# Tagged Questions

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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### 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 ...
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### 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 $A \rightarrowtail B \twoheadrightarrow A$...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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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 x)\,... 1answer 94 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 ... 1answer 62 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 ... 0answers 100 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? 1answer 106 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 ... 1answer 102 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 ... 2answers 156 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? 2answers 1k 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. ... 1answer 412 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 ... 1answer 164 views ### 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, ... 0answers 89 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 ... 1answer 145 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 ... 0answers 70 views ### 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 ... 1answer 70 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 ... 0answers 65 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) is a ... 1answer 39 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 product,... 2answers 120 views ### 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}:$$\begin{matrix}...
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 ...