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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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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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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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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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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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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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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69 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 ...
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60 views

sheaves of rings and maps to classifying topos

Let $R$ be the category of finitely presented commutative rings (but I don't know how necessary the hypothesis of finite presentation is for my question). Let $Set^R=Fun(R, Set)$ be the category of ...
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107 views

Categorical description of equivalence relation generated by a relation?

The notion of an equivalence relation generated by a relation is widespread and useful. How can one categorically describe it?
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154 views

Book for Algebraic Topology- Spanier vs Tom Dieck

A number of times, questions have been asked on this website about good books on Algebraic Topology and the responses have been very valuable. However I need some more specific advice in this matter. ...
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104 views

Universal object

I'm trying to define the concept of universal object in category theory, but I failed to find a universal way to do it. Maybe someone can help ? THe idea is quite simple. Let $\mathcal{C}$ be a ...
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Is there a universal mapping property satisfied by the disjoint union of totally ordered sets?

It is proved in another post that the product and coproduct do not exist in the category of totally ordered sets (except in some trivial cases). (In this post I will only consider the category TOrd, ...
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35 views

Yoneda Embedding into Left Exact Functors

I think I am very confused about something. I've been reading a bit about the Mitchell embedding theorem, and I read that the proof first embeds a given small abelian category $\mathscr{A}$ into the ...
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64 views

Functors Between Functor Categories

I'm currently working through Tom Leinster's Basic Category Theory, and I have searched the internet fruitlessly for examples of functors between functor categories. I haven't yet come up with any ...
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Structural / design / meta optimization - is there mathematical theory. Optimization over categories?

There is huge branch of mathematical optimization theory, but it mostly considers the finding optimal parameter values for the predefined structures. There are variational calculus and optimal control ...
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48 views

Relating morphisms between two objects in category of finite sets and relations

In category of finite Sets and Relations (let's call it FinRel), suppose $A$ and $B$ are two objects (sets), and $b1,b2,b3$ are some morphisms (relations) from $A$ to $B$. Interpreting $b1 .. b3$ as ...
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56 views

Functors that preserve the subset relation on constructs

Is there a name for functors between concrete categories over Set that preserve '$\subset$'? Example, for a functor $F:\mathbf {Top}\to \mathbf {Grp}$, if $Y$ is a subspace of $X$ then $F(Y)$ ...
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26 views

Biproduct in category

Let $(X\oplus X', \pi_X, \pi_{X'}, \iota_X,\iota_{X'})$ and $(Y \oplus Y', p_{Y'},p_Y, j_Y,j_{Y'})$ biproducts in a category $\mathcal{C}$. In MacLane's book, he defines using the structure of ...
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Categorical Pasting Lemma

If I'm not mistaken, the pasting lemma for a two-element open cover $X=A\cup B$ of a topological space is equivalent to saying that the following square is a pushout in $\mathsf{Top}$: ...
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29 views

Associativity of (co)Products

I have proved the following: Let $I=\bigcup _{k\in K}J_k$ be a partition of a set $I$. Consider of famility $(A_i)_{i\in I}$ of objects in a category $\mathsf C$. When all the products involved ...
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34 views

What are coquantales?

A quantale can be defined as a monoid in the monoidal category $\mathbf{Sup}$ of complete join semilattices, equipped with tensor product. What are examples of non-diagonal comonoids in ...
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99 views

Categorical equivalents of Set theory concepts

Update: I am updating my question to be more precise. I am studding Category of finite sets and functions (FinSet). I am aware that for some of the concepts in Set Theory there are well-known ...
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Coequalizers are quotient maps

Is it possible to show that every coequalizer in the category of Hausdorff spaces is a quotient map directly from the universal property of a coequalizer and without use of the set-theoretical ...
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79 views

Query in the definion of abelian category

I am studying the definition of abelian category..Definition says it is a additive category with a)every morphism in category has kernel and co-kernel. b)every monomorphism in the category is the ...
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50 views

The $n$Lab on Currying

First of all, I have absolutely no knowledge in computer science. I am reading this in context with category theory, in particular the general tensor-hom adjunction. Suppose we're living in $\mathsf ...
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30 views

What is the pullback of a central extension?

Suppose we have three objects $A,B,C$ of an (abelian) category $\mathbf{C}$ and a short exact sequence $ 0\to A \to B \to C \to 0 $ such that $B$ is a central extension of $C$ by $A$ ($im(A\to ...
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Understanding the (categorical) Calculus of Fractions

Given a category $\mathsf C$ and a class of arrows $\Sigma$, we say the category of fractions $\mathsf C [\Sigma ^{-1}]$ exists when there's a functor $\varphi :\mathsf C \rightarrow \mathsf C ...
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127 views

Question on adjoint functors

Can someone provide me an enlightenment on the following three statements? (I stumbled on them at the part dealing injective modules in a text of homological algebra.) 1) Let $F \dashv G \colon ...
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Is any morphism with at most one right inverse a mono?

My question is the following: Let $f\colon B \to C$ and suppose that $f$ has at most one right inverse, i.e. if $f\, g = \mathrm{id}_C$ and $f\, h = \mathrm{id}_C$, then $g = h$. Prove that $f$ is ...
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51 views

A question about the definition of additive categories.

What does the following line mean The composition of morphisms is distributive over addition. Does it mean $a+(b\circ c)=(a+b)\circ (a+c)$?
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Why is $S^{-1}A$ the initial object in this category?

Verify that $S^{-1}A$ satisfies the following universal property: $S^{-1}A$ is initial among $A$-algebras $B$ where every element of $S$ is sent to an invertible element in $B$. Won't ...
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What is a “category”? [closed]

I have tried to understand category theory for a while but never been able to get it. I finally found a text I like called Category Theory for Scientists. However, I think it would be easier to read ...
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31 views

Functorial Properties Preserved by Natural Transformation

This question was born from (and is in a sense a continuation of) this one, about functorial properties preserved by natural isomorphisms. What functorial properties are preserved by natural ...
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Functorial Properties Preserved by Natural Isomorphism

Conceptually, functors which are naturally isomorphic should have the same functorial properties e.g exactness, (co)continuity, etc. Thus, ideally, I'd hope for a precise definition of a functorial ...
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91 views

Tensor-Hom Adjunction In Monoidal Categories?

Is there a generalization of the tensor-hom adjunction to monoidal categories, or is it a special property of $\mathsf{Mod}$-$R$?
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1answer
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Adjunctions via Reflections and the Axiom of Choice

I have met two ways of defining adjunctions: via the triangle identities, and via reflections. Proposition 3.1.2 Let $F:\mathsf A \rightarrow \mathsf B$ be a functor and $B$ an object of $\mathsf ...
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Non-Universal Delta Functors

Recall a delta functor on an abelian category is a collection of functors $H^n$, $n \ge 0$ and connecting homomorphisms associated to each short exact sequence so that we get a corresponding long ...
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1answer
34 views

How to calculate number of elements in HomSet

Im giving category theory a chance but have very limited math background, I'm learning from the book "Category theory for the sciences" but got lost on page 16 :) Exercise 2.1.2.12. Let ...
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1answer
45 views

Preferred way to write elements of the direct sum of vector spaces

Suppose $V$ and $W$ are vector spaces over the same field and $V\oplus W$ is their direct sum. Reading through the literature I found essentially two ways of writing elements of $V\oplus W$. 1.) We ...
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Motivation for Definition of Derived Category

On the $n$Lab entry about derived categories, I read the derived category of an abelian category $\mathsf A$ is the localization of $\mathsf{Ch}_\bullet (\mathsf A)$ at the quasi-isomorphisms. My ...
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48 views

Quick question: G-set functor

The Wikipedia page on Representable Functor says: A group G can be considered a category (even a groupoid) with one object which we denote by •. A functor from G to Set then corresponds to a ...
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102 views

Questions about a topological category

Given topological spaces $(X_i,\tau_i)$ with sets $\mathbf S_i=\{\mathcal A\in \tau_i^2|\mathcal A\supseteq\Delta X_i^2\}$, where $\tau^2$ is the product topology and $\Delta$ is the diagonal. The ...
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inverse limit in the plane

What stuff can I say about inverse limits regarding the mapping of $[0, 1]$ onto $[0,1]$ given by $$f(x) = \left\{ \begin{array}{ll} 2x & \mbox{if } 0 \le x \le {1\over2}\\ 1 & \mbox{if ...
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1answer
51 views

Group objects in the category of rings

Are there group objects in: $\text{Ring}$ $\text{CRing}$ If so, why doesn't anyone talk about them? On the other hand, $$ \begin{align} cogroup \ objects \ in \ \text{CRing} &= co(group \ ...
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Functors that has a natural transformation from identity

Let $F:\mathcal{A}\to\mathcal{A}$ be a functor with a choice of $A\to F(A)$ for every $A\in \operatorname{Ob}\mathcal{A}$, such that $$\require{AMScd} \begin{CD} A @>{f}>> B\\ @VVV @VVV \\ ...
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1answer
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Simple question on diagrams in a category

Forgive the simplicity of my question but after running across the definition of a diagram in $C$ of shape $J$ as simply a functor $D:J\rightarrow{C}$ does this require $J$ to be a subcategory of $C$ ...
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1answer
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Product / Co-product in category of sets and relations (Rel) [closed]

What is the co-product and product in category of sets and relations. Thanks
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1answer
71 views

Continuity of Modified Hom Functor

I have been studying category theory and have been exploring hom functors. I've come across an interesting question and after spending several hours thinking about it, haven't gotten anywhere. Let ...
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A Tool to practice Categories / Allegories

Is there any handy tool to practice Categories / Allegories, in the sense that for a defined Category, it is possible to check the result of an operation application. For example, a tool which ...