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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Pushout with principal bundles

I am looking at the wikipedia page on reduction of the structure group for principal bundles (http://en.wikipedia.org/wiki/Reduction_of_the_structure_group) and at the beginning they introduce, for an ...
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1answer
46 views

Why are continuous functions the “right” morphisms between topological spaces?

Recently, someone mentioned to me that given a function $f: X \to Y$ there are two natural functions between the powersets $P(X)$ and $P(Y)$. Namely $f: U \subset X \mapsto f(U)$ and $f^{-1}: V ...
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19 views

The closure of the graph of a certain composite in a topos.

Let $\mathcal{E}$ be an elementary topos with subobject classifier $\Omega$ and let $j\colon \Omega\to\Omega$ be a Lawvere-Tierney topology on it. Assume that, for an object $C$ of $\mathcal{E}$, each ...
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1answer
61 views

Question regarding adjoint functors

Suppose $B \rightarrow A$ is a morphism of rings. If $M$ is an $A$-module, one can create $M_B$ by considering it as a $B$-module. This gives a functor $\cdot_B: \mathrm{Mod}_A \rightarrow ...
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1answer
34 views

Notation for a functor between comma categories

Suppose we have two categories $D$ and $S$, as well as two functors $K,L:D\to S$ and a natural transformation $\varphi:K\to L$. Given another category $C$ and a functor $Y:C\to S^D$, is there a nice ...
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35 views

adjoint of forgetful functor related to localization

Let $A$ be a ring and $S$ a multiplicative subset of $A$ such that $1 \in S$. Let $G$ be the forgetful functor from $Mod_{S^{-1}A} \rightarrow Mod_A$. Taking an $S^{-1}A$-module N and consider it as ...
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1answer
43 views

How do I calculate adjoint functors to forgetful functors?

Suppose I have a forgetful functor $F:Ab\hookrightarrow Grp$ where we forget that we have commutativity. I'm trying to calculate the left and right adjoint functors, so for right adjoint, we have ...
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38 views

Confusion about pullback in $\sf{Set}$.

Given sets $A, B, C$ and set functions $f: A \to C$ and $g: B \to C$, everywhere I look seems to be telling me that the pullback is $A \times_C B = \left\{(a, b) \mid f(a) = f(b)\right\}$ (together ...
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2answers
104 views

The Freyd-Mitchell Embedding Theorem and projective (injective) objects

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell Embedding Theorem gives me a fully faithful exact functor $F:\mathcal{A}\rightarrow R$-$\mathsf{Mod}$, for some unital ring $R$, so ...
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1answer
69 views

Some functorial maps $G\times G\rightarrow G$

Let $G$ be a group. Le diagonal map $\delta:G\rightarrow G\times G$ obviously gives a functorial morphism from the identity functor of $\mathbf{Grp}$ to the functor $P$ sending $G\mapsto G\times G$ ...
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28 views

Derived categories as homotopy categories of model categories

Given an abelian category A, is there a model structure on the category of complexes C(A) (or K(A) ("classical" homotopy category)) such that its homotopy category "is" the derived category D(A)?
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51 views

Where to study $2$-category theory?

Is there any place where I can read about $2$-categories? I am looking for a proper treatment - there is a section in Borceux's Handbook of Categorical Algebra, but it only sketches some parts of the ...
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33 views

What is an easy to read book on category theory including the introduction of some killer apps for the theory? [duplicate]

What is an easy to read book on category theory including the introduction of some killer apps for the theory ?
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13 views

Endofunctors on Cat preserving reflexive coequalizers

Is there any characterization of endofunctors on $\mathbf{Cat}$ preserving reflexive coequalizers?
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2answers
47 views

Must an epimorphism in abelian category have cokernel $0$?

Suppose $\mathcal{A}$ is an abelian category, that is an additive category with 1) a zero object, 2) all binary products and binary coproducts, 3) all kernels and cokernels. 4) monomorphisms are ...
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1answer
37 views

A seemingly simple fact about construction of maps proven categorically, i.e. by universal properties

If I have four sets $A,B,C,D$ and two maps $f_1 : A \to C$ and $f_2 : B \to D$, it is easy to find a unique map $f : A\times B \to C\times D$, namely $$ f(a,b) := (f_1(a), f_2(b)). $$ But now I want ...
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62 views

Transfinite composition and $\kappa$-categories

I wonder whether the following ideas make sense, and whether something in that direction has been written down somewhere; do you know a reference? The last definition is supposed to be a ...
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10 views

Morphisms of $\mathsf{Meas}$ and Dynamical Systems

The morphisms of a category $\mathsf{Meas}$ whose objects are measure spaces are defined to be equivalence classes of a.e-equal measurable maps that pull back null sets to null sets. Why is pulling ...
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3answers
68 views

Does the intersection of sets have a categorical interpretation?

My question is the title, really. I am wondering if the intersection of sets can be seen as a categorical construction on the objects of $\mathbf{Set}$.
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1answer
28 views

Monoidal categories and Generators

Let $\mathcal{C}$ be a Cocomplete Cowellpowered Monoidal category. Does $\mathcal{C}$ need to have a generator? I think it does not, but it seems hard to get a counter example.
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1answer
38 views

Adjoint functors for the power set monad

There is the power set functor, $T$, which gives raise to a monad: For a set $X$, we set $TX:=\mathcal P(X)$ and for $f:X\to Y$, we set $T(f):=S\mapsto f(S)$, where $f(S)$ denotes the direct image. ...
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finite category,idempotents,filtered diagrams

In a finite category A the only filtered diagrams are its idemoptents fof=f. What is the argument?I have completely described my question, but I have to write another sentence so that the question may ...
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1answer
49 views

Question regarding Vakil's algebraic geometry notes

Exercise 1.3 D of Vakil's lecture notes on algebraic geometry asks: "Verify that $A \to S^{−1}A$ satisfies the following universal property: $S^{−1}A$ is initial among $A$-algebras $B$ where every ...
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3answers
51 views

question on fibred products

We are in a category where everything that follows exists. Is the fibred product $A \times_B (B \times_C D)$ isomorphic to $A \times_C D$ or to $A \times_C (A \times_B D)$?
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38 views

Question concerning the meaning of an equality sign in a commutative diagram

$\require{AMScd}$ I have the following question: Let $\mathscr{C}$ be a category, $X,Y,Z\in Ob(\mathscr{C}), \ f\in Mor(X,Y),\ g\in Mor(Y,Z)$ and $h\in Mor(X,Z)$. Question: What does the ...
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14 views

Putting a direct system on a product of direct limits

I was going back through some class notes discussing the direct limit topology (final topology) and we showed that the direct limit of topological groups is a topological group. To do this, we showed ...
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2answers
62 views

Abelian category without enough injectives

What is an example of an abelian category that does not have enough injectives? An example must exist, but I haven't been able to find one. If possible, a brief explanation of why the abelian ...
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1answer
44 views

Can we define near-rings as some kind of a monoid object in the category of groups?

I recently learned about the tensor product of Abelian groups, which can be used to define the concept 'ring.' In particular, a ring is just a monoid object in the monoidal category of Abelian groups, ...
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3answers
80 views

Exact functors and “short” exact functors

Let $A$ be a commutative ring. I thought that an exact functor (from the category of $A$-modules to itself) is defined to be a functor which sends every exact sequence to an exact sequence. But many ...
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3answers
99 views

History of category theory

I am searching some information about the origins of the category theory. Anyone know where can I read about those topics? Thanks!
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2answers
69 views

Extensions of $\mathbb{Z}_p$ by $\mathbb{Z}$ (Hilton & Stammbach III.1.2)

Question is to compute $E(\mathbb{Z}_p,\mathbb{Z})$ i.e., equivalence classes of extensions of $\mathbb{Z}_p$ by $\mathbb{Z}$ By an extension of $A$ by $B$ i mean an $R$ module $E$ such that ...
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48 views

Algebraic K-theory: Categories of modules and their equivalences

I'm currently preparing a presentation on the second chapter, called "Categories of modules and their equivalences", in Algebraic K-theory by Hyman Bass. I have a VERY elementary understanding of ...
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1answer
43 views

How to see whether a category is small or not.

I am new to category theory and I have no experience with set theory and logic. I am having trouble with the notion of smallness. In particular, I can not answer to myself questions of the form "Is ...
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0answers
28 views

Locally Cartesian Closed implies Cartesian Closed

We follow Awodey's definition (page 235, Category Theory, 2nd ed.) of locally cartesian closed categories: A category $\mathcal E$ is locally cartesian closed provided $\mathcal E$ has a terminal ...
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1answer
29 views

Category of pointed manifolds

Let consider the following data: the family of pairs $(M,p)$ with $M$ a smooth manifold and $p \in M$ for every pair $(M,p)$ and $(N,q)$ as above a set $\hom[(M,p),(N,q)]$ whose elements are germs ...
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26 views

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 ...
3
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1answer
44 views

pull back maps in category of sets

Given $\phi : A\rightarrow X$ and $\psi : B\rightarrow X$ in $\mathcal{C}$ (category), a pull back of $(\phi,\psi)$ is a pair of morphisms $\alpha : Y\rightarrow A$ and $\beta : Y\rightarrow B$ such ...
3
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1answer
57 views

Why is every simple object in the category of abelian groups simple in the category of groups?

I'm not asking for a proof that every simple abelian group is simple; that's a fairly trivial question. To lead into my real question, I started thinking about this question: Let $\cal C$ be a ...
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1answer
33 views

Maps from cogroups to groups & Eckmann-Hilton

One way to prove that a topological group has abelian fundamental group is to point out that the two group operations are homomorphisms for each other and apply the Eckmann-Hilton argument. ...
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1answer
49 views

hom-set definition of limit?

I've heard of limits of a diagram as a cone with universal property. But on the ncatlab website, they define it as a hom-set. Specifically, for the limit of a set valued functor $F: D^{op}\rightarrow ...
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1answer
53 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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2answers
50 views

Non-bijective isomorphism in a category of of sets.

I have been commanded on homework to find a non-bijective isomorphism in a category whose objects are sets, whose morphisms are set maps, and composition is the usual function composition. So our ...
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0answers
64 views

If models of set theory can be construed as categories, can notions of forcing be construed as functors?

Consider the following passage from Blass and Scedrov's paper "Complete topoi representing models of set theory"(Annals of Pure and Applied Logic, vol. 57 (1992),PP. 1-26) where 'set theory' in this ...
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1answer
66 views

Proving that the forgetful functor $U:\mathbf{Ring}\to\mathbf{Set}$ is representable.

I am trying to prove that the forgetful (covariant) functor $U:\mathbf{Ring}\to\mathbf{Set}$, sending a given ring to its underlying set is representable. I know this functor is represented by the ...
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43 views

Is there a name for taking the pushforward (ie pushout) over the pullback?

Given maps continuous maps $f:U \rightarrow X$ and $g:V \rightarrow X$ between topological spaces; then there is a unique map $f \sqcup g:U \sqcup V \rightarrow X$ whose restrictions to $U$ & $V$ ...
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1answer
52 views

Do any 2 morphisms from objects $X$ to $Y$ define a chain homotopy equivalence?

I was curious about one thing: Let $A$ and $B$ be abelian categories with enough projectives, let $X$, $Y$ be objects in $A$ and let $P_{\bullet} \rightarrow X$, $P'_{\bullet} \rightarrow Y$ be ...
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95 views

Do schemes have a characterisation as an etale space?

In differential geometry, an etale bundle on a manifold $X$ is a bundle whose projection is a local homeomorphism. This can be equivalently presented as a sheaf of rings on $X$. This can be ...
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1answer
41 views

Idempotents and Adjoints

Suppose $F:C\rightarrow D$ and $ G:D\rightarrow C$ are functors, $\eta :1_{C}\rightarrow GF$ and $\varepsilon :FG\rightarrow 1_{D}$ are natural transformations. Suppose further that $G\varepsilon ...
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33 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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3answers
96 views

Recommendation on Category theory textbook [duplicate]

I had posted a question about category theory some months ago, and I got answered that there are two ways to study Category Theory. One is to treat Category Theory as a logic system independent from ...