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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Comparing Category Theory and Model Theory for Master's Thesis.

I am currently doing a Masters thesis in pure maths, and the two current fields that excite me are Category Theory (CT) and Model Theory (MT). I have been reading up on David Marker's Model Theory: ...
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Pushout in $\mathsf{Set}$ where one of the maps is injective

From I.M. James' book General Topology and Homotopy Theory: Suppose we have a cotriad $$X \xleftarrow{\xi}W \xrightarrow{\eta} Y.$$ ... we might expect the pushout of the cotriad to be a ...
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29 views

Natural transformations and the definition of Monoidal lax functors

The definition of a lax monoidal functor requires the existence of a natural transformation, $\phi$ http://en.wikipedia.org/wiki/Monoidal_functor. A natural transformation relates at least 2 ...
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Why is the $\mathbb Z$-module $\mathbb Q$ projective in category $\sigma[\mathbb Q]$?

We know that $\mathbb Q$ is not a projective $\mathbb Z$-module, but, I've read this paper and it says in Example 2.11 that it's easy to check that "$\mathbb Q$ as $\mathbb Z$-module is projective ...
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71 views

Elementary proof that the category of modules is not self-dual

If $R,S$ are rings such that ${}_R \mathsf{Mod}$ is equivalent to ${}_S \mathsf{Mod}^{\mathrm{op}}$, then $R$ and $S$ are trivial. This is well-known. The usual proof uses of the notions of limit and ...
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How to prove that in this context epimorphisms are 'surjective'

In article AN ELEMENTARY THEORY OF THE CATEGORY OF SETS of William Lawvere I met a proposition left to reader (poor me) and I hope someone can help me with it. It wouldn't surprise me if it is not ...
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46 views

zero object in the category of group schemes

I am currently reading Ravi's lecture notes on AG, and in the introduction of group schemes(Section 6.6), he made a comment after 6.6N that the category of group schemes has a zero object. I can ...
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254 views

Is Category Theory similar to Graph Theory?

The following author noted: Roughly speaking, category theory is graph theory with additional structure to represent composition. My question is: Is Category Theory similar to Graph Theory?
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Categorical Banach space theory

Consider the category $\mathsf{NormVect}_1$ of normed vector spaces with short linear maps$^{\dagger}$ and the full subcategory $\mathsf{Ban}_1$ of Banach spaces with short linear maps. Both ...
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29 views

Definition of a regular category via extremal epi

I know one definition of a regular category saying that a category is regular if it is finitely complete, every kernel pair admits a coequalizer and regular epis are stable under pullback. Now, I am ...
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2answers
76 views

What are some beautiful examples of adjunctions?

Lately I've been very interested in finding examples of adjunctions. In particular, examples that are elementary enough for an undergrad like me to grasp. So I was wondering if perhaps you could share ...
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38 views

What does it mean for pullbacks to preserve monomorphisms?

If two arrows $f_A : A \to C$, $f_B : B \to C$ are monomorphisms, then their pullback arrows $p_A : P \to A$, $p_B : P \to B$ are monomorhisms too. Is that what is meant by pullbacks preserving ...
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Is the product of all objects of a finite category an initial object?

If the product of all objects in a finite category exists, is it an initial object? I presume so, but I'm still learning this subject and I can't make a proof go through. Advice welcome. (Not a ...
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2answers
54 views

Pushout of a subgroup

Let $G$ be a group, $H\subseteq G$ be a subgroup and $\iota: H\to G$ the inclusion map. What is the "pushout along $\iota$? How is it constructed?
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96 views

Characterization of the circle within metric spaces

There are various characterizations of the circle. To be precise, there is not the circle. There are several categories which contain an object we refer to as "the circle". In $\mathsf{Top}$ the ...
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52 views

Looking for info on power set functor

I was reading here about the various functors which take a set $S$ to its power set. In particular, there is the normal contravariant one, and two covariant ones, which the article calls $\exists$ and ...
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2answers
62 views

The inclusion $\mathbb Z \to \mathbb Q$ is an epimorphism

I am supposed to show that the inclusion of the integers in the rationals is an epimorphism in the category of abelian groups. Not only am I unable to find the right argument, I am starting to wonder ...
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65 views

Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?

We are taught that, in general: A type of objects that has nontrivial automorphisms cannot have a fine moduli space. The proof generally goes along the lines of: Take an object $X$ with a ...
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Why do counits go that way?

Imagine you want to motivate for an audience the definition of an adjunction in terms of unit and counit. So you can say: Often two functors $\mathcal{C} \begin{array}{c} \stackrel{\large ...
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29 views

Understanding products and coproducts

I am currently trying to make sense of products and coproducts in different categories, and even in starting with the basic examples (cartesian product and disjoint union in the category of sets) I've ...
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1answer
27 views

Why is the internal hom of a Kan complex also a Kan complex?

Let $X, Y : \Delta^{op} \to \text{Set}$ be simplicial sets. I've seen it stated that if $Y$ is a Kan complex, then the internal hom $Y^X$ is a Kan complex also. How does one show this?
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Subobjects(A) $\cong \operatorname{Hom}(A,\Omega)$ in a topos is a natural transformation?

Let $C'$ be the category with objects $C$ and morphism the monic morphisms of $C$. In any topos, $\phi_A: \operatorname{Sub}(A) \cong \operatorname{Hom}_C(A,\Omega)$ and $\phi_A(m) = \chi_m$. This ...
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45 views

Is GRP a subcategory of SET, or not? [duplicate]

This is the notion of a subcategory $\mathscr{D}$ of a given category $\mathscr{C}$ which I use: it consists of a subcollection of the collection of objects of $\mathscr{C}$ and a subcollection of the ...
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2answers
43 views

What does it mean to say a diagram commutes?

$\require{AMScd}$ In the context of smooth manifolds, the map $F:M\rightarrow N$ is smooth if $G$ on the below diagram is smooth. $\begin{CD} M @> F > > N\\ @V \varphi V V @V V\psi V\\ ...
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Representability criterion with universal element

For a categroy $\mathcal{A}$ we say that a functor $X: \mathcal{A} \to \mathbf{Set}$ is representable if there is some $A\in\mathcal{A}$ and a natural isomorphism $\alpha: \hom(A,-)\to X$. Now as a ...
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Functorial construction with two integral domains

Motivated by this question: Let $\mathsf{Int}$ be the category of integral domains with ring homomorphisms (perhaps only injective ring homomorphisms, if you need this). Is there a functor ...
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About the construction of Quot-Schemes

I am reading in the paper of Nitsure (link: http://arxiv.org/pdf/math/0504590v1.pdf) about the construction of the Quot-scheme $\mathrm{Quot}_{E/X/S}^{\Phi,L}$. After Lemma 5.4. they reduce to the ...
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49 views

Lifting adjunctions

Is there a convenient hom-set proof that an adjunction $ F \colon C \rightleftarrows D \colon G $ where $ F $ is left adjoint to $ G $ can be lifted to an adjunction of functor categories $ F_{*} ...
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75 views

What are monomorphisms in the category of real vector bundles over a fixed base space $X$?

I have a (perhaps stupid) question dealing with vector bundles. In most textbooks, a morphism of real vector bundles $f:\xi\to \eta$ over a fixed base space $X$ is said to be a ...
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1answer
42 views

How can I prove that $Set^I≃Set/I$?

I need help to prove this equivalence. Anyone can do an exhaustive explanation about this? Thank you so much
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1answer
62 views

every category is equivalent to its universal cover

I am just curious how could we show that every category is equivalent to its universal cover. To me, it is not obvious how could we assign to each an object in a category $\mathcal{C}$ to a family of ...
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1answer
38 views

Isomorphic categories

in our lecture notes there is the Statement that the categories of $K$-vector spaces together with linear endomorphisms is isomorphic to the category of $K[X]$-modules. Now I know how to view a vector ...
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89 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley ...
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27 views

Universal Properties and Equalizer

Good evening. At the moment I am looking into category theory and at the moment I am trying to proof the Universal Property of the kernel as described here. I use the definition that the kernel of ...
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1answer
40 views

Functorizing a choice of sections

Take $\mathcal{C}$ and $T$ to be categories (if it helps, assume $T$ is a poset with a minimal element and $\mathcal{C}$ is cartesian closed). Take a functor $P\colon T\to \mathcal{C}$ where the image ...
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Inverse limit of small categories

It is well known that category $\mathcal{Cat}$ of small categories has all small limits and colimits. In particular it has all iverse limits. I am wondering if there is an explicit constraction of an ...
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1answer
65 views

(co)limits in the category of diffeological spaces vs. category of smooth manifolds

I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ...
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1answer
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N-Tuples or N-functions in category theory

When I'm writing out categories of my Haskell programs, I often get stuck whilst trying to describe morphisms that involve functions that involve more than one argument, such as 2-tuple construction. ...
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Predecessor on the final coalgebra (the extended natural numbers): difference of notation

In the Wikipedia article, the predecessor $f$ on the final coalgebra is not defined at 0, it is only defined at $n+1$ and $\infty$. In the $n$Lab article, $\operatorname{pred}(0) = *$, but $*$ is ...
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Isn't this a non-surjective epimorphism on the category of sets?

I am trying to prove that a morphism in the category of sets is epic iff it is a surjective function. Recall that for objects $A,B,C$, $f \in \hom(A,B)$ is epic when $g_1 \circ f = g_2 \circ f ...
5
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1answer
52 views

Prove that the isomorphism between vector spaces and their duals is not natural [duplicate]

In preparation for an introductory talk on category theory, I recently spent some time thinking about natural transformations. The first example, or maybe the second, that everyone gives to motivate ...
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ISO information on powerset functor [closed]

This site has the very bare bones, but I'd like to see more.
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Bijection in the Yoneda Lemma

To prove the Yoneda Lemma one defines a bijection between $[\mathcal{A}^{op},\mathbf{Set}](\hom(-,A),X)$ and $X(A)$ and shows that this bijection is natural in $A$ and $X$. In my textbook this ...
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1answer
19 views

Power set functors preserve monicness

This link discusses power set functors. Proposition 5.7 If $f$ is a epimorphism then so is $\exists_f$. Proposition 5.8 If $f$ is an monomorphism then so is $\forall_f$. A little ...
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In an abelian category is it true that $\ker f \cong \ker (\operatorname{coker} (\ker f))$?

In an abelian category is it true that $\ker f \cong \ker (\operatorname{coker} (\ker f))$? I am teaching myself category and was playing with the definitions of kernel and cokernel and think I ...
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Set of axioms for finite subset of Natural Numbers

I would like to get a set of Peano like first-order axioms for a finite subset of natural numbers $N'$ such that $0 \leq N' \leq Max$, with $Max$ denoting the upper-bound. (So my signature might be ...
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125 views

Name of some category with two objects

Does the category that consists of two objects and exactly one non-identity morphism that connects both objects have a specific name?
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Identify the universal property of kernels

I'm reading Mac Lane's "Categories for the Working Mathematician". I found the following sentence in page 59 of it: Similarly, the kernel of a homomorphism (in $\mathbf{Ab}$, $\mathbf{Grp}$, ...
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Colimits in $Ch_R$, help with a step of the proof

I want to prove that the category of chain complexes of R-modules admits small colimits. I was told to try proving that the chain complex defined degree wise as the colimit of the modules of the same ...
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What's up with this endofunctor $\mathbf{Aff}_k \rightarrow \mathbf{Aff}_k$?

(Work over a fixed field $k$.) The nLab offers a list of definitions of the concept "affine space". Here's two of them: An affine space is a set $A$ together with a vector space $V$ and an ...