Various structures are studied in category theory using properties of objects and morphisms between them. 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 ...

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Alternative Definition of Contravariant Functor

Given two categories, $C$ and $D$, a covariant functor is usually defined as a regular functor $C \to D$, whereas a contravariant functor is usually defined as a regular functor $C^{op} \to D$. ...
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The completeness of a category of additive functors between additive categories

In what follows $\textbf{preadditive}$: a category $\mathscr{C}$ is preadditive when $\forall\ A,B,\ \mathscr{C}(A,B)$ is an abelian group and the morphisms composition is a group homomorphism on ...
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A cartesian diagram?

let $k$ be a field, and $X$ and $Y$ varieties over $k$. Let $L$ be an extension of $k$, and $X_L=X\times_k L$. Is the diagram $$\require{AMScd} \begin{CD} X_L\times Y_L\times X_L @>>> X\...
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Morphisms between biproducts in additive categories

I can't understand the following description of a morphism between biproducts in an additive category, which I found in Borceux, Vol.2 If $A_1,A_2,B_1,B_2$ are four objects in an additive category $\...
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Projective limit of finite dimensional C* algebras

Let $A$ be a separable unital $C^*$-algebra and $A$ = $I_0 \supset I_1 \supset I_2 \supset \ldots$ Be a sequence of ideals in $A$ such that: $I_k$ is ideal in $I_m$ when $k \geq m$ $\bigcap I_k = \...
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$\mathbb{T}\text{-Alg(Set)}$ is complete and cocomplete.

Let $\mathbb{T}$ be a finitary algebraic theory and $\mathbb{T}\text{-Alg(Set)}$ be the category of finite-product-preserving functors $\mathbb{T} \rightarrow \text{Set}$. It is written in my ...
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Functors $1 \to C, 2 \to C, 3 \to C$ (McLane exercise 1.3.2)

This is exercise 2 in Mclane's Categories For the Working Mathematician, chapter 1.3. Show that functors $1 \to C, 2 \to C, 3 \to C$ correspond respectively to objects, arrows and composable pairs ...
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Isomorphism between Category of posets and Category of Alexandrov topological spaces.

Recently I stumbled upon something that seems really intresting, however I couldn't find a proper proof, so I thought that maybe you can help me. Suppose that $\textbf{Poset}$ is the category of $...
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Zero objects in preadditive categories

Let $\mathscr{C}$ be a preadditive category. Let $A,B$ be two objects and assume that $\mathscr{C}$ has a zero object $0$ (an object both initial and terminal). Let $f$ be the unique zero morphism ...
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Why is $\mathrm{Spec}(\mathbb{Z})$ a terminal object in the category of affine schemes?

I've seen this claim repeated in many places (always without source or proof), that $\mathrm{Spec}(\mathbb{Z})$ is a terminal object – however, the most I've been able to prove myself is that for any ...
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Composition in the Arrow category

From a category $\mathcal{C}$ we can construct its arrow category $\text{Ar}(\mathcal{C})$, where objects are morphisms and arrows are commutative squares. But what happens with arrow composition? ...
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Is there a $G$-equivariant bijection $h: G/X \to G/Y$?

Let $X$ and $Y$ be subgroups of a group $G$ such that there are $G$-equivariant maps $f: G/X \to G/Y$ and $g: G/Y \to G/X$. Is there a $G$-equivariant bijection $h: G/X \to G/Y$?
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Is taking a Koszul complex of a function a functor?

While reading about Koszul complexes in Bruns and Herzog, I came across the following proposition: Suppose $L$ and $L'$ are $R$-modules with linear forms $f : L \to R$ and $f' : L' \to R$...
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Problem in understanding some steps in proof

The problem here is about categorical construction of free groups, as in Lang's algebra (p.66-68). Theorem: For any set $S$, there exists free group $(F,f)$ determined by $S$ (here $f:S\rightarrow F$)...
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95 views

A category of relations - or two different?

Objects in the category Rel2 (my notation) are the relations $r\subseteq A\times B$, $r'\subseteq A'\times B'$ (the morphisms in Rel) and morphisms are pair of relations $\alpha\subseteq A\times A'$ ...
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Monomorphisms, epimorphisms and isomorphisms of groups category

I would like to solve the Problem 2.12 in Szekeres, A Course in Modern Mathematical Physics: Show that the class of groups as objects with homomorphisms between groups as morphisms forms a ...
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Motivation for the definition of continuous maps on topological spaces [duplicate]

In any category where the objects are sets equipped with certain relations and operations, the notion of "morphism" arises perfectly naturally. (Generally, a morphism between objects is one that ...
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68 views

Why is the category of finitely generated modules over a non-noetherian ring not abelian?

I am learning about abelian categories for a talk I have to give next week. One of the first questions I had upon learning this definition is "does there exist an additive category that is not abelian?...
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Generating acyclic cofibrations for the Joyal model structure

I was just reading this article by Nikolaus, and at the beginning of Section 4, was surprised to read that there is no explicitly known set of generating acyclic cofibrations for the Joyal model ...
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465 views

Is category equivalence unique up to isomorphism?

Let $\mathbf C$, $\mathbf D$ be categories and $F, F':\mathbf C \to \mathbf D$ and $G, G':\mathbf D \to \mathbf C$ be functors of the shown direction. Is it the case that, if each of $(F,G)$ and $(F'...
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Balanced Tensor Product of Module Categories

Let $C$ be a $k$-linear ($Vect_k$-enriched) monoidal category and consider the 2-category $Mod_{C}$ of $k$-linear $(C,C)$-bimodule categories in the sense of Ostrik (https://arxiv.org/abs/math/0111139....
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On direct sum and direct product of groups

I understood the definitions of direct sums and direct products of groups (rings, modules, etc). The definitions of them can be given in terms of universal properties - certain commutative diagram in ...
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How to obtain operation-version of the principle of equivalence from the property-version

From this link If $X$ is an $\infty$-groupoid, then a property $P$ of objects of $X$ is compatible with equivalence if, whenever $P(a)$ holds for an object $a$ of $X$ and $b$ is equivalent (as an ...
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On a property of generating set for groups

Consider the following paragraph from Lang's Algebra: My question is about converse of one statement, which I was not able to prove. Question: If $f(S)$ do not generates $F$, then can we find ...
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What is significance of this proof of existence of free groups (Lang's Algebra)

There are different proofs of existence of free groups. While reading Lang's Algebra, it caught my attention towards proof of this theorem by first bracket statement in proof: Later I went on ...
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What's wrong with my understanding of the Freyd-Mitchell Embedding Theorem?

It's truly bizarre that there exists no full modern exposition of this theorem, as noted elsewhere. Anyway, I thought I'd poke through and see if I could get the gist of how it works as somebody who ...
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43 views

Is the universal enveloping algebra functor exact?

The universal enveloping algebra is a functor from Lie algebras to unital associative algebras, and is left adjoint to the functor which sends a unital associative algebra to a Lie algebra with ...
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1answer
31 views

Pushout as a functor on an exact sequence

I'm having some problems with the last question in exercise 2.6.4 from Weibel's "An introduction to Homological Algebra". The exercise asks to show that pushout is not an exact functor in Ab (Abelian ...
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1answer
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Free objects in category of algebras

I read in the book "Category Theory By Steve Awodey" the following statement: The axiom of choice implies that all sets are projective, and it follows that free objects in many (but not all!!) ...
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1answer
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Universal property of generating set for vector space

Let $V$ be a vector space over $F$, and $S$ a non-empty subset of $V$. We say that $S$ generates $V$ if every $v\in V$ can be written as finite $F$-linear combination of elements of $S$. I want to ...
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A topological category which is a subcategory of Set

In category theory it is possible to freely chose what are objects and what are morphisms as long as the definitions fulfills the axioms for a category. Now, I'm trying to construct a natural ...
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Does this category explain continuity?

I started study mathematics 1970 in the University of Stockholm and there where no courses in category theory at that time. The students were supposed to do self studies in category theory (and set ...
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Sufficient condition for a monoid

Suppose we have a set $S$ with a single commutative binary operation $*$ and an identity element $i$. Is $(S, *, i)$ necessarily a monoid? According to this answer, a monoid just requires an ...
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Fibered categories, introduction or notes

I would like to learn about fibered categories, I know basic category theory, but not algebraic geometry. Is there a text, or lecture notes, which motivate the definitions from fields other than ...
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When $H$ is a Yetter-Drinfeld module over itself? [closed]

Let $H$ be a bialgebra. When $H$ is a Yetter-Drinfeld module over itself? Thank you very much.
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Problems with subcategories

I've been told that the category of groups isn't a subcategory of Set. How come? Wikipedia: Let C be a category. A subcategory S of C is given by a subcollection of objects of C, denoted ...
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Examples of asymmetrically braided monoid

From nCatlab https://ncatlab.org/nlab/show/braiding : Any braided monoidal category has a natural isomorphism $$B_{x,y} \;\colon\; x \otimes y \to y \otimes x $$ called the braiding. ...
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Adjoint functors and the classical adjoint

Is there any relationship between adjoint functors seen in category theory, and the classical adjoint (as in adjoint matrices)?
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Does taking $\mathbb{C}$-points of a scheme preserve pullbacks or pushouts?

Let $K$ be the field $\mathbb{C}$ of complex numbers and let $X$ be a scheme of finite type over $S=\operatorname{Spec(K)}$. The set $X(K)=\hom_{Sch/S}(S, X)$ of $K$-rational points of $X$ carries a ...
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Counit for the restriction of scalars, extension of scalars adjunction

Let $f: R \to S$ be a morphism of noncommutative rings. Let $$f_!:= S \otimes_R (-) : R \text{Mod} \to S \text{Mod}$$ denote the extension of scalars functor, and $$f^*: S \text{Mod} \to R \text{Mod}$...
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104 views

Basis-free formula for $\mathrm{Hom}_k(V,V)\rightarrow V^*\otimes V$

Let $V$ be a finite dimensional vector space over a field $k$. Then there is a natural map $\phi:V^*\otimes V\rightarrow \mathrm{Hom}_k(V,V)$ given by $$\phi:f\otimes v\mapsto \Big(x\mapsto f(x)v\...
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1answer
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Stronger version of Acyclic Models Theorem

Let $\mathscr{C}$ be an abelian category. If $P_\bullet \in \operatorname{Ch}_{\geq 0}(\mathscr{C})$ is a bounded below complex of projectives, and $C_\bullet \in \operatorname{Ch}_{\geq 0}(\mathscr{C}...
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1answer
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Does $\mbox{Hom}(\bullet,A)$ functor preserves pullbacks?

I know that $\mbox{Hom}(A,\bullet)$ functor preserves pullback (Hom-functor preserves pullbacks) but what could we say about the contravariant functor $\mbox{Hom}(\bullet,A)$?
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Two definitions of double categories?

A double category can be defined as a category object in $\mathbf{Cat}$ the category of small categories. We can also define a double category as four categories satisfying some compatibility ...
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Is there a connection between the concepts of limits in ordinals, functions and categories?

In set theory there is the concept of a limit ordinal: Nonzero ordinals that are the supermum of all ordinals below them. In functional analysis there are the concepts of limits of functions (and ...
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Is there any book similar to “Halmos Naive Set Theory” in Category Theory?

When I wanted to learn set theory in high school, I found Halmos Naive Set Theory book very readable and understandable. But now, at university, I have been searching for a similar book in category ...
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Understanding the tensor-hom adjunction intuitively

I'm currently trying to teach myself some category theory. Recently, I learned that the tensor product is left adjoint to the hom functor in suitable categories, e.g. vector spaces with linear maps, i....
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What do we lose if we allow the Hom sets not to be pairwise disjoint?

In Category Theory the $hom(A, B)$ sets are generally formulated to be pairwise disjoint. We know that set theorists do not agree to this day on whether functions are also defined by their codomain or ...
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A scheme is affine iff the natural map $X\to \operatorname{Spec}\Gamma(X)$ is an isomorphism

We know that the functor $\operatorname{Spec}: \mathsf{Rings}^{\text{op}}\to \mathsf{Schemes}$ is right adjoint to the global section functor $\Gamma: \mathsf{Schemes}\to \mathsf{Rings}^{\text{op}}$. ...
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Simple characterization of integers among abelian groups

This is part of an early exercise in Freyd's abelian categories. Let $\mathscr{G}$ be the category of abelian groups. The group of integers is distinguished, up to isomorphism, by the facts that: ...