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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Categorical version of the Tietze Extension Theorem

In Donald Hartig's short paper An Important Functor in Analysis and Topology, Theorem 1 is preceded by the following statement: Since the spaces we are dealing with are compact, a one-to-one map ...
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Category of ordinal numbers

Let $\Delta$ be the category of finite ordinal numbers with order-preserving maps, i.e., $\Delta$ consists of objects strings $$ [n]: 0 \to 1 \to 2 \to \dots \to n. $$ A morphism $f:[n] \to [m]$ is ...
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Categories for the working mathematician exercises III 1

I'm currently reading Mac Lane's Categories for the working mathematician and I'm having some trouble with the two following exercises from part III. Find (from any given object) an universal arrow ...
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Why is the cartesian product so categorically robust?

The major "broad/natural" categories I encounter in daily life are: sets, groups, topological spaces, smooth manifolds, vector spaces over a fixed field $k$, $k$-schemes, rings, $A$-algebras for a ...
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Some exercises from Introduction to Homological Algebra by J.J. Rotman (category) [on hold]

Please give solutions for these problems: Give an example of a covariant functor that does not preserve coproducts. Prove that every left exact covariant functor $T$: $_RMod$ → $Ab$ preserves ...
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Prove that a transformation of the identity functor of a Group $G$ (seen as a category) into itself is just an element of the center of $G$

I want to prove the follow: Suppose $G$ is a group seen as a category, prove that a transformation of the identity functor of $G$ into itself is just an element of the center of $G$. I'm not ...
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Description of free Lie algebra in Weibel's book

In Exercise 7.3.2 in Weibel's book An Introduction to Homological algebra the following description of the free Lie algebra over some $k$-module $M$ is given (where $k$ is any commutative ring): ...
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Extending actions of Monads on Endofunctors.

Let $X^{X}$ be the category of endofunctors on a category $X$. Then if we define $\otimes :X^{X}\times X^{X}\rightarrow X^{X}$ by $R\otimes S=RS$ on objects and $\tau \otimes \sigma $ to be horizontal ...
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41 views

Definition of Category of Hypergraphs

I have a question concerning the definition of Hypergraphs in category theory, which I adopted from "A category-theoretical approach to hypergraphs" by W.Dörfler and D.A.Waller: ...
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1answer
40 views

Coproduct of groups explanation

Could someone please explain the following? "Let $G=\prod G_{i}$ be a direct product of groups. Then each $G_j$ admits an injective homomorphism into the product, on the j-th component, namely the map ...
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1answer
27 views

equivalence between subcategories in abelian categories keeps exact sequence

Let $A$ be an abelian category and $B$ a subcategory, not necessary abelian. Let $C^\bullet$ be a exact complex in $A$ with $C^i\in B$. Suppose there is another abelian category $A'$ and $B'$ a ...
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Is this a correct way to think about specific examples of groups using the category theory definition?

I'll say now, before anything else, that I probably don't know what I'm talking about. This is more me making a (hopefully) educated guess about a topic I'm not too familiar with. I recently started ...
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66 views

Does the product functor preserve quotient maps?

In Hatcher's Algebraic Topology, he presents a proof that if $(X,A)$ satisfies the homotopy extension property, and $A$ is contractible, then $X \simeq X/A$. Part of Hatcher's proof goes: Suppose ...
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75 views

Prove the isomorphism of categories $Fun(\mathcal{A}\times\mathcal{B},\mathcal{C})\cong Fun(\mathcal{A},Fun(\mathcal{B},\mathcal{C})),$

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
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2answers
39 views

Fiber product of groups

Let $f: G \longrightarrow K$, $g: H \longrightarrow K$ be group homomorphisms. For the fiber product $G \times_K H$ to exist, is it necessary to require that $f$ and $g$ be surjective? I can't see ...
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Proving that disjoint unions of presentations are coproducts of groups

I'm working through Aluffi's Algrebra: Chapter 0 and I need some assistance with an excercise. Aluffi, Ex. II.8.7 Let $(A|R)$, resp. $(A'|R')$, be a presentation for a group $G$ in Grp, resp. ...
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29 views

The coproduct of a family of objects of a Preorder (seen as a category)

If the coproduct of a family of objects of a Poset (seen as a category) is the least upper bound, who is the coproduct of a family of objects of a Preorder (seen as a category)? My intuition ...
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Universality of the Simplex Category. Proving Functoriality of the Map.

Let $B$ be a strict monoidal category, and $\left \langle c,\mu ',\eta ' \right \rangle$ a monoid in $B$. Now suppose we consider the simplex category $\left \langle \triangle ,+,0 \right \rangle$, ...
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108 views

What is the mathematical difference between group and category?

This question is quite similar to the following link: Why learn Category Theory in order to study Group Theory? The above link is nice but I could not find the difference mathematically between ...
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28 views

Simplicial function space and homotopy colimits

I am currently reading the book by Bousfield and Kan, in particular Ch. XII, par. 2, and would like to understand why the functor $hocolim: Top_{+}^{I} \rightarrow Top_{+}$ is left adjoint to $hom(I ...
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1answer
35 views

The pushout of an epimorphism is an epimorphism

I'm reading the book "Handbook of Categorical Algebra: Volume 1, Basic Category Theory" by Francis Borceux, and in page 52 he states that "..."the pullback of a monomorphism is a monomorphism". ...
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74 views

Unifying Connection Between Topological Embeddings and Quotient Maps

In a book on topology I'm reading the following theorem seemed striking to me, not for its proof, which I believe I understand, but because there's some nice symmetry going on that I'd perhaps like ...
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43 views

Is the linear operators must be invertible to from a category?

I am trying to understand the concept of category in mathematics. For example the following link talks about category $Lin$ which is an Abelian category. ...
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Question about homomorphisms

I have a question that asks the following: Let $S,*$ and $T,.$ be binary structures and let the there be a homomorphism betweeen the two. If this is surjective, then if $S$ is a group, so is $T$. ...
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57 views

Exact functors preserve free modules?

Let $R$ be a principal ideal domain. Suppose that $F$ be an exact functor from the category of $R$-modules to itself. If $M$ is a free $R$-module, is $F(M)$ still a free $R$-module? I do not know how ...
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Homology functors defined on $\mathsf{Top} \times \mathsf{Top}$ in Eilenberg-Steenrod axioms?

The Eilenberg-Steenrod axioms state that a homology theory is a sequence of functors $H_n : \mathsf{Top} \times \mathsf{Top} \to \mathsf{Ab}$ satisfying some additional properties. What I don't ...
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280 views

An introduction to algebraic topology from the categorical point of view

I'm looking for a modern algebraic topology textbook from a categorical point of view. Basically, I'd like a textbook that uses the language of functors, natural transformations, adjunctions, etc. ...
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Relationship between differential cohesion and synthetic differential geometry

I was wondering what is the relationship between differential cohesion and synthetic differential geometry? I know the basics of synthetic differential geometry from Kock's text, but I am not ...
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62 views

Associative Law for a Monoid in a Monoidal Category.

Let $B$ be a monoidal category, and $c$ a monoid in $B$. Powers of $c$ are defined by taking $c^{n}$ to be the $\otimes $-string of length $n$ of $c$ in which the parentheses are all in front. We ...
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50 views

Generalize a result to any category.

Consider two categories $\mathscr{C}$ and $\mathscr{D}$ where $\mathscr{C}= Grp$ and $\mathscr{D}= \textbf{Set}$, then we are taking the forgetful and faithful functor $p$ (this is, we have a group ...
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Why is the the double dual functor on finite-dimensional vector spaces naturally isomorphic to the identity?

$\require{AMScd}$ Note: I have already seen this question, which asks about a specific aspect of the construction - here I am trying to construct this functor and failing at a very different stage. ...
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How to show the homotopy category is not abelian [duplicate]

Suppose $K^+(M)$ is the category, whose objects are bounded below complex of abelian groups, morphisms are chain maps modulo homotopy equivalence. How to show the category is not abelian? Exercise ...
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Representation of functions in the simplex category

I am using the following characterization of the simplex category $\Delta$: The objects are finite ordinals and the arrows are weakly monotone functions $f:n\rightarrow m$. $0$ is initial and $1$ is ...
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Pro-completion of finite algebras as Stone algebras

Recall that a profinite algebra (e.g. group, monoid, or whatsoever) is a cofiltered/inverse limit of finite algebra. In Johnstone's Stone space, he showed that finite discrete algebras are finitely ...
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Equivalence between category of $R$-modules and $S$-modules

Is it true that by equivalence between category of $R$-modules and $S$-modules we conclude that $R$ and $S$ are isomorphic?($R$ and $S$ are unital rings.)
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106 views

Yoneda Lemma for 2-categories - lax version

Is there a sort of lax Yoneda Lemma for 2-categories? Here is what I seem to have proven (although I have not checked all the details): If $\mathcal{C}$ is a (weak) 2-category, $A$ is an object of ...
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Is it possible to develop differential geometry without points?

I read about pointless topology and locale theory, and become curious about this topic. For example, there is the concept "differential manifold" corresponds to "topological manifold". As this, are ...
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Heyting algebras and infinite distributive law

I want to prove that "a complete lattice satisfies the infinite distributive law $a\wedge(\vee{S})=\vee\{a\wedge s|s\in S\}$ iff it is a Heyting algebra". I proved "if" part but can't "only if" part. ...
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Is $\coprod \subseteq \prod$ true in any (complete cocomplete) Abelian category?

Consider $A_i, \; i \in I$ a collection of objects in an Abelian category with arbitrary products and coproducts $\mathcal{C}$. Is there always a functorial monomorphism $\coprod_{i}A_i ...
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Why are morphisms of monads lax and not oplax natural transformations

Monads in a bicategory $\mathscr B$ correspond to lax functors $* → \mathscr B$, so one expects morphisms of monads should correspond to nautral transformations between them. A natural ...
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left adjointable functors

When a functor $F$ is left adjoint to some functor $G$, then one usually says that "$F$ is a left adjoint". Is this grammatically correct? Wouldn't it be more accurate to say that "$F$ is right ...
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What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
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Is $\Bbb R$ the soberification of $\mathbb{Q}$?

I'm a beginner. I read about soberification of topological space and thought that if I soberificate $\mathbb{Q}$, for any $x \in \mathbb{R}$, the neighbourhood filter of $x$ in $\mathbb{Q}$ ...
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Proving that the free group on two generators is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$

I want to show that the free group on two elements $F(\{x,y\})$ is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$. The idea is to use the universal property of free groups to prove that ...
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Is any reflector from a presheaf category $PSh(K)$ to a topos $C$ necessarily left exact?

Let $C$ be a topos, $K$ a small category and $$ PSh(K) \leftrightarrows C $$ a reflective subcategory with inclusion $i\colon C\hookrightarrow PSh(K)$ and reflector $T$. Is $T$ left exact? $D$ ...
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Construction of the “swap map” via universal property for products in $Set$

Lemma (Universal Property for Products) let $X,Y$ be sets and let $A$ be a set along with functions $f:A\rightarrow{X}$ and $g:A\rightarrow{Y}$ then there exists a unique function ...
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Endomorphic Function Definition

I need to confirm my thinking on endomorphic functions. Since an endomorphism is just a surjective morphism on an object to itself in a category, can I alter the usual definition of a surjective ...
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What is identity arrow in the category Set? [duplicate]

Given is category Set Given two objects from this category, $A$, and $B$, which are sets without any other structure, there is an arrow $f: A \to B$, from $A$ to $B$, which is any total function from ...
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Is the maximum cardinality of a hom-set $2$? ($\emptyset$ and $1$)

After reading this: A set of morphisms from object $a$ to object $b$ in a category $C$ is called a hom-set and is written as $C(a, b)$ (or, sometimes, Hom$C(a, b)$). So every hom-set in a ...
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Why is it an equivalent definition of a triangulated full subcategory?

We know that an additive full subcategory S of a triangulated category T is called a triangulated subcategory if it is closed under isomorphism, shift and if any two objects in a distinguished ...