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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Why aren't *weak* test categories enough?

Short version : What is the motivation behind the local condition in the definition of a test category ? Why do we want the slice categories $A/a$ to be weak test ? Long version : I start ...
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178 views

Commuting with kernels implies left exactness in Abelian category

I'm following Vakil's notes - chapter on category theory. One issue that is unclear in the notes is the conclusion that right adjoint functors are left exact. The notes define a left exact functor as ...
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40 views

What is the restricted product categorically?

The restricted product is a construction for locally compact abelian topological groups. Let $I$ be an indexing set, with $J$ some finite subset. Let $G_i$ be a locally compact topological group ...
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76 views

What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
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70 views

What are the algebraic and topological properties of a tree (graph theory), and what are any possible connections?

I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup ...
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102 views

Right Kan extension along a diagonal functor.

Denote $\newcommand{\simpcat}{\boldsymbol\Delta} \simpcat$ the simplex category (category of finite ordinals with non decreasing maps). The diagonal functor $\delta \colon \simpcat \to ...
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181 views

Direct limits and pullbacks

Suppose we have three directed sequences of $C^*$-algebras, say $(A_n,\varphi_n)$,$(B_n,\psi_n)$ and $(C_n,\theta_n)$ and $*$-homomorphisms $\alpha_n:A_n\rightarrow C_n$ and $\beta_n:B_n\rightarrow ...
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53 views

Motivation of Strong Monics

Strong monics are defined as: A monic $m$ is strong iff every commutative square $mu=ve$, in $E$ with $e$ epi, has a diagonal i.e. there is a morphism $t$ such that $u=te$ and $v=mt$ in $E$. (where ...
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106 views

Weil restriction - from abstract nonsense to a practical procedure

Let $L/K$ be a finite field extension. Let $X$ be a $L$-scheme, that is $\exists$ a morphism of schemes $\pi : X \to \operatorname{Spec} L$. The Weil restriction of $X$ is a $K$-scheme $W_{L/K}X$ ...
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60 views

Topological characterisations of freeness and separability?

According to wikipedia, a spectral space is defined to be homeomorphic to the spectrum of some commutative ring. They form a category $Spec$ where we take morphisms to be those whose preimage of open ...
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40 views

(Almost) co-free objects?

another naming question from me, which comes up because I try to use category theory as a compass in developing some (new or not?) order-theoretic notions. According to my book (The Joy of Cats, ...
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61 views

For which categories do injections induce surjections in cohomology?

I'll ask a specific question first, but I believe my question might have a rather immediate abstraction, with which I'll finish. Let $H,G$ be finite affine group schemes over an algebraically closed ...
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60 views

An equality of some equalizers of simplicial sets.

$\newcommand{\cosimp}[1]{{#1}^\bullet} \newcommand{\sSet}{\mathsf{sSet}} \newcommand{\Set}{\mathsf{Set}}$ [I realize that the post is imposing.However, all the content of the problem is in the gray ...
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69 views

Existence of a natural transformation

Let $G$ be a group, $S$ a set and $\chi$ an action of $G$ on $S$. This defines a functor $F$ from the group as a single object category to the category of sets. Let $\phi$ be an automorphism of $G$, ...
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65 views

Exact functors in the category of left R-modules - “Fun for the whole family”

The following question has proved troublesome and prompted some deeper questions which I will elaborate on. Our definition of a left exact functor is one which takes exact sequences: $$0 \rightarrow ...
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58 views

Can injective and projective model structures be the same ?

Is there a non trivial example where injective and projective model structures coincide ? By non trivial I mean an example of a functor category on a non discrete base category where I guess ...
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88 views

Understanding Limits and Colimits by Generalized Elements

We want to characterize the limit and colimit of a functor $D\colon J\to \mathcal C$ by generalized elements. The existence of limits theorem states that the limit of $D$ is the equalizer of ...
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64 views

Category of functors not cartesian closed

My question is If $\mathcal{C}$ is a small and cartesian closed category, and $\mathcal{D}$ is some small category, must $\mathcal{C}^{\mathcal{D}}$ be cartesian closed? A counterexample ...
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65 views

Inductive limit of manifolds?

The inductive limit of a direct system of manifolds is a topological space (which I don't think needs be a manifold). But it seems like it should retain some of the structure of manifolds : for ...
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127 views

Category theoretic definition of a “rank” of a subset of an algebraic structure

I am trying to find category theoretic definitions for different possible notions of a rank of a subset (or a subfamily) of an algebraic structure, and to figure out if any of them would be in some ...
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70 views

Is there a topos-like category that classifies regular subobjects?

A quasi-topos is a category that characterised by being finitely complete and finitely cocomplete that is also locally cartesian closed and has a strong sub-object classifier. A topos is finitely ...
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44 views

Size of the collection of morphisms of a category

Suppose we use Grothendieck'universes, at least 2 (named U and W). U has elements called (small-) Sets and its subcollections are called Classes. W has elements called Classes and its subcollections ...
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98 views

Injective model structure

I equip the category of presheaves $[\mathcal{D}^{op},\text{Gpd}]$ with the injective model structure ($\mathcal{D}$ is just any small category). In this structure, weak equivalences and cofibrations ...
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51 views

Seeking information about (category-theoretic) varieties, quasivarieties, and universal Horn classes.

I'm looking for a list of basic facts regarding (category-theoretic) varieties, quasivarieties, and universal Horn categories, as well as information about which forgetful functors preserve what. In ...
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186 views

Definition for a bar resolution for a module over a dg category

Let $ \mathcal{A}$ be a dg category and define a right $ \mathcal{A}$ module to be a dg functor $ M: \mathcal{A}^{op} \rightarrow dif\ k$ where $dif\ k$ is the category of differential $k$ modules ...
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44 views

Are these categories toposes?

Apart from usual examples of toposes, I'd like to know if some of the following categories and some of their subcategories are known to be toposes : the category $\text{Heyt}$ of Heyting algebras ...
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93 views

Natural transformation defined by one element

Let $C$ be a self-enriched category, a CCC, and $F : C \to C$ an endofunctor with strength, that is, $F$ comes with a natural transformation $$st_{A,B} : A \times F B \to F (A \times B)$$ such that ...
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87 views

Morphisms in the derived category

I have just started to learn about derived categories, I am now trying to understand what morphisms look like in some easy examples. Let me describe one for you. Let $D(\mathcal A)$ be the derived ...
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50 views

Do graded modules form an additive category?

I got two conflicting references. On page 466 of Shastri's Algebraic Topology, he claims that graded modules with homogeneous morphisms form an additive cat, whereas Hilton and Stammbach say on page ...
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88 views

Is there such a thing as “combinatorial category theory”?

According to wikipedia, In mathematics, combinatorial group theory is the theory of free groups, and the concept of a presentation of a group by generators and relations Is there such a thing as ...
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194 views

How to prove the equivalence of the two definitions of “equivalence of categories”?

As far as I know, there are two definitions of "equivalence of categories". The first definition of the equivalence of categories $A$ and $B$ is required that there exist functors $F\colon A \to B$ ...
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129 views

Is there a category such that if $\mathbb{R}$ is viewed as an object, we have that $\mathbb{R}^2$ is equipped with the Euclidean distance function?

Viewing $\mathbb{R}$ as an object of the category of metric spaces and metric maps, we have that $\mathbb{R}^2$ is equipped with the distance function $$d(x,y) = ...
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213 views

Are there theorems of mathematics involving category theory that cannot be developed using a Grothendieck paraphrase?

As many people here will be aware, there is a debate within the foundations of category theory as to how to approach the discipline. One can either: Axiomatise the category of all categories ...
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87 views

Are pulation squares “weak equivalences”?

Let $\cal C$ a category where a square is a pullback if and only if it is a pushout. Is there a model structure on $\cal C^\to$ (the category of arrows of $\cal C$) where the class of weak ...
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87 views

Proof in Hungerford about natural isomorphisms of multifunctors

Reading Hungerford's Algebra I encountered the following exercise: (A) Let $\mathcal{C}$ and $\mathcal{D}$ be categories. Let $S,T: \mathcal{C} \to \mathcal{D}$ be covariant functors. If $\alpha: ...
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Directed Colimits exact in the category of abelian groups

Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is ...
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75 views

if $A=B\cup C$, then $A$ is the pushout of $B\leftarrow B\cap C\rightarrow C$

I'm reading the proof of Proposition 1.4.3 in [Johnstone, Sketches of an elephant, Part A]. It take the projection $\pi: C\times D\to D$ and says that $\exists_\pi:\text{Sub}(C\times D)\to ...
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Are there any texts on order theory that treat it as decategorified category theory?

I read this post on nLab and now I want to learn some order theory. What are good texts that contain the above referenced topics, and are there any that are explicitly about order theory as ...
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54 views

Covariant functors represented by schemes

Let $S$ be a nice base scheme and let $F : \mathsf{Sch}/S \to \mathsf{Set}$ be a functor. Are there necessary and sufficient conditions that $F$ is represented by some scheme $X$, i.e. $F \cong ...
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34 views

morphism of monoids and group

let $B,C$ be commutative monoids, $A$ be an abelian group and $B \stackrel{b}{\leftarrow} C \stackrel{a}{\rightarrow} A$ morphisms of monoids. Let $P$ be the push-out of $A$ and $B$ w.r.t. $C$ in the ...
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216 views

Reflection of Exact Sequences

Consider the category of $R$-modules. I am trying to see how i can express a short exact sequence in terms of kernels and cockerels, and how this description can be used to prove that a conservative ...
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135 views

Which (endo)functors of the category of finite-dimensional real vector spaces induce continuous maps between Hom-sets?

Let $\operatorname{Vect-fin}$ be a category of finite-dimensional vector spaces over $\mathbb{R}$. In this category Hom-sets $\operatorname{Hom}(V,W)$ are themselves finite-dimensional vector spaces ...
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116 views

Functors $f^*$ and $f_*$ on the category of sheafs of modules (Hartshorne)

Let $f: (X, O_X) \rightarrow (Y, O_Y)$ be a scheme morphism, $F$ - module over $O_X$, $G$ - module over $O_Y$. How to prove, that $$ Hom_{O_X}(f^*G, F) = Hom_{O_Y}(G, f_* F). $$ Please give the most ...
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How to express, say, the Hausdorff property of a topological space using categories. (And a request for general advise in such practices)

I am new to Category theory and for the sake of the practice, I am interested in revisiting and expressing those concepts that I am familiar with --however basic--, in the language of categories. ...
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221 views

Karoubi envelope and factorization of a functor

I am trying to understand the solution to the following exercise. Let $\mathcal{C}$ be a category with idempotents $e:A\to A$ and $d:B\to B$, and a morphism $f: A \to B$. The Karoubi envelope ...
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64 views

What categories have all the products?

What is the best characterization of categories which have all the products? Is this question related to Axiom of choice in any way?
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289 views

On the bounded derived category of a finite dimensional algebra with finite global dimension

Let $A$ be a finite dimensional $k$-algebra with finite global dimension. How can I prove that the category $D^b(A)$ (bounded derived category of the category of left finitely generated $A$-modules) ...
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141 views

Has the notion of a unique factorization category been defined and studied?

I am using a this notion in a paper that I am writing. The notion is that $(\mathcal{C},\otimes, I)$ (which we will just call $\mathcal{C}$) is a symmetric tensor category, with a unit $I$. Then a ...
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112 views

The category of adjoint functors

Can a category structure be defined on the collection $Adj(\mathbf C,\mathbf D)$ of all pairs of adjoint functors $$ (F\colon\mathbf C\to \mathbf D)\dashv (G\colon \mathbf D\to \mathbf C)$$ in such a ...
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A finite diagram in an abelian category which may not be locally small

This question is motivated by this. I will use the notations of my answer to this. We say a category $\mathcal C$ is locally small if Hom($X, Y$) is small for any $X, Y \in$ Ob($C$). Let $\mathcal ...