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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25 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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0answers
15 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
82 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
18 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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53 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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38 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
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1answer
54 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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27 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
54 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
57 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
47 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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0answers
23 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: ...
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1answer
41 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
26 views

Diagonal morphism and zero [on hold]

Is true that in an additive category an object is a zero object iff the diagonal morphism is an isomorphism?
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0answers
56 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
42 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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0answers
47 views

Exterior Algebra: Characterization [on hold]

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
26 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
39 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
55 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 ...
2
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0answers
34 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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2answers
243 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
65 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 ...
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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? ...
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1answer
84 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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140 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 ...
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0answers
53 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
68 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-, ...
5
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0answers
63 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 ...
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1answer
35 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
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0answers
31 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
32 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
22 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
47 views

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 ...
4
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2answers
103 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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58 views

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 ...
0
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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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2answers
49 views

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 ...
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57 views

If two groups $G$ and $H$ both have a nonzero homomorphism to all other groups, and have only two idepotent homemorphisms, are they isomorphic?

Consider to Groups $G$ and $H$ such that: The only two endo-hom omorphisms (homomorphisms from a group to itself) that are idempotent (any function $f$ such that $f(f(x))=f(x)$ for all x) are the ...
3
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2answers
51 views

Long exact sequence into short exact sequences

This question is the categorical version of this question about splitting up long exact sequences of modules into short exact sequence of modules. I want to understand the general mechanism for ...
1
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2answers
51 views

Demonstrate currying via homomorphism

You can demonstrate currying for two-argument functions is possible showing there's a isomorphism between $(A^B)^C \cong A^{B \times C}$. That is, the set of functions $(C \rightarrow B) \rightarrow ...
3
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0answers
78 views

The Seifert-Van Kampen theorem as a push-out

My question concerns the proof of the Seifert-Van Kampen theorem. The version of such a statement that interests me is the following. Let $X$ be a topological space, and $U, V\subseteq X$ two ...
9
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1answer
207 views

A question about Hom functor in category theory

Suppose $\mathcal{C}$ is a category and $f: A \rightarrow B $ is a morphism from an object $A$ to another object $B$ such that $f_*:Hom(C,A) \to Hom(C,B)$ is bijective $\forall C \in ob(\mathcal{C})$. ...
4
votes
1answer
57 views

Characterization of Schur Functors

A Schur functor (in the thoery of algebraic operads) on the category of vector spaces over a field $k$,(or more generally any abelian symmetric tensor category) is defined by: $$\tilde{M}(V) := ...
2
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1answer
38 views

Diagram commutativity with adjoint functor

Suppose I have a diagram $A \rightarrow B \rightarrow C = A \rightarrow D\rightarrow C$ which I would like to commute. I already obtained commutativity for $\mathcal{F}A \rightarrow \mathcal{F}B ...
3
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1answer
46 views

The injectivity of torus in the category of abelian Lie groups

HI: I have the following question: Definition: A Lie group $T$ is called a torus if $T\cong \prod_{1\leq i\leq k} \mathbb{R}/\mathbb{Z}$ for some $k\in \mathbb{N}$. ${\bf Question}$: Is it true ...
2
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1answer
59 views

Morphisms of sheaves

My question concerns direct/inverse image of sheaves and their properties. Let $\mathfrak{R}$, $\mathfrak{S}$, $\mathfrak{T}$ three sheaves of groups, over topological spaces $X$, $Y$ and $Z$ ...
4
votes
1answer
51 views

Is the existence of finite biproducts a strengthening of commutativity?

Let $T$ denote a monosorted Lawvere theory (call its distinguished object $G$) equipped with a distinguished constant $0 : G \leftarrow 1$ that is "idempotent" in the following sense: for all arrows ...
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0answers
31 views

Applications of Splitting Lemma and Exactness

I'm looking for nice applications of exact sequences, the splitting lemma, and exact functors in algebra and topology (i.e not using the five lemma to get long sequences in homology etc..). For ...