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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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
35 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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40 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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35 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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43 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 ...
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1answer
23 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 ...
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32 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 ...
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1answer
44 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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32 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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237 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}, {-^*}^* : ...
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1answer
60 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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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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82 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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135 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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81 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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45 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 ...
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2answers
63 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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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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34 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 ...
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30 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
30 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 ...
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1answer
21 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
46 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 ...
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2answers
102 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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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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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
48 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 ...
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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 ...
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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 ...
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76 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 ...
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1answer
201 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
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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
37 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
50 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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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 ...
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1answer
50 views

If there monomorphisms $f : A \rightarrow B$ and $g : B \rightarrow A$ then there is an isomorphism $h : A \rightarrow B$

Consider the following set theoretical result of Schröder-Bernstein-Cantor: Let $A$, $B$ be sets. Assume we have injections $f : A \rightarrow B$ and $g : B \rightarrow A$. Then there exists a ...
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1answer
28 views

Borceux - Snake Lemma Question

Below is the statement of the snake lemma from Borceux. My question is which squares are (1) and (2) referring to?
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83 views

Equivalent definition of Schemes

I recall seeing that the category of schemes can be captured by a general construction as follows. Let $\mathbf{Spec}\colon \mathbf{CRing}^{op}\to \mathbf{LRS}$ be the usual functor from the ...
2
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1answer
48 views

Equivalent dfn of Filtered Categories

Let $\mathbb I$ be a small category (i.e., its class of arrows is a set) which satisfies the following: (1) it is nonempty, (2) for each $i,i'\in \mathbb I$ there exists $i''\in \mathbb I$ and ...
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2answers
71 views

pushout of topological Hausdorff spaces is not Hausdorff

$A$, $X$, $Y$ are topological Hausdorff spaces, $f:A\to X$, $g:A\to Y$ continuous maps. I search an example where the pushout $Z$ of the morphisms $f$ and $g$ is not Hausdorff. I thought if I take ...
4
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1answer
29 views

Are units in rigid (autonomous) categories some sort of natural transformation?

In a rigid category $\mathcal{C}$, let us choose left and right duals and left and right (co)units for every object. This gives us, for example, a dualisation functor $-^*:\mathcal{C} \to ...
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1answer
55 views

Limit of groups

My question concerns the properties of special limits of groups. Let $G'$ and $G''$ be two small groups. Suppose that the following diagram in the category $\mathfrak{Grp}$ of small groups and ...
4
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1answer
70 views

Wedge product of a direct sum and the Yoneda Lemma

In a comment to http://math.stackexchange.com/a/344851/58601, Martin Brandenburg suggests that one may prove the existence of the canonical isomorphism $\wedge^n(W_1 \oplus W_2) \to \bigoplus_{p+q=n} ...
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2answers
35 views

Kernel of an arrow that factors through a monic?

Suppose an arrow $A\overset{f}{\rightarrow}B$ factors as $A\overset{q}{\rightarrow} J \overset{j}{\rightarrowtail}B$. When does $\ker f=\ker q$ and how can I prove it?
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Open coverings and (co)limits

My question concerns general topology and category theory. Let $X$ be a topological space, and consider an open covering $\{U_{i}\}$ of $X$. Is it possible to view $X$ as a (co)limit of the ...
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2answers
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Show that function $f: A \to B$ is surjective when there is an implication: $g \circ f = h\circ f \to g=h$ [closed]

Let $f: A \to B$. How can I show that $f$ is surjective if and only if (for every $C$ and every pair of functions $g, h: B \to C$) when there is the following implication? $$ g \circ f = h\circ f \to ...