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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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Monomorphisms and epimorphisms in the category of morphisms

Let $\mathcal{A}$ be (an abelian, I don't think it should matter?) category. Then let $\operatorname{Mor}(\mathcal{A})$ be the category of morphisms, that is the category with objects given by ...
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How to prove that two fusion systems are equal?

How to prove that two fusion systems $F$ and $F'$ are equal? the collection of objects of $F$ = the collection of objects of $F'$ and for any morphism $f$ in $F$, it is a morphism in $F'$, and ...
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Categorical general topology — reference request

I am looking for literature describing general topology in terms of category theory. I would prefer literature which does not assume too much familiarity with category theory, but would appreciate any ...
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structure-preserving: a history?

I am familiar with the occurrence of structure-preserving morphisms in Category Theory, but I would love to know more about the history of the concept, where it started and so on? I suspect it might ...
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About the complement of a subobject in a topos

Let $\mathcal{E}$ be a topos and let $X$ be an object of $\mathcal{E}$. Let $S \to X$ be a subobject of $X$. We only know that the category of the subobject of $X$ is a Heyting Algebra, so we do not ...
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What are the domains of the multiplication and unit morphisms of a monoid object?

I'm trying to understand what a Monoid is from category theory perspective, but I'm a bit confused with notation used to describe it. Here is Wikipedia: In category theory, a monoid (or monoid ...
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Left adjoint for forgetful functor

I am aware that forgetful functors usually have a left adjoint, but I'm more interested in developing my technique. Let $\mbox{Cat}$ be the category of small categories and $\mbox{Set}$ the category ...
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Inversion on an internal groupoid

Let $\mathcal{C}$ be a category with pullbacks, with $\mathscr{G}=({\bf Ob}_\mathscr{G},{\bf Hom}_\mathscr{G},cod,dom,{\bf 1}, \circ_{\mathscr{G}},-^{-1})$ an internal groupoid in $\mathcal{C}$. I'm ...
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Image in category of sets, existence of pushouts

First question: According to definition from wikipedia: definition What is $v$ in category $Set$? Second question: Let $C$ be a category. Show that binary coproducts and coequalizers in $C$ exist of ...
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Two questions about the Hom functors

Given a locally small category $\mathcal{C}$, Wikipedia defines the Hom functors as At my lectures (for the more specific case of $\mathcal{C}=\operatorname{R-Mod}$ -- the category of R-modules for ...
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Suggestion for seminar about rings of continuous functions [on hold]

I have to do a seminar about the rings of continuous functions, it will be a part of a course in topology. The main topic of my seminar will be the functor from the topological space and the ...
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How to define pivot columns?

When you use Gaussian elimination to solve a homogeneous system of linear equations, you end up with "pivot variables" and "non-pivot variables". The non-pivot variables have the property that they ...
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Simplicial Sets as a Lawvere Theory

A Lawvere theory is a category $T$ with finite coproducts in which every object is isomorphic to a finite coproduct $\amalg_{i = 1}^n x$ of a distinguished object $x$. Then a $T$-theory is a functor $...
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Group structure on the set of maps in a model category

Let $B$ be an object equipped with two maps $ \mu: B \times B \to B$ and $\rho : B \to B$ in a cocomplete model category $\mathcal{M}$ such that $(B, \mu, \rho )$ is a group object. These two maps ...
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Proving group objects in the category of sets are group [on hold]

I'm having difficulty proving that in the category of sets, the group objects are just groups. I know that for a category, C with finite products, an object,G, is a group object with the morphisms m,e,...
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IS CGWH+Meas cartesian closed?

Consider the convenient category of topological spaces - the space of Compactly Generated Weak Hausdorff spaces. This category is a cartesian closed category. Now for every obect in this category, ...
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Smooth real line and Dedekind cuts

I am reading Bell's A primer of infinitesimal analysis, and the real numbers he considers have certain properties for doing synthetic differential geometry. He calls this object the smooth real line. ...
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1answer
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Hopfian modules and equivalence of categories of modules

For a ring with unity (not necessarily commutative) $R$, let $R$-$Mod$ denote the category of left $R$-modules. Let $R,S$ be two rings with unity and $T: R$-Mod $\to S$-Mod be an equivalence of ...
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The pullback functor between categories of etale sheaf on an affine space satisfies solution set criterion

Let $Sh_{et}(Spec(A),\mathbb{F}_p)$ be the category of $p$-torsion affine etale sheaf on $Spec(A)$, which is a contravariant functor from AFFINE etale covers of $Spec(A)$ to $\mathbb{F}_p$-modules ...
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Is (co)product is a bifunctor?

I am learning category theory from Bartosz's blog, where he mentioned that: If the product exists for any pair of objects, the mapping from those objects to the product is bifunctorial. Based on ...
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Uniqueness of projection map of categorical product

This proof shows that products are unique up to unique isomorphism. I can understand the part that objects in product must be unique up to isomorphism, i.e. $(P, \pi_1, \pi_2)$ and $(Q, \pi_1', \pi_2'...
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1answer
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Idempotence of Lawvere-Tierney topology induced by Grothendieck topology

I'm hoping someone can elucidate a step in the proof of V.1.2 in Mac Lane & Moerdijk's Sheaves in Geometry and Logic. Let $\mathcal{C}$ be a small category and $J$ a Grothendieck topology. $J$ ...
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how are characterized categories that are determined by they graph up to isomorphism?

I know that thin categories have this property, but i also know that they are not the only ones. What are other kinds of categories that have this property? Is there any characterization of that fact?(...
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What is name we need to use for endofunction if in its definition endomorphism were not in the category of sets?

An endofunction (or self-mapping or other variation) is an endomorphism in the category of sets, that is a function from a set X to X itself. Ok, what new names we have now for endofunction if ...
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Lemma for showing that presheaves are colimits of representables

$\newcommand{\PShv}{\text{PShv}}$ $\newcommand{\Fun}{\text{Fun}}$ $\newcommand{\C}{\mathcal{C}}$ $\newcommand{\Hom}{\text{Hom}}$ $\newcommand{\ra}{\rightarrow}$ $\newcommand{\op}{\text{op}}$ $\...
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Why is an arrow going out from a terminal object in natural number definition on nLab?

I have just started studying category theory. In section 2 of the nLab page on "natural number objects", it talks about a morphism $z: 1 \rightarrow \mathbb{N}$. But it calls the $1$ object a terminal ...
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Function that maps strings from one formal language into string of another formal language?

Is there branch of mathematics and mathematical theories, that considers mappings from strings of one language into strings of another formal language? Example. Let's consider two languages that can ...
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Category of (small) categories has finite limits

I'm trying to prove that the category of (small) categories $\mathcal{Cat}$ contains finite limits. I know that this is equivalent to saying that $\mathcal{Cat}$ contains finite products and ...
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1answer
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Proving that the free abelian group $G$ for the set $\mathbb{N}$ is isomorphic to its product with itself $G \times G$

So let $G = \mathbb{Z}^{\oplus\mathbb{N}}$. We need to prove $G \cong G \times G$ (Aluffi ex. II.5.9). Here's my stab at it. First, denote the set-function from $\mathbb{N}$ to $G$ that's the part of ...
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1answer
60 views

What does type constructor in type theory correspond to in category theory?

If types themselves correspond to object or for example Unit type corresponds to terminal object, then what does type constructor correspond to in a category?
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Exchanging objects in distinguished triangles

Let $\mathcal{D}$ be a triangulated category and assume we are given two distinguished triangles $$ \begin{align} A \rightarrow B \rightarrow C \rightarrow +1\\ D \rightarrow C \rightarrow E \...
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Connected scheme of dimension $0$ which admits a monomorphism to an affine scheme

Let $X$ be a connected scheme of dimension $0$ (not necessarily Noetherian). If there exists an affine scheme $Y$ such that there exists a monomorphism (in the category of schemes) $X \to Y$, then is $...
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Connected scheme which admits a monomorphism to an affine Noetherian scheme of finite Krull dimension

Let $Y$ be a Noetherian affine scheme of finite Krull dimension. Let $X$ be a connected scheme and suppose there exists a monomorphism (in the category of schemes) $X \to Y$. Then is it true that $\...
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1answer
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Ind Category of Finitely Generated k-Algebras

Let $C$ be the category of finitely generated algebras over a field $k$. Let $\hat{C}$ be the category of functors from $C^{op}$ to $\text{Set}$. Let $k \text{-alg}$ be the category of $k$-algebras. I ...
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In what sense is a fully faithful functor equivalent to the inclusion of a full subcategory

On the nLab page for sieves and elsewhere it is asserted that a fully faithful functor is 'equivalent' to the inclusion functor of a full subcategory -- what is this intended to mean, explicitly? ...
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Illustrative example of applying the Nichols-Zoeller theorem

I've been looking through the literature of Hopf algebras, and although everyone mentions the Nichols-Zoeller theorem as a very important one in the study of Hopf algebras and lists a few corollaries, ...
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How to categorically represent natural numbers and connection between them only if their difference is 1?

How can one categorically represent natural numbers (as objects) with connection between each two of them only if their difference is 1? Obviously the above mentioned connection(relationship) can not ...
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category theory, $F$ is free object with basis $X$ iff $ (F,f) $ is initial object in $D$

Let $C$ be a concrete category, let $F$ be an object in $C$, let $X$ be a nonempty set, let $f : X → F$ be a function between sets. Let $D$ be a category whose objects are pairs $(B, g)$, where $g : X ...
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How to construct a natural isomorphism of hom-sets from an adjunction of functors [duplicate]

Let $F: C \rightarrow D,\ G: D \rightarrow C$ be an adjunction of functors. My goal is to construct an isomorphism $$Hom_D(F(x), y) \simeq Hom_C(x, G(y))$$ So given a morphism $f(x): F(x)\rightarrow ...
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2answers
60 views

Simple examples of non-representable functors

I am looking for examples of non-representable functors, to see how the Yoneda lemma works in these cases. Here is one: let $\mathbf{C}$ be the category of finite-dimensional Euclidean spaces, with ...
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1answer
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Does the category of finite dimensional free modules over a principal ideal domain have all finite colimits?

A paper I'm reading states this fact without proof, and I can't seem to find a proof for it.
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Example of a mathematical structure in which morphisms do not compose, if any? [closed]

In category theory the morphisms have to be able to compose. Do we have a mathematical structure in which morphisms do not compose? If yes then what is the simplest example of such? I understand that ...
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Do homotopy pullbacks always compose?

Classical pullbacks compose, as is easily checked with the universal property. More precisely, if $\require{AMScd} \begin{CD} A @>>> B\\ @VVV @VVV\\ C @>>> D \end{CD}$ and $\require{...
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1answer
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Kernel and Image definition in general category theory.

Let $A \xrightarrow{f} B$ be a morphism of objects $A,B$ of a category $\mathcal{C}$. Then We say that a morphism $K \xrightarrow{\iota_K} A$ of objects of $\mathcal{C}$ is a $\mathit{kernel}$ of $f$ ...
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1answer
120 views

What is the shape group?

I'm familiar with topology and category theory a little bit as well as inverse limits of inverse systems. I would like to understand the shape group better. For example, one could describe the ...
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Is there an official terminology for functors that are adjoint up to a given functor?

Assume that we are given categories $\mathcal{A}, \mathcal{B}, \mathcal{C}$ and functors $U:\mathcal{A}\to\mathcal{B}$, $L:\mathcal{B}\to\mathcal{C}, R:\mathcal{C}\to \mathcal{B}$ such that $$\mathsf{...
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3answers
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Any isomorphism $f: x \rightarrow x$ in a category has to be the identity morphism?

Let $C$ be a category and $f: x \rightarrow x$ be an isomorphism. Is it true that $f$ must be $id_x$? If no, what would be a good counterexample?
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What's an example of a non-strong monad on the category of sets?

If I understand the definition correctly, it seems to me that all the monads I know of on the category of sets can be viewed as strong monads in a natural way. For example, if $T$ is the free group ...
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1answer
54 views

Coend and End of this Functor

Take a ring $R$ and view it as being a category enriched over the category $\text{Ab}$ of abelian groups. Note: The Yoneda embedding $R \rightarrow \text{Fun} (R^{op}, \text{Ab})$ corresponds to the ...
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1answer
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Is there a way to show Yoneda equality visually instead of compositional algebraic symbols?

The proof that $u \in F(A)$ is sufficient to define a natural transformation $\alpha : \text{Hom}(A, \cdot) \Rightarrow F(\cdot)$ goes like this: Let $\alpha_X : g \in \text{Hom}(X, Y) \mapsto F(f)(u)...