Questions tagged [category-theory]
Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.
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Subcategory determined by composition series
Suppose $A$ is an artin algebra and take the category $\operatorname{mod}A$ of finitely generated $A$-modules. Consider the following construction. Let $M$ be an $A$-module. Since $A$ is artinian, $M$ ...
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What are some examples of injective sheaves?
Injective objects in the category of abelian groups are precisely the divisible groups. However, despite using injective resolutions of sheaves everywhere, I realized that I did not know a single ...
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Hom set between profinite groups can be written as limit of Hom sets
Problem:
Consider profinite groups $\mathcal{G}=\varprojlim_IG/U_i$ and $\mathcal{H}=\varprojlim_JH/V_j$, prove that
$$\operatorname{Hom}_\text c(\mathcal{G},\mathcal{H})\simeq\varinjlim_I\...
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What is the name for a matrix which is generated by a recursive sum whose form equals a recursive product when replacing the sums with products?
In this answer to the question "Do these series converge to logarithms?" it is shown by George Lowther that each Dirichlet series involving the pattern of divisors converge to $\log(n)$ in ...
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Injective object in category $\mathcal{K}$, which has an object $a$ and $\text{Hom}(a,a) = S $ is a semigroup with unit.
We consider the category $\mathcal{K}$ with an element $a$ and $\text{Hom}(a,a) = S$, where $S$ is a semigroup with unit.
Is $a$ an injective object of $\mathcal{K}$, if $S = \{ 1,\alpha, \alpha^2\}$ ...
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Tools to investigate unusual algebraic structure
I will begin with a mostly motivational thought about the projective plane. In this plane, every two lines intersect at a singular point. Let's mark the lines set as $\mathcal{L}$ and the points set ...
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Model Structure on Constant-free Symmetric Operads
I am currently trying to find a reference for the assertion that the category of positive / constant-free (meaning $\cal{O}(0)=\emptyset$ is the initial object) symmetric operads $\operatorname{Opd}_\...
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How many universal cones from $\Delta_{A\times A}$ to D?
I have drawn the picture like the following.
Let $C$ is a category (suppose $Set$), $I$ is a category with only two objects (1 and 2), $\Delta_{A\times A}$ and $D$ are constant functors from $I$ to C....
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Constructions in the Category of Vector Bundles
Generally I know that the Category of Vector Bundles is not a nice one. I know that we have the pullback bundle. Now if I have a map $f \colon X \to Y$ between topological spaces $X$ and $Y$ and a ...
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Showing functor for composition of morphism two variable functor with one fixed variable?
The following question is taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes
$\color{Green}{Background:}$
$\textbf{(1)}$ $\textbf{Definition:}$ A ...
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Questions about the definition of normal category and wording of an exercise.
The following is taken from 'Theory of Categories" by Barry Mitchell
$\color{Green}{Background:}$
$\textbf{(1) Definition:}$
If $\alpha;A'\to A$ is a monomorphism, we shall call $A'$ a $\textbf{...
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Comparing Vector Bundles
Does it make sense to compare two vector bundles $\pi_F\colon F \to X$ and $\pi_E\colon E \to X$ over the same space $X$ ? By inclusion for example? Or like elements in a Poset?
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Vector Bundles Categorically
Definition
I understand the first two points of this definition. However I need assistance in understanding further the sentance:
It is required that the topology of $E_x$ as a subspace of $E$ ...
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Epimorphism and Surjective Homomorphsim [duplicate]
Here is a question I came across recently:
If a morphsim in Grp (category of groups) is an epimorphism, then it is a surjective group homomorphsim.
I believe it boils down to show that for any group $...
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Other things with operators
There is the notion of "groups with operators" which signifies a group $G$ together with a morphism of sets $X \to \operatorname{End}(G)$.
It is easily observed that the endomorphism monoid ...
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Construction of left adjoint
A very standard example of left adjoints would be the functor $F:Sets\to Grps$ that maps sets to their free groups is a left adjoint to $G:Grps \to Sets$ which forgets the group structure. My question ...
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Definition of functor
This is a question addressed to those familiar with category theory, since it concerns the preferability of two similar definitions, which I suppose is revealed only by further experience. In sum, the ...
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The intersection of $\mathcal{O}_X$-submodules is an intersection as the categorical sense?
I have simple question in studying algebraic geometry.
First, next is the definiiton in the Gortz's Algebraic geometry book (p.174)
Let $\mathcal{F}$ be an $\mathcal{O}_X$-module and let $(\mathcal{F}...
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Minimally sufficient condition for composability of quiver morphisms?
Given three quivers $(X_0, X_1, \sigma, \tau)$, $(Y_0, Y_1, \phi, \psi)$, and $(Z_0, Z_1, \chi, \omega)$, and two morphisms $F := (F_0 : X_0 \to Y_0, F_1 : X_1 \to Y_1) : (X_0, X_1, \sigma, \tau) \to (...
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Applications of Category Theory in Abstract Algebra
Almost every text on Category theory uses categories such as Ab, Grp, and so on as examples to work with but can category theoretic methods actually help us understand the structures better? In ...
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Example about endomorphism ring of an indecomposable object, which have non-trivial idempotent element in an additive category.
Recently I read some about Krull-Remak-Schmidt category.
If $A$ is an additive category in which every idempotent splits, every object is the biproduct of finitely many indecomposable objects and the ...
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Notation related question about a two variable functor fixing one of its variables.
The following question is taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes
$\color{Green}{Background:}$
$\textbf{(1)}$ $\textbf{Definition:}$ A ...
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Different categories and isomorphisms of categories
When a mathematician says that two categories are the same thing, they may mean there is an equivalence or an isomorphism between them. I am wondering if there is a precise way we can say that two ...
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why universe is a set?
Recently, I study algebraic geometry, and the first time I doubt the axiom.
There exists a universe defined to be a set $U$ with these properties:
$x \in u \in U \Rightarrow x \in U$
$u \in U$ and $...
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What does it mean to induce a functor?
I have a text ("Sheaf Theory Through Examples") that just drops some weird notation and the concept of inducing a functor here:
"In other words, composing with $t$ induces a functor $\...
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Left-adjoint to Yoneda embedding
Let $C$ be locally small. Consider the Yoneda embedding $Y:C\rightarrow [C^{op},Set]$. Since limits in functor categories are computed pointwise and since the hom-functor preserves limits, the Yoneda ...
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Two definitions of categorical limits
For $C$ a locally small category, $J$ an essentially small category and $F\colon J\rightarrow C$ a functor, the limit of $F$, if it exists, can be defined as a representation of the functor
$\...
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How are not all objects in category sets?
Recently I saw a video talking about the Yoneda lemma from category theory being used in neuroscience. It was my first introduction to category theory. In category theory we have objects and maps ...
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What are the names for the following “anti-ideal”-like properties?
Let $C$ be a semigroup (or analogously a category). A family $A ⊆ C$ is called
*subsemigroup* if $a, b ∈ A \implies ab ∈ A$,
*left ideal* if $b ∈ A \implies ab ∈ A$ (analogously we define right ...
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How Free and Initial algebras relate?
I was wondering what is the precise relation between initial $F$-algebras and free $F$-algebras. These concepts seem very similar, as initial algebras can be used to define recursive constructions ...
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$M$ being the direct sum of submodules $(M_i)_{i \in I}$ is equivalent to a certain map (between $\operatorname{Hom}$-sets) being an isomorphism.
I am currently trying to prove a remark in Bosch: Algebraic Geometry and Commutative Algebra (chapter 1.4):
Consider a family $(M_i)_{i \in I}$ of submodules in $M$. Then the inclusion maps $\iota_i \...
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Commutative diagrams in Opposite Category
I am working on the kernels and co-kernels of morphisms in categories with zero objects. I came across the following: $(C,j:B\to C)$ is a cokernel of $f\in \text{Hom}_{\mathcal{C}}(A,B)$ $\iff$ $(C,j^{...
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When to use definition or theorem concerning showing existence of left/right adjoint.
The following are taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes.
$\color{Green}{Background:}$
$\textbf{Definition for “free over an object with ...
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Generalisation of "discrete" and "indiscrete" left/right adjoint to the forgetful functor for general "ordered" Categories with monotone. morphisms.
The following is taken from "An introduction to Category Theory" by Harold Simmons
$\color{Green}{Background:}$
$\textbf{Example (Galois connection):}$ We modify the category of $\textbf{...
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When is the unit map $\eta$ for a free object an epimorphism?
The following is taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes.
$\color{Green}{Background:}$
$\textbf{Definition for “free over an object with ...
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Details on the explanation of why there is no functor $\mathsf{Group}\to\mathsf{Ab}$ sending groups to their centers? [closed]
The hint to the problem is to consider homomorphism $S_2 \rightarrow S_3 \rightarrow S_2$
I found this answer by Arturo Magidin
here Why is there no functor $\mathsf{Group}\to\mathsf{AbGroup}$ ...
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Morita theory for presheaf (functor) categories
In this 2013 paper, in Proposition 3.14, the author notes
$ \mathcal L(1) $ embeds in $ \mathbf L $, which embeds in $ \mathbf R $ the category of retracts of $ \mathcal L(1) $, so $ P \mathbf R \...
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What is the most diagrammatic proof that $gf = 0 \implies \text{im } f \subset \ker g$?
The proof of the following fact is trivial to most Arrow theorists / Linear algebraists, but I'm developing software that needs to "understand" in a sense this basic fact, because it is ...
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Reconstruction of commutative differential graded algebras
Let $k$ be an algebraically closed field of characteristic $0$.
Let $A,B$ be commutative differential graded algebras (cdga) over $k$ such that $H^{i}(A)=H^{i}(B) =0 \ (i>0)$.
Here, differentials ...
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Functor $\text{Ext}^{1}(A,-)$, equivalent interpretations (i) group of equivalence classes of short exact sequences, (2) measurement of non-exactnesss
All objects in the following live in an abelian category. Consider the short exact sequence $0\rightarrow B\rightarrow C\rightarrow D\rightarrow 0$. Apply the Hom-functor $\text{Hom}(A,-)$ where $A$ ...
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Can categories with suspension be considered as Cat-enriched presheaves over certain strict 2-category?
I came across the concept of a category with suspension $(\mathcal{C},\Sigma)$ which is defined as a category $\mathcal{C}$ together with an endofunctor $\Sigma:\mathcal{C}\rightarrow \mathcal{C}$. We ...
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Knowledge graphs, logic and categories recommendation
I recently started to be more interested about "classical AI" and in particular about knowledge graphs/ontologies. I was looking for a modern (written after 2015 if possible) and highly ...
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Category of Vector Bundles
I am doing Category Theory and part of my project is to understand the category of vector bundles. Is there any reference(s) for vector bundles from a categorical perspective?
I also aim to understand ...
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Size-issues in the definition of categorical limits
I know (at least) two definitions of categorical limits. The setting is always as follows. Let $F:J\rightarrow C$ be a functor between not necessarily small nor locally small categories.
For the first ...
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Are all endomorphism identities in a category?
I am currently studying categories without products. I have found that the category of fields has no products Examples of a categories without products, in the proof an endomorphism appears and while ...
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Proving comonad identities related to internal category
I'm going through the Elephant but I'm having a hard time verifying a given structure satisfies the comonad conditions.
Let $\mathbb{C}$ be some internal category in $\mathcal{S}$ and $\mathbb{D}$ an $...
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Duality in extension groups of $k$-linear abelian categories
In a $k$-linear abelian category $\mathscr{A}$, where $k$ is a field, two objects $A,B$ are given. The extension group $\text{Ext}^1_{\mathscr{A}}(A,B)$ is a group consisting of the exact sequences $$...
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Proving that there is a functor $F: Grp \rightarrow Ab$ s.t $F(G)=G_{ab}$- How to deal with the quotient?
Let $G$ be a group. $f$ a group hom. $f:G\rightarrow H $
I define:
$F(G)=G_{ab}$
and
$F$ : $G$ $\rightarrow$ $G_{ab}$ a homomorphism, where $G_{ab} : = G/[G,G]$ is the abelianization of group $G$....
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Small question about contravariant functors
I am new to Category Theory and it was recommended to me to start my understanding of basic concepts by reading $\textit{Basic Category Theory}$ by Tom Leinster.
But I am finding myself struggling a ...
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How do I complete the proof of the composition rule for the functor that sends a ring $R$ to its group of units $R^×$?
I have to prove that there is a functor Ring $\rightarrow$ Grp that sends a ring $R$ to its group of units $R^×$. In order to to that :
Let $R,S,T$ be rings, $h$ a homo of rings from $R$ to $S$ and I ...
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