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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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Understanding the morphism category (arrow category)

$\require{AMScd}$ I want to understand how the morphism category (arrow category) of $\operatorname{Mod}(A)$ works. The material I have found online have not really helped me that much with proper ...
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Why is $f^{-1}f(U)=U+\ker f$?

We are working in some abelian category. Given subobjects $Z, Z'\hookrightarrow X$ of $X$, we can define $Z\cap Z'= Z\times_X Z'$ as the pullback over $X$ and $Z+Z'= Z\amalg_{Z\cap Z'} Z'$ as the ...
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Identity map on coproduct as 'linear combination' [duplicate]

Let $\mathcal{C}$ be a category, and suppose for all objects $X, Y$ of $\mathcal{C}$, Mor$_{\mathcal{C}}(X,Y)$ is equipped with the structure of an Abelian group, such that the composition of ...
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Show that the functor that takes $R$ to the set of invertible elements of $R[X]/(X^2-a)$ is representable.

The following question is from the Fall 2016 UCLA algebra qualifying exam: Let $F$ be a field and $a\in F$. Show that the functor that takes $R$, commutative $F$-algebras to the invertible elements ...
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I am stuck with an example of direct limits and need help to prove $\mathbb{Q/Z} = \varinjlim \mathbb{Z}/(i)$.

Let $N$ denote the set of positive integers that are ordered as $m \le n \iff m | n$. Let $X_m = \mathbb{Z}/(m)$ denote the set of integers $x \mod m$. I want to show that the direct limit of $X_i$, ...
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Category of multisets, spans and pullbacks

I want to define a category of multisets, $Multi$. To do this, I take the ambient category $SET$, and represent the multisets as functions. So, the objects $Multi$ are functions. We define ...
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Why is $\mathbb{Z}$ not an inital object of Gr or AB?

Why is $\mathbb{Z}$ not an inital object of GR or AB? Claim 1: for every group $G$ there exists a groups morphism from $\mathbb{Z}$ to $G$. PF: Let $f:\mathbb{Z} \rightarrow G$ be given by: $f(n) = ...
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Category where morphism sets are Abelian groups

Let $\mathcal{C}$ be a category, and suppose for all objects $X, Y$ of $\mathcal{C}$, Mor$_{\mathcal{C}}(X,Y)$ is equipped with the structure of an Abelian group, such that the composition of ...
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Under what conditions, one of completeness and cocompleteness can imply the other in a category?

When I tried to prove the completeness and cocompleteness of the category of small categories $\mathbf{Cat}$, I thought that proving either one of them could imply the other by taking the dual ...
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Representing multisets

Multisets are like sets, but the "elements" can have multiplicities. An example is $M = \{ a,a, a, b,c,b,c \}$. We can present the multiset by giving the multiplicities for each set element. Can we ...
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What is the Krull dimension of the Burnside ring of $\mathbb N$?

A contravariant functor $F$ from monoids to commutative rings was defined there. Question. What is the Krull dimension of $F(\mathbb N)$? (Here $\mathbb N$ denotes the additive monoid $(\mathbb N,+...
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The opposite category of the category of all small categories

For the category of small categories $\mathbf{Cat}$, to prove it has finite limits and colimits, is it enough to prove the existence of either finite limits or colimts and then somehow apply it to its ...
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How to find initial object of the category of pointed rings?

I have the category of pointed rings. Objects are all pairs $(R, r)$ where $R$ - ring (with 1) and r is the element of R. Morphisms are homomorphism of rings. Morphism $(R, r) \longrightarrow (R', r')$...
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Bijection between hom$(A, B)$ and hom$(1, B^A)$

Suppose that $A, B$ are objects of a category with all finite products and exponentials, $\mathbb{C}$. Show that there is a bijection between hom$(A, B)$ and hom$(1, B^A)$ where $1$ is a terminal in $\...
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How does a map between inverse systems induce the inverse limit of its components?

I encountered the following definition in the book Profinite Groups by Ribes and Zallesski: Let $\{X_i, \varphi_{ij} \}$ and $\{ X_i', \varphi_{ij}' \}$ be inverse systems of topological spaces over ...
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How to produce uncountable ordinal from universe?

A universe $U$ is defined as the following. (1) If $x\in u\in U$, then $x\in U$. (2) If $x\in U, u\in U$, then $\{x,u\}\in U,x\times u\in U$. (3) If $x\in U$, then power set $\mathcal{P}(x)\in U,\...
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From monoids to commutative rings

We shall first define a functor $$ F:\mathsf{Mon}^{\text{op}}\to\mathsf{CRing}, $$ where $\mathsf{Mon}^{\text{op}}$ is the category opposite to the category of monoids and $\mathsf{CRing}$ is the ...
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Ind-completion of a 2-category

If $\mathcal{C}$ is a category, there is a well known construction called the Ind-completion of $\mathcal{C}$, indicated by $\text{Ind}(\mathcal{C})$. This can be equivalently defined in several ways: ...
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A bijection between the regular closed subsets of Hausdorff space and the clopen subsets of it's projective cover.

Let $X$ be a compact Hausdorff space and let $e(X)$ be it's projective cover, i.e. the Stone space of the Boolean algebra of the regular closed subsets of $X$. Let $e:e(X)\rightarrow X$, $u\mapsto\...
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$A\times B \cong B\times A$ in a category $\mathbb{C}$

I work in a category $\mathbb{C}$, and use definitions and notation from the book 'Category Theory', by Steve Awodey. I'm learning some basic category theory from Awodey's book as part of self ...
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Algebra operations as natural transformations

Apologies in advance if the following makes little to no sense, but here goes .. Denote $m_G : G\times G\to G$ the multiplication of a group $G$. Does it make sense to think of the map $m_G$ as some ...
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What Is The Non-Extra Equivalent Of Coends?

As far as I understand, the coend of a diagram $X:I^\text{op}\times I\to D$ is an object $x\in D_0$ together with a natural isomorphism $\alpha:[[I,D]](X,\Delta^e_*)\cong D(x,*)$ in $[D,\text{Set}]$ ...
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Question about $Q = \varinjlim \frac{1}{n} \mathbb{Z}$

I am a little confused by one of Vakil's exercises, which reads Interpret the statement $Q = \varinjlim \frac{1}{n} \mathbb{Z}$ Does it mean $Q = \bigcup \frac{1}{n} \mathbb{Z}$? But what is the ...
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Currying in a locally small category with coproducts

While studying for category theory course I stumbled upon the following question taken from a previous exam: Let $\mathcal{D}$ be a locally small category with all coproducts. Show that for every ...
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Proving a lemma about the interpretation of regular logic in a regular category

I'm trying to prove and I'm having trouble for the case $\phi \equiv (x = s)$ where $x$ is a variable and $s$ is a term, both of sort $S$. This is what I have done so far, nothing much except ...
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Defining along pullback preserves image in a regular category.

In a regular category we can factor any morphism $A \xrightarrow{g} X$ as $A \xrightarrow{e} E \xrightarrow{m} X$ where $e$ is regular epi and $m$ is mono. This factorization is unique up to ...
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Why is it interesting that the fundamental group induces a functor?

The fundamental group of a pointed topological space $(X,T,x)$, is the group whose elements are homotopy equivalence classes of paths from $x$ to $x$, and whose composition is the concatenation of ...
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Proving the equivalence of different definitions for idempotent monads

Let $T \in C^C$ be a monad with multplication $\mu : T^2 \to T$ and unit $\eta : 1_C \to T$. I am trying to prove that the following definitions for an idempotent monad are equivalent, The arrows $(\...
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Triangulated category associated to a subcategory of an abelian category

As I know that if $\mathcal A_0$ is a thick subcategory of an abelian category $\mathcal A$, then I can define a triangulated subcategory $D^*_{\mathcal A_0}(\mathcal A )$ of $D^*(\mathcal A)$ ( $*= b,...
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Definition of triangulated functor

In Huybrechts' book "Fourier Mukai transforms in algebraic geometry" he defines (def 1.39) an exact (or triangulated) functor between triangulated categories as follows. An exact functor is an ...
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Are diagrams of a quiver the same as small diagrams

I've just started to wrap my head around category theory, and came across two (from my perspective not obviously equivalent) definitions of a (small) diagram in a category $\mathcal{C}$: Definition 1 ...
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Example of computing a direct limit

Let $(\mathbb{N},\leq$) be a directed set where $m\leq n$ if an only if $m$ divides $n$. We define a directed system of groups where $G_{n}=\mathbb{Z}$ for all $n\in \mathbb{N}$ and $f_{mn}\colon \...
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Formalizing the idea of this “equivalence of data” in the category of vector spaces, and how does this generalize to other categories?

My category theory is almost nonexistent, but this seems like a "categorical idea". So I'm looking to formalize this idea: Given the data of $U,V,W$ vector spaces the following are "equivalent" $$B:U\...
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Is there a fiber bundle/espace-etale interpretation of sheaves on a Grothendieck site

Whenever I'm doing sheaf things and I have a construction that involves sheafifying, I find it convenient to think "the thing that has the same stalks but sections must be locally trivial sections to ...
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Is the transpose of the projection under the exponentiation adjunction a constant morphism?

Consider a cartesian closed category $\mathbf{C}$ and fix an object $B \in \mathbf{C}$. For any $X$, we have the product $X \times B$ and a projection $\pi_B : X \times B \rightarrow B$. Under the ...
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In R-Mod, all monomorphisms are equalizers

I want to prove that in the category of R-Modules, all monomorphisms are equalizers. We start by assuming that if $f : A \to B$ is mono and if $f$ equalizes $g : B \to C$ and $h : B \to C$ (i.e. $ g \...
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Is a limit of profinite spaces profinite?

Here is the statement of Lemma 5.22.3 in the Stacks Project: Lemma. A cofiltered limit of profinite spaces is profinite. And here is the proof: Proof. Let us use the characterization of profinite ...
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Johnstone, Topos Theory Exercise 7.3

I need to use the following result about essential points from Johnstone's Topos Theory: Let $\mathcal{E}$ be a Grothendieck toposes and $\mathcal{S}$ be a topos with a natural number object. Show ...
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Product in category of cyclic groups.

I know that if $ G_1, G_2 $ are cyclic groups then $ G_1 \times G_2 $ is cyclic if and only if $ |G_1| $ and $ |G_2| $ are coprimes. But I have to reponds a similar question in the context of category ...
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Do subobjects in concrete categories correspond to subsets?

A concrete category is a category $C$ endowed with a faithful functor $U:C\rightarrow Set$. And if $a$ is an object in $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with ...
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Canonical isomorphisms to vector space duals

Let $V$ be a finite dimensional vector space. The issue of there being no canonical isomorphism between $V$ and its dual $V^*$ is commonly explained by stating that such an isomorphism would require ...
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Characterization of left invertible functors

What are the split monomorphisms in $\mathbf{Cat}$, forgetting 2-categorical structure? So far I know that they are faithful and injective on objects, since those functors are the monomorphisms in $\...
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Exact sequence involving pushout

I think I have a bit of a misunderstanding of pushouts. Let $U, X, Y$ be objects of an abelian category $A$ with morphisms $f: U \rightarrow X$ and $g: U \rightarrow Y.$ Let $Z$ be the pushout of this ...
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Is there a notion of a transversal of subobjects?

If $a$ is an object in a category $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with codomain $a$. The subobjects of all the objects in $C$ partitions the class of ...
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Does an adjoint of an internal Hom functor of a prounital closed category define a tensor product?

A closed category is a category equipped with internal Hom functors along with a unit object. Now this answer shows that if $C$ is a closed category whose internal Hom functor has a left adjoint, ...
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Is “monoidal category enriched over itself” the same as “closed monoidal category”?

If $M$ is a monoidal category, an enriched category over $M$ is a category $C$ whose hom-sets are viewed as objects in $M$. And a monoidal category $M$ is said to be closed if the tensor product ...
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What is actually objects in the big and small étale site over $S$

Let's first consider the small sites $S_{étale}$. In the stack project, they defined the objects in it are schemes $T$ s.t... So the objects are schemes? But in description they also said it is a full ...
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What Idea(s) is natural isomorphism trying to convey

Before I proceed, I would like to first acknowledge that there are similar posts to my question, but I didn't seem to understand fully. So allow me to elaborate on my inquiry. If I view the term "...
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Natural transformation = parametric polymorphic function in “structure categories”?

By “structure category” I mean a concrete category that contains as objects all spaces of a particular type of structure, and as morphisms, functions that preserve that type of structure. I.e. the ...
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Abelian group structure of the additive category

Let $A$ be an additive category. We have that for all $X, Y$ objects in $A, Hom_A(X, Y)$ is an abelian group. However, what is the abelian group structure? If $f, g: X \rightarrow Y$ are two morphisms ...