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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Does the category of $k$-algebras have terminal objects?

Consider the category of $k$-algebras with $k$-algebra homomorphisms. It's clear that the field $k$ itself is the initial object, since any $k$-algebra morphism must fix $k$. Do terminal objects ...
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1answer
30 views

Example of colimit of Hausdorff spaces which is not Hausdorff

In http://mathoverflow.net/questions/195248/co-hausdorffification, it is mentioned that the subcategory of Top consisting of Hausdorff spaces is not closed under colimits. The simplest colimit I ...
2
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1answer
23 views

Proposition 4.2 in Goerss & Jardine's *Simplicial homotopy theory*

I'm having trouble filling in a detail in Goerss and Jardine's book Simplicial homotopy theory. Their Proposition 4.2 claims that two classes of monomorphisms $\mathbf{B}_2\subset\mathbf{B}_3$ in the ...
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0answers
34 views

Doubt about proof of factorization $f=pi$, where $i$ is acyclic cofibration and $p$ is fibration

I try to understand a proof in More Concise Algebraic Topology: Localization, completions and model categories by May & Ponto (pdf). The proof is on page 262, and it is for the statement Any ...
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1answer
26 views

Restriction functors between Categories O over semi-simple Lie algebras

I have the following question: Let $\mathfrak{g}:= \mathfrak{gl}(3)$ be the general linear algebra and $\mathfrak{gl}(2) \cong \mathfrak{a} \subset \mathfrak{gl}(3)$ a sub-algebra. Let ...
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1answer
46 views

Galois comodules

I tried to figure out why Galois comodules is a generalization of several Galois aspects, but I could not? I am really interested in Comodule theory, and I am very curious to know the answer for ...
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0answers
13 views

Full subcategory of objects equals to the sum of all their noetherian subobjects

Let $\mathcal{C}$ be a Grothendieck category, consider the full subcategory $N(\mathcal{C})$ consisting of the objects of $\mathcal{C}$ which are the sum of their noetherian subobjects. Is ...
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29 views

Reference for closed categories and monoidal categories

I'm looking for a book that: Defines closed categories separately from monoidal categories, and then proves in detail that the structure induced by a left adjoint to the internal hom is closed ...
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0answers
32 views

Fiber product of non-abelian groups.

I am trying to understand whether surjectivity is needed for a fiber product of non-abelian groups to exist. I seem to have checked that the usual construction works for groups without any ...
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0answers
32 views

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ as a coproduct?

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ is defined grading-wise by $$(A\Rightarrow B)_n=\prod_{i\in \mathbb Z} \text{Hom}_R(A_i, B_{i+n})$$ Intuitively, I would have defined the ...
2
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1answer
30 views

Universality of tensor product from monoidal structure

As a follow-up to this previous question of mine, I'm trying to understand how to obtain tensor products from internal homs. I'm having a lot of difficulties and have found myself stuck already in ...
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52 views

What kind of a thing is the forgetful thing $\mathrm{Hom}_\mathbf{C}(-,X) \rightarrow \mathbf{C}/X$?

Let $\mathbf{C}$ denote a category and suppose $X$ is an object of $\mathbf{C}$. Then intuitively, there should be a "forgetful something" from the hom-functor $\mathrm{Hom}_\mathbf{C}(-,X)$ to the ...
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29 views

Definition of Formally Smooth from Stack Project

$T, T'$ are affine schemes. What is meant by $F\leftarrow T$ (or $G \leftarrow T')$
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2answers
142 views

Tensor products from internal hom?

Monoidal categories come with tensor products, and sometimes, these categories are biclosed, i.e each restriction of the tensor bifunctor has a right adjoint. If the category happens to be symmetric, ...
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1answer
24 views

Free product and direct sum as coproduct

In the proof of the coproduct of groups being the free product, it seems that we haven't made any assumption that the groups are non-Abelian. However, we know that for Abelian groups, the coproduct is ...
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69 views

Where can I find linear algebra described in a pointfree manner?

Clearly, some of linear algebra can be described in a pointfree fashion. For example, if $X$ is an $R$-module and $A : X \leftarrow X$ is an endomorphism of $X$, then we can define that the ...
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1answer
58 views

Coproduct of groups

Can anyone explain why the coproduct of groups are the free product? For finite groups, the products are direct products which are also finite. But the free groups are infinite? So the coproduct is ...
2
votes
1answer
60 views

Are there any theorems about functors that reflect exactness?

Suppose $F:\mathbf{A}\to \mathbf{B}$ is an additive functor between two abelian categories, we say $F$ is exact iff it preserves short exact sequences. Is there a name for a functor $F$ that ...
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29 views

“Absolute retracts” in arbitrary category

Is there a standard notion of something like "absolute retract" in arbitrary categories that generalizes absolute retracts in topology? I am mostly interested in categorical approach to Hausdorff ...
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1answer
83 views

Question about general comma categories

Let $F:\cal{A}\to\cal{C}$ and $G:\cal{B}\to\cal{C}$ be functors, and let $(F\downarrow G)$ be the comma category of $F$ and $G$. My question is, how do we know that the Hom-sets are pariwise ...
2
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3answers
61 views

What is a conventional name for a set of values having no properties except that values are distinct?

I know essentially nothing about math but I'm interested in very low-level concepts. I'm thinking of something like a finite or infinite set (although I'm not married to consider sets per se, maybe ...
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1answer
49 views

Presheaf image of a monomorphism of sheaves is a sheaf

EDIT: The original version regarding coker is false. Consider a morphism of sheaves of abelian groups $\Phi:\mathscr{F}\to\mathscr{G}$. It is not in general true that Im$_{pre}\Phi$ is a sheaf. ...
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29 views

An object $S$ is a separator if and only if the set $Mor(A,S)$ is a monosource?

I have started self studying category theory from a set of lecture notes and I am struggling with the following excercise. Let $\mathbb{A}$ be any category. First I will provide the definitions: ...
3
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1answer
47 views

Wide equalizers

If $(f_i : A \to B)_{i \in I}$ is a family of morphisms in a category, we may declare their wide equalizer as a universal morphism $\iota : E \to A$ which satisfies $f_i \iota = f_j \iota$ for all ...
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1answer
28 views

Diagonal morphism and zero [closed]

Is true that in an additive category an object is a zero object iff the diagonal morphism is an isomorphism?
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62 views

Image sheaf is the sheafification of the image presheaf

This is an exercise in Vakil's notes on foundations of algebraic geometry. Suppose $\Phi:\mathscr{F}\to\mathscr{G}$ is a morphism of sheaves of abelian groups, show that the image sheaf Im $\Phi$ ...
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1answer
44 views

Greatest common factor in a category

I'm looking for a name (or references or search terms) for the following construction. In a category $\mathcal C$, let a doodle mean a pair $\langle D,\Omega\rangle$ where $\Omega$ is an object and ...
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48 views

Exterior Algebra: Characterization [closed]

Just a short question: Given a vector space. Why is the exterior algebra characterized as: The largest anticommutative integer-graded algebra with identity linearly embedding the vector ...
3
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1answer
32 views

The irreducibility of representations of parabolic induction over $\mathfrak{gl}_n(\mathbb{C})$

I have the following problem: Let $\mathfrak{g} = \mathfrak{gl}(n)$ be the general linear Lie algebra over $\mathbb{C}$. Denote $\Pi = \{\alpha_i:=\epsilon_i-\epsilon_{i+1}\}_{i=1}^{n-1}$ by the ...
2
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1answer
50 views

$\mathfrak{Top}$ and injective objects

My question is very simple. Let $\mathfrak{Top}$ be the category of all topological spaces and continuous functions between them. Does such category have enough injectives? Is there a simple way to ...
2
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1answer
57 views

Colimit preserves monomorphisms under certain conditions

I know that colimit preserves epimorphisms. Consider the special case where The diagrams are indexed by a directed set $I$, We are in the category of certain algebraic structures, such as ...
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0answers
36 views

Coproduct of Lie algebras

Fix a commutative ring $k$ and look at the category of Lie algebras over $k$. How do coproducts in that category look like? Notice that what is usually called the "direct sum" of Lie algebras is not ...
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247 views

Finite-dimensional space naturally isomorphic to its double dual?

The example that a finite vector space is naturally isomorphic to its double dual seems to be the canonical example of natural isomorphisms. Concretely, there are two functors $\mathsf{Id}, {-^*}^* : ...
3
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1answer
68 views

Invariance of the Fredholm index under finite-dimensional perturbations

Let's call a linear map $f : V \to W$ between vector spaces over some field Fredholm if $\ker(f)$ and $\mathrm{coker}(f)$ are finite-dimensional. (Equivalently, it represents an isomorphism in the ...
0
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1answer
38 views

SupLat and InfLat

I just read in ncatlab that SupLat and InfLat are equivalent. But, it seems to me, that they should actually be isomorphic categories. Am I correct or is ncatlab correct and they are not isomorphic? ...
5
votes
1answer
87 views

In which cases does pullback commute with the Hom-sheaf?

Assume $f: (X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ is a morphism of locally ringed spaces and E and F are two locally free $\mathcal{O}_Y$-moduels of finite rank. I was wondering if we have ...
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142 views

Do hom-sets really live in the category Set?

In familiar introductory books on category theory, one of the first examples of a category given is Set. And what category is that? Typically no explanation is given at this stage. But of course ...
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83 views

What is the “correct” way of making $\mathcal{P}(X)$ into a topological space?

If $X$ is a topological space, I want to know the "correct" way of making the powerset $\mathcal{P}(X)$ into a topological space. So let $\mathsf{SupLat}$ denote the infinitary Lawvere theory whose ...
2
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0answers
57 views

Connecting morphism in an abelian category

I'm trying to understand how one gets the long exact sequence in homology from a short exact sequence of chain complexes in an arbitrary abelian category. So far I have the commutative diagram ...
2
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2answers
71 views

Relating categorical properties of arrows

Diagram is from some book on categories shelf (ID?)- It's a good visual to rembember. What are some obvious extensions of categorical properties? Ie valid in every category, eg: strong-, extremal-, ...
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64 views

Examples of categorical adjunctions in analysis and differential geometry?

In a lot of introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis and ...
1
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1answer
37 views

A necessary and sufficient condition for contravariant auto-equivalence on module categories

I have a problem about the condition of contravariant auto-equivalence on module categories. Let $R$ be a algebra over a field. Let $\mathcal{C}$ be a abelian subcategory of $R$-modules, and assume ...
3
votes
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33 views

The bijection between central characters and linkage classes over a semisimple Lie algebra

I have a question about the modules over a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let $\mathfrak{h} \subset \mathfrak{g}$ be a Cartan subalgebra. For a given $\lambda \in ...
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1answer
33 views

Non-integral blocks of category $\mathcal{O}$ over $\mathfrak{sl}_2$ are semisimple.

Hi: I have a problem as follows. Consider the category $\mathcal{O}$ of $\mathfrak{g}: = \mathfrak{sl}_2(\mathbb{C})$. Let $r\in \mathbb{C}$ but $r\notin\mathbb{Z}$. Let $s_\alpha$ be the simple ...
2
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1answer
24 views

Common kernel for compositions of epis?

The following proposition is an excerpt from Osborne's *Basic Homological Algebra: Regarding the proof: Why does there exist an arrow $j$ which is simultaneously the kernel of both $\pi$ and ...
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1answer
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Abelian categories with tensor product

Is there a standard notion in the literature of abelian category with tensor product? The definition ought to be wide enough to encompass all the usual examples of abelian categories with standard ...
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2answers
106 views

Toy examples for Kan extensions

Background: If $\mathcal{C}$ is a cocomplete category and $f : I \to J$ is a functor between small categories, then $f^* : \mathrm{Hom}(J,\mathcal{C}) \to \mathrm{Hom}(I,\mathcal{C})$ has a left ...
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1answer
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When is it possible to interpret composition as a natural transformation?

First note that for any objects $X$, $Y$, and $Z$ in a category $C$, we can get a morphism $\bigcirc: Z^Y \times Y^X \rightarrow Z^X$ as following. We define $\bigcirc$ as $\lambda (eval_{Z^Y} \circ ...
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1answer
49 views

$\operatorname{Im}f\cong A/\operatorname{Ker}f$ in abelian categories

Let $f:A \rightarrow B$ be an arrow in some abelian category. There is the usual epi-mono factorization of any such arrow, but can we go further and prove isomorphism of the objects: ...
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Projection between quotients by related subobjects

For a subobject $A\overset{a}{\rightarrowtail} B$ we define the quotient object $B\twoheadrightarrow B/A$ as the cokernel of any monic representing the subobject. Suppose we have another subobject ...