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Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Most mathematical structures can serve as objects of a category, with structure morphisms as arrows. 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 theory, too.

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A coend in the category of vector spaces

Let $Vect_k$ denote the category of (not necessarily finite-dimensional) $k$-vector spaces. Clearly, this category is closed symmetric monoidal with internal hom $[X,Y]=Hom_k(X,Y)$. Is it true that ...
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Multivariate Conditional Entropy as a test of correlation between random variables

I use the word columns to mean the data from which a random variable can be estimated. It is a sample of a random variable. I am working with $N$ columns of weakly correlated data. Furthermore, I ...
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Why is a slice category a category?

I don't understand why the Hom-sets in a slice category are disjoint. Let $C$ be a category and $A$ an object of $C$, then the slice category $C_{A}$ has as objects all morphisms $f\colon Z \to A $ ...
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Two questions regarding the convention concerning concrete categories

Definition of Concrete Categories Let $\mathbf{X}$ be a category. A concrete category over $\mathbf{X}$ is a pair $(\mathbf{A}, U)$, where $\mathbf{A}$ is a category and $U : \mathbf{A} \to \...
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Can you express this easy theorem in fancy categorical terms?

Here is a theorem (of homological algebra): Given $A \rightarrow B \rightarrow C$ in an abelian category $\mathcal{A}$. If for all $D \in \mathcal{A}$ we have that $Hom(D,A) \rightarrow Hom(...
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Proof verification: An arrow which is monic under a faithful functor is itself monic

Context: To introduce some symbols and such, what I'm seeking to prove is this: Let $F$ be a faithful functor. Suppose $F(f)$ is a monic arrow. Show $f$ is monic. This came up as part of a class ...
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Distinction between a “strictly typed function” and a “not strictly typed function”?

Let $f$ be the identity function for the real numbers. In the vernacular, we'd say that $f$ is a function from reals to reals, or that $f:\mathbb{R}\to \mathbb{R}$. Let $g$ be the inclusion map ...
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Stuck with Category Theory notation. What is the meaning of 'Corner brackets' 「 」?

While reading an article, I encountered this expression. Expression I was wondering if anyone knows what does the corner brackets 「(upper) and 」(down) in this expression do? Thank you.
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category theory terminology: “divergent objects”?

Suppose $\mathcal C$ is a category, $A,B$ are objects of $\mathcal C$, and there do not exist $X,f,g \in \mathcal C$ such that $f : A \to X$, $g : B \to X$. Is there a word to describe a pair $A,B$ ...
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Category of Abelian Groups: Limits

Let the category of Abelian Groups. I know that product and coproduct of a finite number of objects are the same in this category. Then, it follows that the projective and injective limits of finite ...
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Filtered colimits in the category of sets: an equivalence relation

A category $\mathsf{I}$ is filtered if it is nonempty, for any $i,j \in \mathsf{I}$ there is $k \in \mathsf{I}$ and morphisms $f\colon i\to k$ and $g\colon j\to k$, for any pair $f,g\colon ...
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General colimits and filtered colimits in the category of sets

A category $\mathsf{I}$ is filtered if $\mathsf{Ob(I)} \neq \varnothing$, for any $i,j \in \mathsf{Ob(I)}$ there is $k \in \mathsf{Ob(I)}$ and morphisms $f\colon i\to k$ and $g\colon j\to k$...
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What does it mean for two morphisms with different sources and targets to be isomorphic?

The standard definition of a subobject relies on the following definition: we call two morphisms $f : X \rightarrow A, g : Y \rightarrow A$ with the same target isomorphic if there exists an ...
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Finding a colimit in the category of presheaves

I have the following problem as part of my exam preporation and I need an idea how to approach it at least: Let $\mathfrak C$ be a small category and $F: \mathfrak C^{opp} \rightarrow \mathfrak S ...
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Why the empty set isn't a terminal object in the $\mathcal{SET}$ as well as it is an initial object?

It is said that the $\{\}$ is the initial object of the $\mathcal{SET}$ - category of sets and functions. By definition, it implies that for all $S \in Obj(\mathcal{SET})$ there is exactly one ...
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Can I define a functor F and a “ΔF” of sorts, which will uniquely determine a new functor?

Let $F: \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Natural transformation between $F$ and some other functor is defined as an assignment of a morphism in $\mathcal{D}$ to each object in $\...
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Exercise about map object (“Conceptual Mathematics Second Edition”, p.315, Article V Map objects, Exercise 6)

I feel difficult to show the existence of a map $\gamma$ which corresponds to the composition of two maps. $\gamma$ is the following map. $$\gamma: B^A \times C^B \to C^A$$ And I want to show that ...
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group objects in the category of groups and Eckmann–Hilton argument

Let $\mathscr C$ be the category of groups and $G\in \mathscr C$. Let $$m:G\times G \to G,e:1\to G, i:G\to G$$ be associated morphisms satisfying the commutativity diagrams. I wonder whether it is ...
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Books/Lecture notes about 2-categories.

Are there good books or lecture notes just about 2-categories?(not about higher categories nor $\infty$-categories) (I'm studying fibered categories for the descent theory of quasi-coherent sheaves. ...
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The coherence theorem for monoidal categories

I am reading the coherence theorem of monoidal categories. However I am confused by the following paragraph on page 165 of the book "Categories for the working mathmatician" $\bf{Here \ are \ my \ ...
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Example of a finite category without inverse or injective limit

I recently started studying categories theory and i need help to understand the concept of limit. Please, tell me an example of a finite category without inverse or injective limit and why. Thanks.
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Importance of local smallness when checking functor diagrams?

I have a question concerning the importance or non-importance of assuming local smallness for a category $\mathcal{C}$. There are two procedures I have done while working through an exercise that ...
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Is this a 'relative' colimit or some other categorical construction?

Let $\mathcal{A}$ be a closed covering of some space $X$, then let $\Sigma[\mathcal{A}]$ be the category of intersections of subsets of $\mathcal{A}$, further let $F: \Sigma[\mathcal{A}] \to \mathsf{...
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Functor category of the module category $\mathrm{Vect}(\mathbb{C})$

My current exercise is the following: Take $\mathcal{M}=\mathrm{Vect}(\mathbb{C})$ as a module category over $\mathcal{C}=G\text{-}\mathrm{Vect}(\mathbb{C})$, the $G$-graded vector spaces over $\...
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About a specific step in a proof of the fact that filtered colimits and finite limits commute in $\mathbf{Set}$

I'm currently working on the following theorem from Emily Riehl's Category Theory in Context: Theorem 3.8.9. Filtered colimits commute with finite limits in $\mathbf{Set}$. I understand most of ...
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Countable Completion is Isomorphic to Full Completion (Lang Algebra)

This is from Lang's $Algebra$, revised third edition, page 52. I will state my understanding of the problem. Suppose $G$ is a group, and $F$ is a family of normal subgroups of $G$, partially ordered ...
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Well-powered/subobject functor?

Is there a construction which maps each object in a category to the set of its subobjects? Concretely, I'm interested in mapping an object $M$ in the category of manifolds $\mathbf{Man^1}$ to the set ...
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A new category $C^*$ from a given category $C$

For a given category $C$ define the category $C^*$ as follows: the objects of $C^*$ are those of $C$; for given objects $u,v$, the $C^*$-morphisms $u\to v$ are all finite sequences $(a_1,\dots,a_n)$ ...
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Is there a category whose morphisms between A and B are diagrams of type “A -> B -> A”?

I'm reading about spans in Category theory, which denote categories with finite colimits whose morphisms are equivalence classes of diagrams "A <- X -> B". There's also their categorical dual, ...
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Fixing and pointwise-fixing of a structure in a suqare

Suppose we have this square in the context of Abstract Elementary Class ${\frak K}=(K,\leq_\frak K)$: $$ \require{AMScd} \begin{CD} N_1 @>f_1>> N_3 \\ @A{\preceq_\frak K}AA @AA{f_2}A\\ M_0 @&...
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Why products do not exist in the category of measurable spaces and probability kernels?

Consider a category whose objects are measurable spaces and morphisms are probability kernels (also called stochastic kernels or Markov kernels). That is, objects are pairs $(X,\mathbf{A})$ where $\...
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On the canonical map $\text{colim}_I \text{lim}_JH(i,j) \longrightarrow \text{lim}_J\text{colim}_IH(i,j)$

I'm working through a proof of the existence of a canonical mapping $$ \mu: \text{colim}_I \text{lim}_JH(i,j) \longrightarrow \text{lim}_J\text{colim}_IH(i,j) \tag{1} $$ induced by a cone $(\mu_i: \...
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Composition of monadic functors isn't monadic

Disclaimer: this question already has a solution here: Composition of monadic functors may not be monadic . However, I would like to understand how to solve this using another characterisation of ...
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Twisted arrow category - convention

I have found two slight different definitions of twisted arrow category. The part about the arrows in $Tw(C)$. In nCatLab it is something like given two objects $f:a\to b$ and $f':a'\to b'$ in $Tw(...
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The category of vector spaces over $\mathbb{R}$, Vect$_{\mathbb{R}}$, is equivalent to the category of $T$ algebras for some monad $T:$ Set $\to$ Set.

Prove that the category of vector spaces over $\mathbb{R}$, Vect$_{\mathbb{R}}$, is equivalent to the category of $T$ algebras for some monad $T:$ Set $\to$ Set. My attempt: First I know that ...
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What do Monic and epic morphisms imply?

If I were the person creating category theory, I wouldn’t have been interested in left and right cancellation. I may or may not notice it, but it doesn’t feel fundamental enough for me to define it ...
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Are equivalence classes of subobjects of some X in Set just equivalence classes of subsets of X with a specific cardinality?

I'm trying to understand what would be the subobjects of $\{0, 1\}$. Would they be $\{\emptyset, \{0\}, \{1\}, \{0, 1\} \}$? Or are $\{0\}$ and $\{1\}$ somehow identified together? Because I can map ...
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How does a category have to be connected

A multi question here: Could I have an isolated object in a category that is neither a source or target (identities excluded)? I have a suspicion that everything in a category has to be connected, and ...
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Showing that $F$ is not representable [closed]

As I'm trying to find (counter)examples of representable functors, I tried looking up some instructive examples. One of the counterexamples I'm having trouble with, is the following: Show that the ...
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Meaning of index of a multiplication symbol in a Cartesian product

I've just started to read about Category Theory. More precisely nLab, opposite category. I'm trying to understand the expression: $$ \circ_{C^{op}} : C_{\mathrm{mor}}^{op} {}_{s^{op}}\times_{t^{...
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Struggling to understand why $Hom(-, -)$ is a bifunctor and how exactly it suppose to map?

Seems like I do understand covariant the $Hom(A, -)$ and contravariant $Hom(-, A)$ functors. But when it comes down to the $Hom(-, -)$, then I am getting lost: 1) How does it work? What does it do ...
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Are Morphisms of a Category Order Isomorphisms?

Let the objects be all partially ordered sets $(S,\le)$ in a Category $\mathscr{C}$. A morphism $(S,\le) \to (T,\le)$ is a function $f: S \to T$ such that for $x,y \in S, x \le y \implies f(x) \le f(y)...
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The ampleness of canonical sheaves and the proof of “$X \simeq \mathrm{Proj}\left(\bigoplus_k H^0(X, \omega_X^k)\right)$”.

In Bondal and Orlov's paper, ''Reconstruction of a variety from the derived category and groups of autoequivalences'' (http://www.mi-ras.ru/~orlov/papers/Compositio2001.pdf), I think that they use the ...
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How to prove that $A^T y=0\implies y=0$ if and only if $A$ defines a surjective linear transformation?

I know that in vector spaces, the cokernel of a linear transformation $f$ is isomorphic to the set of all $y$ such that $f^T(y)=0.$ (This is the transpose of $f$.) Then, how can I prove that $\text{...
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Ternary function in axiomatization of category theory

I am interested in axiomatizing category theory (for example, the category of categories or category of sets just like Lawvere did). I went through some sources and everywhere I looked I saw that the ...
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quotients by extended and contracted ideals as tensor products?

Let $f:A\to B$ be a morphism of commutative rings and let $I\vartriangleleft A,J\vartriangleleft B$ be ideals. The left square below is a pushout. Question 1. When is the right square a pushout as ...
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What constructions of “elementary” mathematics are actually functors?

I'm not looking for the usual simple examples of functors like the fundamental group or forgetful functors, what I'm looking for is some interesting examples of constructions from "elementary" ...
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Why are categories and monads called that way?

The words "category" and "monad" existed already in philosophy. The usage of the terms in category theory seems to be slightly influenced by the philosophical meaning, but actually the concepts are by ...
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Resources for Algebraic Machine Learning

I have humble, around-undergraduate level understanding of mathematics. I enjoy abstract algebra and statistics the most. After stumbling upon Michael Izbicki's paper Algebraic classifiers, I decided ...
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Motives and representations

Assuming standard conjectures, the category of motives is equivalent to the category of representations of a certain group over $\mathbb Q$, but I don't understand the abstract construction. Now, if ...