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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 ...

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Why is finiteness necessary in definition of connected category

I am reading through Category Theory in Context by Emily Riehl. On page 33 she defines a connected category as one in which any pair of objects can be connected by a finite zigzag of morphisms (...
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Objects of the $\mathcal{Pno}$ category?

The objects of $\mathcal{Pno}$ catergory are structures $(S, \lambda, s)$ where $S$ is a set, $\lambda: S \mapsto S$ is a functions and $s \in S$ is a nominated element. Given two such structures, a ...
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Specifying composites of morphisms in a localisation of a triangulated category

I am reading the book Triangulated Categories by Neeman. I have come across a sentence and I'm not really sure what it is trying to say. For those with access to the book, it is Remark 2.1.23. Let $\...
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Is a continuous functor the generalization of a continuous function? [on hold]

See title, and if so, what is the intuition? If not, how does category theory generalize continuous functions?
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Mistake in Mac Lane's presentation of the “universal natural transformation”?

On page 39 of Category Theory for Working Mathematicians, Mac Lane makes a claim that seems to me to be false. Let $\mathcal{C}$ be a category, and let $\mathbf{2}$ be the category with exactly two ...
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Categorical product cancellable

It is pretty easy to prove that $A\cong B\Rightarrow A\times X\cong B\times X$ (assuming the products exist). However, I haven't been able to prove the other direction, that is, $A\times X\cong B\...
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Why do the $-1$-categories form a categeory rather than merely a set?

The collection of $n$-categories naturally has the structure of a $(1+n)$-category. For example $\mathbf{Set}$ is a $1$-category and $\mathbf{Cat}$ is a $2$-category. Therefore we would expect the $-1$...
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What is a function $g$ called, which, composed with a certain function $f_2$, yields a given function $f_1$ (i.e. $f_1 = f_2\circ g$)?

Let $X_1$, $X_2$, and $Y$ be non-empty sets. Let $f_1:X_1\rightarrow Y$ and let $f_2:X_2\rightarrow Y$. What's the terminology for a function $g:X_1\rightarrow X_2$ that satisfies: $f_1 = f_2\circ g$? ...
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In a groupoid G, is G(x,y) called a coset of the automorphism group G(x,x)?

In isomorphism problems one often has a pair $(x,y)$ of objects in a category $\mathcal{C}$ and wants to describe the set $\mathrm{iso}(x,y)$ of isomorphisms $x\to y$ in the category. This is ...
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1answer
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Coproducts and products are the same

I was playing aroudn with finite dimensional vector spaces, and I found that coproducts are the same thing in $\textbf{FinDiVec}$. What other categories exist where this is the case, and what are ...
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1answer
43 views

Equivalence of categories of cones

I've still got a problem with this exercise: I've posted question with my old approach, which didn't unforunately get any answers. I have then come up (after getting a huge hint) with different and ...
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1answer
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Precondition “small category” in functor category

I am currently working on a exercise sheet about categories. There are two exercises: In the first parts I have to show that the vertical composition and the horizontal composition of two natural ...
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352 views

“categorical” proof of a seemingly symmetric statement about Noetherian/Artinian modules

There are two statements which to me seem rather symmetric: Let $A$ be a ring, $M$ an $A$-module, and $f : M \to M$. If $M$ is Noetherian and $f$ is surjective, then $f$ is injective. If $M$ ...
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1answer
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Not sure whether I understand certain adjoint impilcation correctly.

My textbook: The naturality axiom implies that from each array of maps $A_0 \rightarrow... \rightarrow A_n$, $F(A_n) \rightarrow B_0$, $B_0 \rightarrow... \rightarrow B_m$ it is possible to ...
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1answer
33 views

In Finite Dimensional $\mathbb{C}[[x]]$ Modules, $x$ is Nilpotent

Please check my proof of the following: If $V$ is a finite dimensional $\mathbb{C}[[x]]$-module (over $\mathbb{C}$), then the action of $x$ is nilpotent. Let $X \in M_{n \times n} (\mathbb{C})$ be ...
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Vector Spaces are Free Objects

Warning: I know little linear algebra and my assertions below may all be incorrect. I am interested in lists --i.e., free monoids-- and my interest has led me to [finite-dimensional] ...
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Show that $M$ admits a projective cover

I was asked to show. Let $R$ an Artinian ring and $M$ a left $R$-module finitely generated. Show that $M$ admits a projective cover. My Attempt: I take any $L$ $R$-module projective finitely ...
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Do I have to convert myself to a formalist to learn algebra? Can an intuitionist success in advanced algebra? [on hold]

Recently, I stumbled on the topic of the division between geometers and algebraists in mathematics. I really want to research in Category theory but my college doesn't have a minor in this branch so ...
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2answers
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Injectivity class of connected spaces in Top

Consider this one-line statement and its proof. All connected spaces do not form an injectivity class in Top. Proof: let $m:A\to A'$ be a continuous map such that every connected space is $\{m\}-$...
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3answers
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Presheaves are sheaves for the trivial topology

I've just started learning about toposes and I have a stupid question to ask. Suppose we are given a small category $\mathcal{C}$ with trivial topology $T$ on it, (where the trivial topology $T$ on ...
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1answer
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Basic Misunderstanding of Commutative Diagram - Kernel

In A Categorical Introduction To Sheaves p. 3 (2), it says that (categorically) a kernel of a homomorphism is defined by the following diagram: I don't see how this characterizes the kernel. In ...
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2answers
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Universal property of the product of two smooth manifolds

I'm reading Ravi Vakil's book Foundations of Algebraic Geometry, and in the first chapter, which discusses category theory, he starts the discussion by showing that one can define the Cartesian ...
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Question regarding notation in path homology paper

In the paper, https://www.math.uni-bielefeld.de/~grigor/quivers.pdf I have two questions regarding the definitions on Page $5$. What is $\sum c$ in Equation $3.1$ ? Is this just a new edge ...
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Is there a name for the inverses of the functions (i.e for the left-unique, right-total binary relations)?

The right-unique, left-total binary relations are the functions, while the left-unique, right-total ones are their inverses. So, they are the morphisms of $\mathsf{Set}^\mathrm{op}$. Do they have a ...
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Terminal Objects in the category $htop$

Consider the Category $C$ of topological spaces with Homotopy classes of continous maps as Morphisms. An Object $T$ is terminal if for every object $X\in C$ there exist a single morphism $X\...
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Matrices over noncommutative rings?

In chapter 1, section 2 of Categories for the Working Mathematician, Mac Lane says: For each commutative ring $K$, the set $\mathbf{Matr_K}$ of all rectangular matrices with entries in $K$ is a ...
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corollaries and applications of Yoneda's famous lemma

English: I am doing a paper on Yoneda's lemma and its corollaries, for example the corollary to see when two objects in a category are isomorphic. I want to know if there are other applications of ...
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What does the category of RDF models look like in Institution Theory? [closed]

This question has been represented on cstheory.stackexchange.com where it is getting answers. The Question in short Here is the question in its pure form. Details of my reasoning can be found below. ...
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connected components of homotopy pullbacks of nerves of categories

I am not sure if the question makes sense. Given three categories $\mathcal M_1$, $\mathcal M_2$ and $\mathcal M_3$ and a zig-zag $$\mathcal M_1\xrightarrow{f} \mathcal M_3\xleftarrow{g} \mathcal M_2$...
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1answer
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Is the geometric realization of a pointed category contractible?

Given a pointed category $X$, is $|N(X)|$ contractible ($N(X)$ is the nerve of $X$)? Or are there counter-examples?
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Construction of a monad from an operad is in the CGWH category

If $\mathcal{C}$ is an operad and if $X\in\mathcal{J}$ then $CX\in\mathcal{J}$, where $\mathcal{J}$ is the category of compactly generated weakly Hausdorff spaces well-based. I'm studying the ...
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1answer
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Some questions about Enrichement Definition in Category Theory

Here is the definition of enrichment captured from Borceux. My questions: It seems to me we cannot define enrichment over any monoidal category, because: First, take the 3rd requirement, the ...
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1answer
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Yoneda Lemma in Vakil’s FoAG 1.3.Y

I am trying to do the exercise that reads as follows, Suppose you have two objects $A$ and $A'$ in a Category $D$, and morphisms $i_C:Mor(C, A) → Mor(C,A′)$ that commute with the maps $Mor(C,A) → ...
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1answer
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Long exact sequence of cohomology group “without” Snake lemma

Let a short exact sequence $$ 0 \to L \to M \to N \to 0 $$ is a short exact sequence of $G$-modules, then a long exact sequence is induced: $$ 0\longrightarrow L^G \longrightarrow M^G \...
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Modularity vs Antipode In the category $Bord^{\operatorname{or}}_{1,2,3}$

Supposedly the category $Bord^{\operatorname{or}}_{1,2,3}$ carries with it the structure of a Hopf algebra. If that's the case I would like to understand what the antipode is. To that end, I've ...
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1answer
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Nerve of a simplicial category

Here a simplicial category is a simplicial object in $\textbf{Cat}$ (that is, a functor from $\Delta^{op}$ to $\textbf{Cat}$). I wonder why the nerve of a simplicial category is a simplicial set? For ...
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1answer
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Definition of algebraic K-theory space

Let $(C,wC)$ be a Waldhausen category. The algebraic K-theory space is the loop space of the classifying space of the simplicial pointed category $wS_*C$, i.e. of the topological realization of the ...
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How to prove, in general, the categorical non-equivalence

Suppose you have two categories $\mathcal{F}$ and $\mathcal{G}$ and a faithful forgetful functor $\phi\colon \mathcal{G} \to \mathcal{F}$. If you suppose that $\mathcal{G}$ and $\mathcal{F}$ are not ...
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1answer
51 views

Cocontinuous functors as an equalizer

Is it possible to express the full subcategory of colimit-preserving functors between categories $A,B$ as an equalizer of a single pair of maps? More explicitly, I would like to find an equalizer ...
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35 views

A 2-group $\mathbb{G}$, so that always exists $0 \to BG_b \to \mathbb{G} \to G_a \to 0?$

If $\mathbb{G}$ is a 2-group, does there always exists a short exact sequence for this $\mathbb{G}$, such that $$ 0 \to BG_b \to \mathbb{G} \to G_a \to 0? $$ where both $G_a$ and $G_b$ are nontrivial ...
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36 views

Extensive + Exact implies coherent

A pretopos (an extensive and exact category) is coherent. I am looking for a reference of this well known fact.
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3answers
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Free vector space over a set

Given a set $S$ and a field $F$ we can construct the $F$-free vector space over $S$ in the following way. Consider the set of formal sums $$FS:=\left\{\sum_{s\in S} \alpha_s s\,:\, \alpha_s=0\, \text{...
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1answer
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Degreewise cofibration in $S_nC$

Given a category $C$, we have $S_nC=Fun(Ar[n],C)$ and given $A,B\in Ob(S_n(C))$, (i.e. A,B are functors from $Ar[n]$ to $C$), then what does a morphism $f:A\to B$ in $S_nC$ mean by degreewise ...
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1answer
36 views

Exercise on categories of $G$-set

The following is exercise 6 from chapter I of Mac Lane and Moerdijk's Sheaves in Geometry and Logic: I'm currently having trouble with point $(b)$, probably because of my little ability with group ...
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1answer
32 views

map on connected components is injective

Consider the usual model structure on $SSet$ (which is proper). Let $X$ and $Y$ be two fibrant simplicial sets and $f:X\rightarrow Y$ a fibration. If for any two vertices $x,x'$ of $X$, the homotopy ...
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60 views

Show that if the arrow $f: A \to B$ is an epimorphism then the arrow $id \times f: C\times A \to C \times B$ is an epimorphism as well

I'm following the book Sets for Mathematics by Lawvere. As an exercise in section 7 it's stated that if we have an arrow $f: A \rightarrow B$ that is an epimorphism then the arrow $id\times f: C\...
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1answer
27 views

The direct sum of any family of objects

Suppose in an Abelian category $\mathscr C$, the direct sum of any family of objects exists, then is $\bigoplus_{i\in\varnothing}A_i$ equal to 0 or meaningless?
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Is there a category theoretic characterisation of the exponential map from differential geometry?

While I have only a shallow understanding, I like category theory. I find definitions and proofs in terms of category theoretic concepts to be very clean and deep, often cutting to the core of a ...
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1answer
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Is there a category similar to Set, but having the morphism binary relations instead of functions?

I call this hypotetic category BigSet. The definition of BigSet is: Objects are sets Morphisms are binary relations (including functions) Composition is the composition of relations Identities are ...
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The understanding of category of groups

When we say the category of groups, do we refer to the same category of groups? For example, in Tom's mind, there is a category of groups; in Jack's mind, there is a category of groups. However, one ...