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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115 views

Relations as arrows and as objects - what are the arrows in the latter?

Since a relation $R$ from $X$ to $Y$ defined as a subset of $X \times Y$, the category of sets and relations is just that: the objects are sets and the arrows are the relations. Is there a generally ...
2
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227 views

Inverse of power-set functor?

In his answer to this Q: How to interpret $1 \to 0$ in $\mathbf {Set^{op}}$, and $\mathbf {Set^{op}}$ itself? Zhen Lin proposed that $\mathbf {Set^{op}}$ is naturally equivalent to the category of ...
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185 views

Final object in fields of characteristic $ 0 $?

In his answer to this question: Category of Field has no initial object, Arturo Madigin indicated that the field of rational numbers is the initial object in the category of fields of characteristic $ ...
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244 views

Group of Isomorphisms of a Groupoid

Write a careful proof that every group is the group of isomorphisms of a groupoid. In particular, every group is the group of automorphisms of some object in some category. First can someone tell me ...
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145 views

Internalising the functor action on morphisms (e.g. to exponential objects)

Part of what it means to be a functor between two categories is to have a map of morphisms e.g. $F$ sends $f: A \to B$ to $Ff: FA \to FB$. Suppose $F$ is a functor from a category to itself, and that ...
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222 views

Eilenberg Moore category

I've been trying to code up the Eilenberg-Moore category for a monad in Haskell. As I understand it, given a category $C$ and a monad $(T,\eta,\mu)$ on $C$ we build the Eilenberg-Moore category $C^T$ ...
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4answers
2k views

What is category theory useful for?

Okay so I understand what calculus, linear algebra, combinatorics and even topology try to answer, but why invent category theory? In wikipedia it says it is to formalize. As far as I can tell it sort ...
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154 views

Long Exact Sequence on Homology in an Abelian Category

Let $\mathcal A$ be an abelian category and let $0 \xrightarrow{} X \xrightarrow f Y \xrightarrow g Z \xrightarrow{} 0$ be an exact sequence of chain complexes in $\mathcal A$. I am using the ...
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2answers
193 views

Coproducts and products of modules

I've just looked at the book "A Course in Homological Algebra" ( by Hilton and Stammbach) . They show the universal property of the direct sum (coproducts) using injections and the universal property ...
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416 views

Completion as a functor between topological rings

In the following all rings are assumed to be commutative and unitary. Preliminaries: For any topological ring $R$ we can form its completion $\widehat{R}$ by taking all Cauchy sequences modulo null ...
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4answers
266 views

Can we modify ETCS to handle structures directly, as objects in their own right?

I have completely rewritten this question; thus, some of the comments/answers may no longer be relevant. The elementary theory of the category of sets (hereafter, ETCS) is an axiomatic approach to ...
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1answer
99 views

Are bicategories and lax 2-categories the same?

My question is that whether the definition of bicategories is the same as the definition of lax 2-categories. I heard that they are both week versions of 2-categories. Are they the same? If not, how ...
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60 views

Is this thing K-finite?

This is related to this question: Freyd's Geometric Finiteness : An Example Computation I've essentially reduced the problem to the following question: Equip $\mathbb{N}$ with the discrete ...
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2answers
129 views

Is there a “partial function” approach to subobjects in category theory?

Given a relation $f : X \rightarrow Y$, lets define that the source of $f$ is $X$, and that the domain of $f$ is the set of all $x$ such that there exists $y \in Y$ satisfying $(x,y) \in f$. Thus the ...
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567 views

Examples of categories where morphisms are not functions

Can someone give examples of categories where objects are some sort of structure based on sets and morphisms are not functions?
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167 views

Pullbacks and transpose map

Given maifolds $M,N$ and a smooth map $\phi:M \to N$, and a smooth function $f:N \to \mathbb{R}$, we have the pullback of $\phi$ by $f$ to be the function $\phi^* f = f \circ \phi : M \to \mathbb{R}$. ...
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1answer
106 views

Isomorphism between (source of) kernels of parallel arrow of a pullback square, by adjunction

Let $\mathcal A$ be an abelian category, $\alpha_1 \colon A_1 \to B$, $\alpha_2 \colon A_2 \to B$ two morphisms and $A_1 \leftarrow A_1\times_B A_2 \to A_2$ their pullback (with morphisms, say, $p_1, ...
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513 views

Products and pullbacks imply equalizers?

I was reading Herrlich & Strecker's Category Theory, and there is a theorem called The Canonical construction of Pullbacks which states that if a category has products and equalizers, then it has ...
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107 views

Freyd's Geometric Finiteness : An Example Computation

In his paper "Numerology in Topoi" available here: http://www.tac.mta.ca/tac/volumes/16/19/16-19abs.html Peter Freyd defines an object $A$ in a topos $\mathcal{E}$ to be geometrically finite if ...
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102 views

Sections, Transversals and Quotient Maps

Here I read: Given a quotient space $\bar X$ with quotient map $\pi\colon X \to \bar X$, a section of $\pi $ is called a transversal. I asked myself how such sections are possible. It must be a ...
4
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1answer
333 views

why split epi and mono implies iso?

I was doing some exercises on the definitions of epics, monos, split monos, etc..., and I asked myself that if you could take, for instance an epi which is mono, and then deduce it is an iso, which is ...
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110 views

MacLane's Abelian Categories Chapter

I have just started reading MacLane's chapter Abelian Categories in Categories for the Working Mathematician. I am stuck in the second page, where he discusses the equivalence relation on a set of ...
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3answers
101 views

Is there a category-theoretic perspective on induced functions?

Let $X$ and $Y$ denote sets. Given a function $f : X \rightarrow Y$ and a natural number, there is an induced function $g : X^n \rightarrow Y^n$ defined by $g(x_1,\cdots,x_n) = (f x_1,\cdots,f x_n).$ ...
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3answers
248 views

In what sense is the forgetful functor $Ab \to Grp$ forgetful?

One sometimes hears about "the forgetful functor $Ab \to Grp$." Given that the image of an object under this functor is still abelian, in what sense is this "forgetful"?
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99 views

Property kept under base change and composition is preserved by products

The following is true? Why? Let $P$ be a property of morphisms preserved under base change and composition. Let $X\to Y$ and $X'\to Y'$ be morphisms of $S$-schemes with property $P$. Then the unique ...
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146 views

adjunction relation

Assume I have simplicial sets $X:\Delta^{op}\rightarrow Set$ and $Y:\Delta^{op}\rightarrow Set$, then I can form the simplicial set $\operatorname{Hom}(X,Y)_n := ...
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1answer
112 views

How we can understand one category is small

"A category is said to be small if its objects form a set." Now one question is in my mind and that is although we know lots of sets and always working with them, but how we can show a class of ...
2
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1answer
110 views

Is there a categorical construction of the general linear group?

This question is related to the answer of Qiaochu in this one. Since the object $X=\mathbb{F}_2^2$ generates the category of vector spaces of dimension $2^n$ over $\mathbb{F}_2$, and since we know ...
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157 views

Projective objects in the category of rings

What are the projective objects in the category of rings with identity ? Remarks: The only projectives I could find so far are $\{ 0\}$ and $\mathbb{Z}$. If $R$ is projective and ...
3
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1answer
70 views

What does it mean for a Category to have equalizers or/and pullbacks?

I know the definitions of what pullbacks and equalizars mean, but I don't know what it means that a given category $\mathfrak C$ has pullbacks or equalizers. Thanks
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197 views

Terminology Question: Precompose vs Compose?

I was wondering if there was a standard convention on what 'precompose' means compared to 'compose', as I am often confused between the two when all sorts of text casually use both terminologies. For ...
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Good reference for co-groups, perspective of co-algebra applications

There are lot of applications of state transition systems STS (computer science, planning problems in robotics and so on) and lot of algorithms are devised, but the mathematical background for STS is ...
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153 views

A pedantic question about defining new structures in a path-independent way.

Sometimes there are multiple equivalent ways of defining the same structure; for example, topological spaces are determined by their open sets, but also by their closed sets. I'm looking for a way of ...
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1answer
106 views

Name/notation for the subgroup generated by all stabilizers

Say we have a group $G$ acting on a set $X$. I'm interested in the subgroup generated by all isotropy groups $G_x$, and looking for a designation for it. Thanks in advance! PS1: I thought about the ...
1
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1answer
126 views

Definition of monoids in slice category

Can someone please tell me what would be an appropriate definition of an internal monoid in the slice category? Or better yet, suppose you have an object $p : X \rightarrow A \in \mathcal{C}/{A}$ ...
2
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2answers
166 views

Does an adjoint pair fix a unit/counit pair?

From Ravi Vakil, Fundations of Algebraic Geometry. I want to ask if anyone can give a hint in how to prove Execrise 1.5.B(page 43). I tried to draw the diagram for half an hour but the resulting ...
4
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2answers
242 views

If $gf$ is an equalizer , is $f$ an equalizer?

Suppose $gf$ is an equalizer in a category $\mathfrak C$, I think that $f$ not necessarly is an equalizer, but I don't know how to come up with a counterexample; i've really tried it so hard. Thanks ...
5
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1answer
141 views

What is the minimum required background to understand articles in the nLab?

I am interested in learning more about the nLab categorical perspective on several mathematical subjects such as topology and logic, but found that my understanding of category theory was not ...
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63 views

Are finitely additive measures 'topological'?

The category of measurable spaces are topological over $Set$ in that they support initial & final structures similarly to that topological spaces. A measurable space is a set supporting ...
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1answer
91 views

Why $Kom(\mathcal{A})$ may not be triangulated, while $D(\mathcal{A})$ may not be abelian?

Let $\mathcal{A}$ be an abelian category, let $Kom(\mathcal{A})$ be the category of complex with a shift functor $T$, and Let $D(\mathcal{A})$ be the derived category of $\mathcal{A}$. Why: (1) ...
3
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2answers
145 views

Category-Theory and the modelling of $n$-ary functions (especially the $0$-ary functions)

Hi have a questioning regarding the modelling of $n$-ary functions and constants. First in the category of sets I know that the empty set $\emptyset$ is initial because for every other set $X$ there ...
4
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2answers
184 views

Is duality an exact functor on Banach spaces or Hilbert spaces?

Let $V,V',V''$ and $W$ be vector spaces over $k$. Then, it is known that $\operatorname{Hom}(\cdot,V)$ is a contravariant exact functor, i.e. for each exact sequence $0\to V'\to V\to V'' \to 0$, and ...
13
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0answers
266 views

Closed model categories in the sense of Quillen [1969] vs the modern sense

The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition: Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
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1answer
76 views

Mono/Epi of sections of presheaves

Let $\mathsf{C}$ be a category with initial and terminal objects, and $\phi:\mathscr{F}\to\mathscr{G}$ a morphism of presheaves on $X$ taking values in $\mathsf{C}$. I have a rather messy proof that ...
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3answers
173 views

Reference request - being rigorous about a common abuse of notation.

I've completely rewritten this question, in accordance with this advice. As a motivating example, suppose we're working in ETCS. Let $\bar{1}$ denote the canonical singleton set, and assert that by ...
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Real world applications of category theory

I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if ...
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Saying $a \in b$ in category theory

Suppose I have a category $C$ of sets, and $a,b \in C$. How can I express, in the language of category theory, that $a \in b$? (To clarify: the objects of $C$ are actually sets, and I want to express ...
2
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1answer
67 views

Is there any non-trivial relationship between kernels & kernel pairs?

Kernels are inspired by group theory, and kernel pairs by a similar concept in monoids where kernels aren't sufficient to capture the information necessary for the first isomorphism theorem. When a ...
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140 views

When is the relationship between kernel pairs and kernels an isomorphism?

Kernel pairs can be taken in any category with pullbacks, when there is a zero object we also have kernels. Then there is a morphism from the kernel to the kernel pair (via pullback uniqueness). What ...
5
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intersections in abelian category

Let $\mathcal{A}$ be an abelian category. We fix an object $A$ and we consider the category $mono(A)$ whose objects are the monomorphisms $u:B\rightarrow A$ and where a morphism from $u:B\rightarrow ...