation induced on presheaves

I just read that every functor $\phi : C \to D$ induces the well-known adjoint situation $\phi_! \dashv \phi^\ast : \hat{D} \to \hat{C}$. Well, this is not so well-known to me, so could someone ...
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2answers
55 views

Functor that preserves zero object

My question is not dificult, but i'm confused with something. The question: Prove that if $F: \mathbb{C} \rightarrow \mathbb{D}$ is a equivalence of categories and $Z \in \mathbb{C}$ is a zero ...
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1answer
89 views

Categorical proof of the existence of the real numbers

In his answer to my question on the real numbers SE/839848, Qiaochu Yuan mentioned that the real numbers are the terminal archimedian ordered field. I wondered, what can be said about the category of ...
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797 views

Category-theoretic description of the real numbers

The familiar number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ all have "natural constructions", which indicate, why they are mathematically interesting. For example, equipping $\mathbb{N}$ with ...
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0answers
55 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 ...
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2answers
39 views

A question regarding interpreting the statement “the operation $A\amalg B$ is well-defined up to isomorphism”

I have to prove that the operation $A\amalg B$ is well defined up to isomorphism. I thought the phrase well-defined is associated only with mappings. OK let us assume the "operation" $A\amalg B$ is ...
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2answers
366 views

Difference between being faithful and being injective on arrows

I'm studying the concept of faithful functors, but I cannot grasp the difference between being faithtful and being injective on arrows. Could someone explain the difference and provide some examples? ...
3
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1answer
49 views

Can I take Inverse Limits as Cauchy sequences literally?

I have been told to think of inverse limits as "Cauchy Completions" under some metric, for instance through the construction of the p-adic numbers. This got me thinking, though, and I wonder if the ...
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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 ...
2
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2answers
56 views

What is meant by a “structure map”?

The title is the question. Somehow I should know the answer, but I am by no means sure what is meant exactly by it. Perhaps it doesn't have a definite meaning and only in context, could someone ...
5
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1answer
63 views

Name for a property in a brutally elementary presentation of a monad

For evil reasons of my own, I'm trying to give a presentation of a monad in primitive terms, assuming only the notion of a category. More honestly, I looked at this post and got intrigued by the ...
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1answer
61 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\\ ...
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1answer
80 views

What Is The Product Functor [closed]

I was reading a paper which refrenced something called a "product functor". The was a functor mapping from $\bf{Sets}^{C^{op}}$ for some category $C$. I was wondering what this was? Thanks for any ...
6
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2answers
100 views

Ways to formalise $\text{Ring}\approx \text{Group}\times \text{Monoid}$.

In a (unit) ring, elements (of a set $S$) are able to operate on each other via $\cdot,+$. If we are to consider the maps $M:S\times S\to S:(a,b)\mapsto a+b$ and $G:S \times S \to S: (a,b )\mapsto ...
4
votes
1answer
88 views

Two definitions of homology

Let $f,g$ be arrows in an abelian category such that the composite $gf$ is defined and is given by the zero arrow. I shall try to find a definition for the quotient $\ker g /\operatorname{im} f$, ...
4
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0answers
45 views

Presheaves in a quasi-topos.

I do believe it is a trivial question. But unfortunately I don't know where I can find an answer. Where could I find the answer to the following question? If $S$ is a small category and $X$ is a ...
4
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0answers
80 views

Completeness in a Category of Metric Spaces.

Is there a way to describe completeness within a category of metric spaces? The point is that I'd like to have a description of compactness in metric spaces by something of the form totally bounded + ...
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 ...
3
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1answer
59 views

Does a Cocomplete Cowellpowered Additive Category have Generator?

Let $\mathcal{C}$ be a Cocomplete Cowellpowered additive category. Does $\mathcal{C}$ need to have a generator?
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1answer
31 views

Coproduct of $C^*$-algebras

I want to prove that the free product $A*B$ of two unital $C^*$-algebras $A$ and $B$ is a coproduct in the sense of category theory. Remember the construction of $A*B$: Take generators $\{a:a\in ...
6
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1answer
86 views

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
94 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
54 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] + ...
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1answer
54 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
71 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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2answers
76 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
116 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 ...
4
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1answer
97 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
140 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: ...