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 ...

learn more… | top users | synonyms (2)

0
votes
0answers
10 views

A localization of a topos is still a topos

I am trying to see how a localization of a topos is a topos. That is, localization of a cartesian closed category is cartesian closed, and if a category has a subobject classifier any localization of ...
0
votes
0answers
16 views

Proper Lie Groupoid confusion

I was studying the book Introduction to Foliations and Lie Groupoids by I. Moerdijk and J. Mrcun and I have a doubt. On page 140 they give the following definition for a proper Lie groupoid. A ...
0
votes
1answer
36 views

Build sheaf from stalks

If I have a topological space $T$ and for each $p \in T$ I have an object $A_p$ in some category $\mathscr{A}$, then how can I define a sheaf out of this? In other words can I build a sheaf with ...
1
vote
1answer
44 views

No natural transformation

I'm having trouble with part b. Intuitively, it is clear that there is no canonical way of assigning a permutation of $X$ onto an ordering of $X$, but I've failed to prove it rigorously thus far. ...
0
votes
2answers
74 views

Should an object in the category always be a formal mathematical structure?

Today I heard during a popular lecture about the applications of category theory, than an object in the category should always be some kind of mathematical structure e.g. a set, ring, monoid, etc. So ...
1
vote
2answers
69 views

Left G-Sets category

Let $G$ be a group, and $\mathbf{G\text{-}Sets}$ the category whose objects are left G-Sets and whose morphisms are G-Set homomorphisms, that is functions $f:X\to Y$ such that $f(ax) = af(x)$, $a\in ...
5
votes
2answers
71 views

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: ...
1
vote
0answers
37 views

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 ...
3
votes
1answer
35 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 ...
1
vote
1answer
31 views

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 ...
0
votes
1answer
85 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 ...
7
votes
1answer
69 views

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 ...
1
vote
1answer
47 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 ...
5
votes
2answers
257 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?
4
votes
1answer
55 views

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 ...
1
vote
0answers
30 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 ...
3
votes
2answers
78 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 ...
3
votes
1answer
39 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 ...
5
votes
1answer
85 views

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 ...
1
vote
2answers
55 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?
8
votes
1answer
97 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 ...
3
votes
0answers
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 ...
3
votes
2answers
63 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 ...
1
vote
1answer
66 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 ...
0
votes
5answers
127 views

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 ...
2
votes
1answer
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 ...
2
votes
1answer
28 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?
1
vote
0answers
44 views

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 ...
2
votes
1answer
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 ...
1
vote
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\\ ...
0
votes
1answer
26 views

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 ...
2
votes
1answer
77 views

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 ...
1
vote
0answers
31 views

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 ...
4
votes
1answer
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_{*} ...
5
votes
2answers
76 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 ...
0
votes
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
0
votes
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 ...
2
votes
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 ...
5
votes
1answer
92 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 ...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
0answers
17 views

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 ...
3
votes
1answer
67 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 ...
1
vote
1answer
45 views

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. ...
1
vote
1answer
32 views

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 ...
5
votes
1answer
56 views

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
votes
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 ...
2
votes
1answer
40 views

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 ...
2
votes
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 ...
2
votes
0answers
65 views

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 ...