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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What should this definition be?

This is from Advanced Linear Algebra by S.Roman. Definition (on page 356). Referring to the figure below, let A be a set and let $\mathcal S$ be a family of sets. Let $$ \mathcal F = \{g\colon ...
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50 views

Quotients vs equivalence relations

In a recent preprint I defined the category of quasi-frames (qframes for short) as follows: a qframe is a modular and upper continuous complete lattice; a morphism of qframes $f:L_1\to L_2$ is a map ...
4
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2answers
91 views

Objects that send only monomorphisms

I just re-learned that fields can have only 1-to-1 homomorphisms from them. Is this a common trait in other categories? Can we extend, for instance, many topological spaces to spaces that have only ...
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1answer
39 views

Is Module Category over Monoidal category Monoidal?

let $\mathcal{C}$ be a monoidal category and $\mathcal{M}$ a $\mathcal{C}$-module category. Does $\mathcal{M}$ need to be a monidal category? I know it is true for certain categories, but is it true ...
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26 views

Fibrations over topoi

Let $\mathcal{S}$ be an elementary topos. What is (exactly) the relation between $\mathcal{S}$-indexed categories and fibrations over $\mathcal{S}$? Where can I read about this? (Or even find the ...
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Are groups with all the same Hom sets already isomorphic?

I was thinking about the following: Say we got two groups $A$ and $B$, and we know that for any group C, there is a bijection $$Hom(A,C) \to Hom(B,C).$$ Are $A$ and $B$ already isomorphic? If the ...
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1answer
39 views

How to read “realize the mapping $x \cdot -: T \rightarrow T$”

This question is about Category theory for the sciences (by David Spivak). In Exercise 3.1.2.4-a the set $T = \{x \in \mathbb{R} \; | \; 0 \leq x < 12\}$ needs to be defined using a coequalizer. I ...
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2answers
98 views

Unique extendable functions… Is there a theory?

Motivation: Take a continuous function $f$ from a topological space $X$ to a hausdorff space $Y$. If $g$ is a continuous function from $X$ to $Y$ that coincides with $f$ in a dense set, then $g=f$. ...
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51 views

Monomorphism preservation by pullback

I'm working through Category theory for the sciences (by David Spivak) and I'm a bit stuck in section 2 - my problem is with Proposition 2.7.5.5 and Exercise 2.7.5.6, I hope someone can help :) The ...
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69 views

“Lossy” v.s. “Lossless” Monads

Let us look at two monads on $\bf Set$. The first will be the finite sequence monad (from the free forgetful adjunction with $\bf Mon$.) $$\eta(x)=[x]$$ $$\mu([[a,b, \dots, z],[\alpha,\beta, \dots, ...
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1answer
44 views

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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1answer
62 views

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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1answer
69 views

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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4answers
817 views

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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2answers
42 views

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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55 views

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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1answer
55 views

If there is in a category $\mathcal{A}$ finite products and equalizers then it has pullbacks

My homework consist in showing that "If there is in a category $\mathcal{A}$ finite coproducts and coequalizers then it has pushouts" based on the proof that "If there is in a category $\mathcal{A}$ ...
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19 views

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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1answer
65 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
59 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
39 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 ...
2
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2answers
51 views

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 ...
5
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1answer
101 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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1answer
95 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
47 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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2answers
61 views

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. ...
3
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1answer
33 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 ...
3
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1answer
59 views

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 $\left \langle \triangle ,+,0 \right \rangle$, ...
0
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1answer
126 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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34 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
40 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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1answer
95 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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45 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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2answers
104 views

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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1answer
71 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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1answer
71 views

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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4answers
328 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. ...
3
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37 views

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 ...
4
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1answer
66 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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1answer
55 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 ...
2
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1answer
39 views

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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0answers
25 views

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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33 views

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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1answer
65 views

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.)
3
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1answer
118 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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1answer
154 views

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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0answers
56 views

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. ...
4
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1answer
68 views

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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0answers
56 views

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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1answer
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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 ...