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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Is there a universal property in this proposition (which regards field extensions)?

Since learning a bit of category theory, I am trying, as an exercise, to state results that I come across in categorical language. I am trying to do this with the following: Let ...
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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 ...
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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 ...
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1answer
48 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...
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28 views

induced functor between weighted limits

I'm working on some special weighted limits in the category of categories. And I have to prove that the induced functor between two special weighted limits is faithful. But,before working in the ...
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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 ...
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85 views

Distinguishing sets according to more fine-grained notions than cardinality.

I'm interested in distinguishing sets according to more fine-grained notions than cardinality. Now I don't know a thing about computability theory, but it seems to me that considering sets up to ...
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0answers
45 views

Objects without extensions

How do you call an object $X$ for which every monomorphism $i : X \hookrightarrow Y$ has a retract (i.e.\ a morphism $r : Y \rightarrow X$ such that $r \cdot i = 1_X$)? I think of Y as an extension ...
3
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1answer
45 views

Help defining the $\mathrm{supp}$ function on free algebraic structures.

Given an algebraic structure $F(K)$ freely generated by a set $K$ with underlying set $U(F(K))$, I'm trying to define a "support" map $\mathrm{supp} : U(F(K)) \rightarrow \mathcal{P}_\mathrm{fin}(K).$ ...
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45 views

Adjunction between cocomplete categories

Let $C$ be a small category. Let $D,E$ be cocomplete categories. Let us denote by $\hom$ (resp. $\hom_c$) the category of (cocontinuous) functors. Then there is an equivalence of categories ...
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1answer
74 views

An example of when a product in a category may not exist.

My question is about the category of finitely generated Abelian groups; in particular, I want to show, by definition, that there exists a set of objects $T$ in this category for which there is no ...
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1answer
66 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 ...
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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
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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 ...
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1answer
62 views

References for a notion of “restricted adjoint”

A construction that I've been finding all over the place in studying the category of NF (Quine's New Foundations) sets and functions is a situation like the following: there's a functor ...
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0answers
104 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have ...
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2answers
101 views

On the category of Sets as an example of an algebraic category

What follows comes from Algebraic Theories, pag. 7. Definition An algebraic theory is a small category $\mathcal{T}$ with finite products. An algebra for the theory $\mathcal{T}$ is a functor ...
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2answers
140 views

Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
2
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1answer
49 views

“Elements” of algebraic structures

Let $T$ denote an algebraic theory, and $e$ denote the $T$-algebra freely generated by a singleton set, and write $U$ for the forgetful functor. Now suppose we're given a $T$-algebra $A.$ We might say ...
2
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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 ...
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2answers
49 views

Understanding morphisms in FinSet

I'm looking at the skeletal FinSet category ; there is one object for each natural number $n$ (including $n=0$), and a morphism from $m$ to $n$ is an $m$-tuple $(f_0, \ldots, f_{m-1})$ of numbers ...
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1answer
85 views

What is the left adjoint of the forgetful functor from fields to integral domains?

I quote from Wikipedia, regarding the construction of the field of fractions of an integral domain: "There is a categorical interpretation of this construction. Let $\mathcal{C}$ be the category ...
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1answer
49 views

Collection vs set in this textbook about category theory, and some related questions.

What is the meaning of collection in this context ? Is it here a synonym of set ? Can someone please explain what the author means by "A moment's though shows that, as sets of functions, these two ...
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3answers
185 views

Simplifying a categorical proof of constructive dilemma

In axiomatic propositional calculus the following axiom schema captures constructive dilemma: $\newcommand{\lif}{\supset} \renewcommand{\land}{\&}$ \begin{equation} (a \lif c) \lif ((b \lif c) ...
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169 views

Definition of forgetful functor [duplicate]

Is there an actual definition of "forgetful functor?" Wolfram MathWorld defines it in terms of functors from algebraic categories to the category of sets, but then says, "Other forgetful functors..." ...
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1answer
103 views

Can I define a category as a monoid with partially defined multiplication?

A groupoid can either be thought of as a category whose morphisms are isomorphisms, or as a generalization of a group whose multiplication is only partially defined. Can I do a similar thing with ...
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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
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1answer
94 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 ...
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50 views

Polynomial Functor is ω-Continuous

Suppose $P:Set\rightarrow Set$ is given by $X \mapsto \sum_{i=0}^{n} C_{i}\times X^{i}$. I want to prove that $P$ is ω-continuous. Three questions: 1). Is my following argument correct? 2). Is there ...
5
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1answer
62 views

Defining algebras over noncommutative rings

A definition of "algebra" (that is, an associative algebra, in the sense of ring theory) generally requires a commutative base ring. But there are cases where it's reasonable to consider algebras ...
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1answer
85 views

Arrows-only implication & disjunction in $\mathbf{Set}.$

Just before the truth-arrows in a topos subsection of Goldblatt's "Topoi: A Categorial Analysis of logic," descriptions of the truth functions $\Rightarrow$ and $\smallsmile$ are given in ...
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42 views

Commutativity of direct and inverse limits

In exercise 5.34(iv) of Homological Algebra book by Rotman one is asked to prove that direct limits and inverse limits do not necessarily commute. I have two questions : 1.) Is it true that ...
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2answers
121 views

Categorial definition of free products?

If $X$ and $Y$ are objects of a concrete category $\mathcal{C}$, is there an accepted definition of "free product of $X$ and $Y$," generalizing the in the special case where $\mathcal{C}$ is the ...
5
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1answer
59 views

What is the coproduct in the category of Banach spaces and continuous linear maps?

In the category of Banach spaces, where the objects are Banach spaces and the morphisms are continuous linear maps, what are there coproducts? Are they the typical direct sum of Banach spaces? If so, ...
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1answer
37 views

Strange monomorphism of sheaves

What is an example of a Grothendieck topology $J$ on a small category $\mathcal{C}$ such that the category of sheaves $\mathsf{Sh}(\mathcal{C},J)$ has a monomorphism which is not a monomorphism of ...
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3answers
148 views

Why does Fld not have an initial object?

My Algebra book says that the category Fld of fields has no initial object. Why would $\{0,1\}$ not be an initial object? Does it not have a unique homomorphism to every other field?
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1answer
44 views

When does a functor commute with colimits?

Is it true that an additive functor between abelian categories commutes with colimits if it's right-exact and commutes with (arbitrary) direct sums? If yes, does someone know a good source of a ...
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1answer
78 views

The n-dual of a vector space

Studying linear algebra for my exam, I doubt arose. If I define the category $\mathcal{V}$ of all the vector spaces, and the functor $\mathcal{F}:\mathcal{V}\rightarrow \mathcal{V}$ given by ...
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0answers
58 views

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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1answer
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
57 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
65 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
230 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
96 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
34 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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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 ...
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1answer
64 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 ...
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0answers
103 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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70 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?
2
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152 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 ...