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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51 views

Is the existence of finite biproducts a strengthening of commutativity?

Let $T$ denote a monosorted Lawvere theory (call its distinguished object $G$) equipped with a distinguished constant $0 : G \leftarrow 1$ that is "idempotent" in the following sense: for all arrows ...
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31 views

Applications of Splitting Lemma and Exactness

I'm looking for nice applications of exact sequences, the splitting lemma, and exact functors in algebra and topology (i.e not using the five lemma to get long sequences in homology etc..). For ...
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1answer
50 views

If there monomorphisms $f : A \rightarrow B$ and $g : B \rightarrow A$ then there is an isomorphism $h : A \rightarrow B$

Consider the following set theoretical result of Schröder-Bernstein-Cantor: Let $A$, $B$ be sets. Assume we have injections $f : A \rightarrow B$ and $g : B \rightarrow A$. Then there exists a ...
2
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1answer
28 views

Borceux - Snake Lemma Question

Below is the statement of the snake lemma from Borceux. My question is which squares are (1) and (2) referring to?
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83 views

Equivalent definition of Schemes

I recall seeing that the category of schemes can be captured by a general construction as follows. Let $\mathbf{Spec}\colon \mathbf{CRing}^{op}\to \mathbf{LRS}$ be the usual functor from the ...
2
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1answer
48 views

Equivalent dfn of Filtered Categories

Let $\mathbb I$ be a small category (i.e., its class of arrows is a set) which satisfies the following: (1) it is nonempty, (2) for each $i,i'\in \mathbb I$ there exists $i''\in \mathbb I$ and ...
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2answers
71 views

pushout of topological Hausdorff spaces is not Hausdorff

$A$, $X$, $Y$ are topological Hausdorff spaces, $f:A\to X$, $g:A\to Y$ continuous maps. I search an example where the pushout $Z$ of the morphisms $f$ and $g$ is not Hausdorff. I thought if I take ...
4
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1answer
29 views

Are units in rigid (autonomous) categories some sort of natural transformation?

In a rigid category $\mathcal{C}$, let us choose left and right duals and left and right (co)units for every object. This gives us, for example, a dualisation functor $-^*:\mathcal{C} \to ...
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1answer
55 views

Limit of groups

My question concerns the properties of special limits of groups. Let $G'$ and $G''$ be two small groups. Suppose that the following diagram in the category $\mathfrak{Grp}$ of small groups and ...
4
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1answer
71 views

Wedge product of a direct sum and the Yoneda Lemma

In a comment to http://math.stackexchange.com/a/344851/58601, Martin Brandenburg suggests that one may prove the existence of the canonical isomorphism $\wedge^n(W_1 \oplus W_2) \to \bigoplus_{p+q=n} ...
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2answers
35 views

Kernel of an arrow that factors through a monic?

Suppose an arrow $A\overset{f}{\rightarrow}B$ factors as $A\overset{q}{\rightarrow} J \overset{j}{\rightarrowtail}B$. When does $\ker f=\ker q$ and how can I prove it?
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43 views

Open coverings and (co)limits

My question concerns general topology and category theory. Let $X$ be a topological space, and consider an open covering $\{U_{i}\}$ of $X$. Is it possible to view $X$ as a (co)limit of the ...
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2answers
43 views

Show that function $f: A \to B$ is surjective when there is an implication: $g \circ f = h\circ f \to g=h$ [closed]

Let $f: A \to B$. How can I show that $f$ is surjective if and only if (for every $C$ and every pair of functions $g, h: B \to C$) when there is the following implication? $$ g \circ f = h\circ f \to ...
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0answers
43 views

Quillen groupoid of a groupoid.

For any category $\mathcal{C}$ we can define its Quillen's groupoid, denoted $\mathcal{Q}(\mathcal{C})$, as the category which have the same objects than $\mathcal{C}$ and the arrows between two ...
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2answers
50 views

Linear maps on tensor products

Short question. Suppose we have vector spaces $V_1,V_2,V_3,V_4$ and a linear map $f: V_1\otimes V_2 \to V_3 \otimes V_4$. Are there always linear maps $f_1: V_1 \to V_3$ and $f_2: V_2 \to V_4$, such ...
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0answers
23 views

Does the 2-functor $PsAlg\to \mathfrak{X} $ reflect equivalences?

Consider a $2$-monad $ T: \mathfrak{X}\to \mathfrak{X} $ and consider its 2-category of pseudoalgebras $PsAlg$. There is a forgetful functor $ U: PsAlg\to \mathfrak{X} $. Does this forgetful functor ...
2
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0answers
74 views

Categorical proof that subgroups of free groups are free?

Is there a categorical proof that the subgroups of free groups are free? Also that abelian subgroups of free abelian groups are free.
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34 views

Coequalizers in categories of relations and partial functions

Let $\mathbf{Rel}$ be the category whose morphisms are triples $(X, R, Y)$ where $X$ and $Y$ are sets and $R$ is a relation between $X$ and $Y$ and let $\mathbf{PFun}$ be the subcategory of ...
2
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1answer
55 views

Group Duality with respect to Generators and Relations

Although the following question is not phrased in the most accurate way, I would like to ask it in the same way it rushed to my mind: "Looking at some basic examples of group theory with geometrical ...
3
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0answers
41 views

What kind of objects are both subobjects and quotients?

Fix an object $B$ in some category. What does the existence of a diagram $A \rightarrowtail B \twoheadrightarrow A$ imply about $A$ and $B$? What if $A \rightarrowtail B \twoheadrightarrow ...
5
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1answer
51 views

What can we say if $A\twoheadrightarrow B$ and $A \rightarrowtail B$?

In some category, suppose there are two objects $A$ and $B$ such that the arrow-class $\mathsf{Hom}(A,B)$ has a monic $A \rightarrowtail B$ and an epic $A\twoheadrightarrow B$. Can we say anything ...
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32 views

A hierarchy of arrows by “monicity/epicity”?

In some familiar categories, we can think of the kernel $\ker f$ of an arrow $A\overset{f}{\rightarrow}B$ as a subset $\mathrm{Ker} f$ of its domain. In these cases, we usually think of the kernel as ...
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3answers
501 views

Group Theory via Category Theory

I have previously done a course on group theory and now I am doing a reading course on category theory. So as an interesting exercise I have been asked to write an exposition of group theory for ...
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2answers
31 views

Definition of codiagonal in a category

I'm confused by the definition of a codiagonal in a category with coproducts. The definition on nLab is as follows. Let $\mathcal{C}$ be a category with coproducts and let $X \in \mathcal{C}$. Then ...
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2answers
171 views

Colimits glue. What do limits do?

The informal but very useful way to think of colimits is as 'gluing things'. This intuitive view is very helpful to me, and I'd like one for limits aswell, but I haven't stumbled opon such a thing ...
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0answers
112 views

Tensor Product Construction, Solution Set Condition.

I am developing the basic properties of tensors, using categories. Let $R$ be a commutative ring. Fix $A, B\in R-Mod$ and define $K:R-Mod\rightarrow Set$ by $KC=\left \{ \beta :\left | A \right ...
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votes
1answer
38 views

Does a bijective homomorphism in Category theory have to be a bijection of arrows as well as objects?

In giving an example of two non-isomorphic posets related by a bijective homomorphism the following was proposed. $A=\{a,b,c\}$ with $a\leq b$, $a\leq c$ and $b$ and $c$ not comparable. ...
0
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1answer
47 views

Are Boo and BooRng really isomorphic?

In ACC on the top of page 34, I read: The construct Boo of boolean algebras is isomorphic to the construct BooRng of boolean rings and ring homomorphisms. This contradicts my intuition since ...
1
vote
1answer
36 views

Quotient objects as constructions from subobjects?

A quotient object of an object $A$ is usually denoted $A/B$ (we're talking about equivalence classes of epis). It seems that in categories like $\mathsf {Grp}$ and $\mathsf {Ab}$ one can associate ...
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0answers
48 views

An equivalence of categories that is not adjoint.

Is there a good example of an equivalence of categories $F:\mathcal{A}\to\mathcal{B}$ such that $F$ is neither left nor right adjoint to its inverse $G:\mathcal{B}\to\mathcal{A}$?
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1answer
52 views

A functor that has both left and right adjoints

What can we say about a functor that has both left and right adjoints? I vaguely recall hearing that it is then an equivalence of category. Is it true? If not, then under what conditions it is true? ...
2
votes
2answers
185 views

Construction of Yoneda extension

In "Category Theory" by Steve Awodey, there is a suggestion for the reader to construct a left adjoint in the proof of the UMP of the Yoneda embedding. Namely, for any small category $\mathbb{C}$, the ...
5
votes
1answer
62 views

Proving the 'letters' of a free group generate the group

A group $F$ is free over a set $X$ if there exists an injection $\sigma: X \to F$ such that for any function $\alpha: X \to G$ to any group $G$ there exists a unique homomorphism $\phi : F \to G$ such ...
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97 views

Is there an accepted term for those objects of a category $X$ such that for all $Y$, there is at most one arrow $X \rightarrow Y$?

In category theory, I have seen "weakly initial object" used as follows: $X$ is weakly initial iff for all objects $Y,$ there is at least one arrow $X \rightarrow Y$. Of course, another way of ...
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0answers
85 views

Computing $f^{*}\mathscr{O}_X$ directly via colimit

I want to prove that, for affine schemes $X = \text{Spec} (A)$, $Y = \text{Spec} (B)$ and $f: Y \rightarrow X$ morphism of schemes ($\varphi: A \rightarrow B$), $f^{*}\mathscr{O}_X \cong ...
0
votes
1answer
53 views

Rel is a concrete category over Sets, but how to concretize that?

The traditional denotation of a structured set object is something like $(1)\qquad(X,\mathcal S_1,\dots\mathcal S_n)$ for some "structures" $\mathcal S_1,\dots\mathcal S_n$ on a set X. The modern ...
4
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1answer
75 views

Faithful functors from Rel, the category of sets and relations?

Are there examples of faithful functors $F:\mathbf{Rel}\to \mathbf C$, where C is a concrete category over Set? Or can it be proved that such functors don't exists?
4
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1answer
78 views

Why are there only limits and colimits?

Part of my intuition about the construction of limits and colimits is based on the idea that they are initial and terminal objects in the appropriate category: The limit of a diagram $D$ is of course ...
3
votes
1answer
43 views

“Any epi into a projective object clearly splits”

I am reading Category theory by Steve Awodey. So that we are all on the same page regarding the definitions I am using, I will repeat the definitions described in the book (starting on pg. 28): ...
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24 views

Difficulties with right-adjoint-right-inverse in $\mathsf{Top}$

I'm having difficulties with section $9$ of chapter $\mathrm V$ of CWM. There is the following proposition: Proposition 1. If $G:\mathsf C\rightarrow \mathsf D$ is a faithful functor, if $\mathsf ...
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1answer
30 views

Compact objects and locally finitely presentable categories (the Category of Groups)

I am trying to understand the concept of locally finitely presentable categories. I have discovered the concept of compact object here. I have discovered that for groups, the finitely presented ...
4
votes
2answers
188 views

What's the explicit categorical relation between a linear transformation and its matrix representation?

There several questions about linear transformations and its respective matrices in some basis, but I'm particularly interested in the explicit definition of this relation in the category $Vect$ (of ...
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2answers
50 views

inverse system vs inverse sequence

I am wondering about such problem. Let $\{X_i,\phi_{ij},I\}$ be an inverse system, where the directed set $I$ has such property that there exists a sequence $i_1 \leq i_2\leq\cdots\subset I$ such that ...
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1answer
60 views

nicer proof of basic functor category fact?

In functor categories, there's a nice isomorphism ${\mathcal C}^{\mathcal A \times \mathcal B} \cong ({\mathcal C}^{\mathcal B})^{\mathcal A}$. Proving this is a good exercise. It's not exactly hard, ...
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1answer
69 views

Intuition for the Discrete$\dashv$Forgetful$\dashv$Indiscrete Adjunction in $\mathsf{Top}$

Section 9 of CWM's chapter on limits beings by introducing the adjunctions $D\dashv U \dashv I$ where $D$ is the forgetful functor, $D$ equips sets with the discrete topology, and $I$ equips sets with ...
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1answer
71 views

Boolean algebra gives rise to a ring

There is a well-known result that every boolean algebra $(R,0,1,\land,\lor,\neg)$ can be made a boolean ring by defining $$x\cdot y=x\land y\qquad\text{and}\qquad x+y=(x\land\neg y)\lor(y\land\neg ...
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1answer
80 views

Common conditions on functions to be morphisms. [closed]

When coming in contact with the concept of morphism one may start to wonder what makes different structured objects of the same kind to be similar in a "morphical" way. At least I did. Below ...
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1answer
47 views

Explicitly describe colimits in $\mathsf{Set}$

I just started learning category theory a couple months ago. In my understanding, there is a nice fact about the category of sets that one can explicitly describe limits. If $F:J \to \mathsf{Set}$ is ...
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68 views

Categorical description of Alexander horned sphere?

Looking at the construction of the Alexander horned sphere: Is it possible to describe as some kind of direct limit of spaces? How can one formally see this?
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38 views

Direct way to show that 2-out-of-6 holds for weak equivalences in a model category?

In a model category, when weak equivalences are inverted, nothing else gets inverted. It follows that weak equivalences satsify 2-out-of-6. But the first sentence takes some work to show. Is there a ...