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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category of linear maps between fixed vector spaces [duplicate]

Let $V,W$ be fixed $k$-vector spaces. Let $C$ be the category whose objects are linear map $f:V\to W$ and morphisms from $f$ to $g$ are pair of linear maps $(\alpha,\beta)$ where $\alpha:V\to V,\beta ...
2
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69 views

About details of the Fakir theorem proof (associated idempotent triple)

On the ncatlab work http://ncatlab.org/toddtrimble/published/Associated+idempotent+monad+of+a+monad Todd Trimbe quote the Fakir theorem about the associated idempotent triple, and this is based on ...
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2answers
164 views

What is the product and coproduct of Morphism category(Arrow category)?

Given category C, Its morphism category D means a category that has 1) "morphisms of C" as its objects 2) "pair (f,g) s.t. the diagram(square) commutes" as its morphisms The above definition is ...
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418 views

Which books to study category theory?

I am an amateur math researcher in the field of general topology. I've set the purpose to learn enough category theory for my research. After reading Steve Awodey, "Category Theory", 2010, is it ...
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1answer
54 views

Factoring of a Pro-$\mathcal{C}$ morphism

Let $\mathcal{C}$ be a category, $X$ an object of $\mathcal{C}$, and $p:I^{\circ}\rightarrow \mathcal{C}$ a projective system in $\mathcal{C}$. Let $\alpha\in \mathrm{Mor}_{Pro(\mathcal{C})}(p,X)$, ...
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102 views

Completing a Partially Defined Associative Binary Operation

This is more like a question about terminology. I would like to hear some recommendations of books that discuss algebraic structures with one partially defined associative binary operation, and the ...
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45 views

Help with formalizing a diagram in category theory

I'm working on a particular type of diagram using category theory, but I'm confused right now about a part of this diagram and how to properly formalize it. The situation is the following: I have two ...
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1answer
82 views

$\textbf{C}$-Monoids and products

i have a question about $\textbf{C}$-Monoids. We can make a new category $\textbf{Mon(C)}$ from the category $\textbf{C}$, namely the category of all $\textbf{C}$-monoids. A $\textbf{C}$-monoid is a ...
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282 views

Catagorical Definition of Coproduct and Abelian Groups

I have the definition of a coproduct which is as follows: A coproduct of $\{A_\alpha\}$ in $\mathcal{G}$, where $\mathcal{G}$ is a category and $\{A_\alpha\}$ a collection of objects, is an object ...
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146 views

category of linear maps

Let $V,W$ be vector spaces. Let's define a category whose objects are linear map $f:V\to W$ and morphisms from $f$ to $g$ are pair of linear maps $(\alpha,\beta)$ where $\alpha:V\to V,\beta :W\to W$ ...
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61 views

How to see a 2-group as a 2-category with only one object?

We'll take the following definition of a 2-group: A 2-group $\mathsf{G}$ is a category internal to $\mathsf{Grp}$ Namely, it is a group $\mathsf{G_0}$ of objects, a group $\mathsf{G_1}$ of ...
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49 views

Examples of non fibrations

What are some illustrating examples of functors $\mathcal{E} \to \mathcal{B}$ which are neither a fibration nor an opfibration? I've found many positive examples but I'm blanking out on negative ...
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83 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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208 views

Can we found mathematics without evaluation or membership?

In some sense, composition generalizes evaluation. The trick is, instead of writing $f(x)$ for $x$ an element of the domain of $X,$ we write $f \circ x$ for $x$ a function $1 \rightarrow X$. ...
3
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2answers
686 views

Injectivity of the dual map

Suppose V and W are vector spaces of possibly finite and infinite dimension over a field K. Show that if a linear map $L : V → W$ is surjective the its dual is injective. Also prove the converse of ...
3
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2answers
107 views

How can I quantify over the class of all cardinalities?

I'd like to quantify over all cardinalities of sets. My end goal is to make a category-theoretic arguement: For all cardinalities of sets, in the category of sets with maps as morphisms: the ...
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2answers
70 views

How are expressions like $(Df)(x)$ treated in categorial set-theories (in a foundational context)?

In the categorial set-theory approach to the foundations (like ETCS), an element of a set $X$ is usually taken to be a function $x : 1 \rightarrow X,$ where $1$ is a singleton set. This is analogous ...
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114 views

Simple clarification - $\operatorname{Hom}_{\mathsf{Set}}(X,Y)$

I'm currently working through David Spivak's Category Theory for Scientists, and I'd just like to verify that I am understanding $\def\homset{\operatorname{Hom}_{\mathsf{Set}}}\homset(X,Y)$ correctly. ...
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233 views

Pullbacks and pushouts in the category of digraphs?

By definition the category of digraphs is: Objects are endomorphisms of the category $\mathbf{Rel}$ (that is sets equipped with a binary relation on that set). Morphisms from an object $\mu$ to an ...
3
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1answer
115 views

Why is an epic equalizer an isomorphism?

I've seen in several places without further comment that if an equalizer is epic, it's an isomorphism. I've only proved one half of this: Suppose $e:X \rightarrow A$ is an epimorphism and an ...
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1answer
122 views

Elementary Category Theory

On page 4 of Categories for the Working Mathematician Mac Lane writes that "the definition of a monoid is more general because the cartesian product may be replaced by another operation". He then ...
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1answer
50 views

Monos in $\mathsf{Mon}$ are injective homomorphisms.

Let $f: M \to N$ be a mono in $\mathsf{Mon}$. I want to prove that the underlying function $U(f): U(M) \to U(N)$ is injective. How can this be done ? Thanks in advance.
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1answer
109 views

Pushouts and Pullbacks in Category Theory

How would one prove existence of pushouts and pullbacks where the objects are vector spaces and the morphisms are linear transformations?
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71 views

What is the use of locally connected spaces?

One of the main properties of locally connected spaces is that their connected components are clopen and thus, they are homeomorphic to the colimit of their connected components. This is good to ...
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122 views

Question on category theory

So I have an introductory knowledge of category theory but there is one concept I can't get my head around and would like some help: When my class had categories defined we said a category ...
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1answer
88 views

Is there a generic definition of “strongly indistinguishable”?

This is related to a previous question. Consider the quasiordered set $Q = \{\bot, q,q', \top\}$ with $q \lesssim q'$ and $q \lesssim q',$ such that $\bot$ is the unique least element and $\top$ the ...
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176 views

Equivalence of the definitions of “image” in category theory

The concept of "image" in Category Theory is, depending on source, defined in two possible ways: Either as a factorization of a morphism, or as the kernel of a cokernel. More precisely, let some ...
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96 views

Categories, whose objects are morphisms [closed]

I am interested in categories, whose objects are morphisms (in an other category). I want to see examples of such categories. I have examples of this in my research: Such as the categories of ...
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79 views

explicit proof of pullback stability of epics in a topos

As in Explicit construction of a initial object in a topos I'm looking an elementary proof of the fact that, in a topos, epimorphisms are stable under pullback or, equivalently, that images are ...
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1answer
43 views

${\rm Fun}^R(C_1^\text{op},C_2)$ is a presentable category

I am stuck in proving Lemma 5.25 in Moritz Groth's notes: I am asked to prove that for any two presentable categories $C_1,C_2$ the category of limit preserving functors ${\rm ...
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1answer
84 views

Should the first be the last by composition of paths?

Given two paths $f,g:\mathbb{I}\rightarrow X$ with $f\left(1\right)=g\left(0\right)$ there is a composite $f.g$ defined by $t\mapsto f\left(2t\right)$ if $2t\leq1$ and $t\mapsto g\left(2t-1\right)$ ...
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161 views

Can abstract nonsense be helpful here?

Here a question for those among you, who teach Homotopics/Algebraic Topology at university. I encountered some questions that were in my view quite easier to solve in category hTop instead of Top ...
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165 views

Can logic be significantly geometrised?

I've already asked this question on philosophy.stackexchange, I'm hoping for a different answer here: Descarte has been lauded for putting together geometry and algebra, and his achievement allowed ...
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1answer
645 views

On a joke of Yoneda embedding

I have heard a joke like this: The Yoda embedding, contravariant it is. And a joke concerning "How to put an elephant into a refrigerator", a comment from "Category Theorist" says Isn’t this ...
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1answer
97 views

Category of fractions: transitivity and cancellation property

Given a category $\mathcal{C}$, and a right calculus of fractions $\Sigma$. We can construct the category of fractions $\mathcal{C}[\Sigma^{-1}]$ which has the same objects as $\mathcal{C}$, and ...
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125 views

Can an object be universal with respect to several properties?

Does anyone know an example of an object $A$ in some category $\mathcal C$ such that $A$ satisfies at the same time more than one universal property? Of course, when I say that $A$ is universal with ...
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1answer
42 views

what is this categorical construction?

This is a terminology question. Consider two morphisms $X \to Y, X \to Z$. Consider all such "wedges" of morphisms $X' \to Y, X' \to Z$ dominated by $X$, i.e. endowed with a morphism $X \to X'$ (and ...
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1answer
48 views

Equiv. of Cats. Preserves Product

Show that an equivalence of categories sends products to products and coproducts to coproducts. That is, if $X_i$ are a family of objects in $\mathcal{C}$ with coproduct $X$ then $F(X)$ is the ...
2
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2answers
96 views

Characterization of epimorphisms of sheaves on a site

I'm stuck with a detail in the proof of the characterization of epimorphism of sheaves on a site in the Mac Lane & Moerdijk book "Sheaves in Geometry and Logic". I want to prove that: "If ...
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3answers
171 views

Symmetric monoidal products that preserve limits and colimits

Are there common examples of a symmetric monoidal product $\otimes$ that preserves both limits and colimits in each variable? This question is worded incorrectly, I now realize: (A) I originally ...
2
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3answers
232 views

Why are simplicial categories useful?

By simplicial category here I mean simplicially enriched category, i.e. all $Hom$-sets are simplicial sets and compositions are morphisms of simplicial sets. My question is the following. Suppose I ...
2
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104 views

Does $\mathsf{Man}$ possess countable products?

In our lecture we defined a category $\mathcal C$ to have arbitrary products, iff every diagram $A:\mathcal J \to \mathcal C$ with $\text{Morph}(\mathcal J)=\{\text{id}_j\}_{j \in Ob(\mathcal J)}$ has ...
2
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2answers
114 views

Proof by induction in categorical terms

Given a category cartesian closed $C$ and a functor $F : C \to C$, I consider the initial object in the category of $F$-algebras. This initial object $\mu F$ seems to codify an "inductive object" in ...
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3answers
294 views

What does it mean to have exact derived functors?

Let $F:\mathcal A\to \mathcal B$ be a functor between abelian categories. Suppose $F$ is, say, left exact (plus additive and covariant). We have built its right derived functors $R^iF$. I see no ...
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3answers
113 views

From where comes the horizonal composition in a 2-category?

Viewing a 2-category as a category enriched in $\mathsf{Cat}$, I can see from where comes the vertical composition: morphisms of a 2-category are objects of $\mathsf{Cat}$ and 2-morphisms of this ...
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1answer
59 views

Replacing a morphism with composition with this morphism

I have a certain category. I feel it is better to study the functor $x\mapsto f\circ x$ (where $\circ$ is the composition in my category) than the morphism $f$ itself. How is it called when a ...
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103 views

Derive adjoint unit from counit

As an exercise (2.4.12#5) in Pierce's Basic Category Theory for Computer Scientists, I'm trying to derive the unit natural transformation $\eta : I_{\textbf{C}} \xrightarrow{\cdot} G \circ F$ given ...
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155 views

Understanding Equivalence of Categories

An equivalence of two categories $\mathcal{C},\mathcal{D}$ consists of a pair of functors $F:\mathcal{C} \rightarrow \mathcal{D}$, $G:\mathcal{D} \rightarrow \mathcal{C}$ and natural isomorphisms $FG ...
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1answer
60 views

$GL(-)$ as a k-group functor

My question is essentially may lye simply in a notational obstruction. For a k-algebra M, Jantzen J. defines the k-group functor $GL(M)$ as: $GL(M)(A):=(End_A(M\otimes_{\mathbb{k}} A)^*$. My ...
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associativity of composition

Perhaps a bit silly, but I can't find a really convincing answer: why is associativity of composition so fundamental that it's included in the axioms for a category? I suppose that the lack of ...