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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2
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36 views

How to show that in the category of Hausdorff spaces every epimorphism is a continuous function with dense image? [duplicate]

How to show that in the category of Hausdorff spaces every epimorphism is a continuous function with dense image ? that is if $X,Y$ are Hausdorff spaces and $f:X \to Y$ is continuous such that for any ...
13
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4answers
191 views

Honest application of category theory

I believe that category theory is one of the most fundamental theories of mathematics, and is becoming a fundamental theory for other sciences as well. It allows us to understand many concepts on a ...
1
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0answers
12 views

A morphism to the mapping cone?

In the second part of the proof for the Proposition 2. in Derived Categories by Daniel Mufet, one finds the following: A collection of morphisms $f^n:Q^n\to X^n\oplus Y^{n-1}$ with components ...
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0answers
32 views

What does the statement “x is a diagram of classical logics” mean?

From the Wikipedia entry on quantum logic A more modern approach to the structure of quantum logic is to assume that it is a diagram – in the sense of category theory – of classical logics (see ...
1
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1answer
39 views

Limit as universal arrow

I was interested in the definition of limit as universal arrow from $\Delta$ to $F$ given in Mac Lane's CWM book. Let us begin with some notation: Let $C$ be a category, and $J$ be an index ...
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1answer
31 views

Functor from category of group representations to space of $G$ invariants

For a representation $(V,\rho)$ of a group $G$, define the subspace of $G$-invariants by $$ V^G=\{v\in V: \rho(g)v=v\quad \forall g\in G\} $$ and want to prove the following: $V\mapsto V^G$ ...
7
votes
1answer
54 views

Intuitive explanation of Four Lemma

In the Short Five Lemma where the rows are exact, it is a fact that $$\alpha \text{ and }\gamma \text{ injective (surjective) }\implies \beta \text{ injective (surjective)}.$$ I've heard this fact ...
4
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2answers
60 views

Examples of functors that preserves products but not equalizers, and vice versa.

What are simple examples, for student consumption, of A functor which preserves products (or at least finite products) but not equalizers. A functor which preserves equalizers but not products. ...
4
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2answers
33 views

About the presheaf used to define the inverse image sheaf.

Let $f \colon X \to Y$ be continuous and $\mathcal{F}$ be a sheaf on $Y$. Then the inverse image sheaf $f^*\mathcal{F}$ is defined to be the sheafification of the presheaf on $X$ given by $$ U \mapsto ...
0
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1answer
41 views

Every epimorphism in Sets is split: why is it equivalent to axiom of choice?

Suppose that $f: A \rightarrow B$ is epic in Sets. One can construct a section $s: B \rightarrow A$ of $f$ as follow: Let us define an equivalence relation $R$ on $A$ as follow: $aRa'$ iff $a, a' \in ...
-1
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1answer
26 views

Homotopy split monomorphisms [on hold]

Let $C$ be a model category. Recall that a morphisme $f : X \to Y$ is called a homotopy monomorphism if the diagonal $X \to X \times^h_Y X$ induces an isomorphism in the homotopy category. Suppose ...
2
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1answer
46 views

How to “lift” a path to the tangent bundle?

Given a path $c: (-\epsilon,\epsilon)=I \to M$ in a manifold. Define $\widetilde c:I \to TM$ (a kind of "lift") as the curve $t_0 \mapsto [c(t+t_0)] \in T_{c(t_0)}M$. Is there a nice categorical ...
2
votes
1answer
34 views

When is an object in a linear or abelian category simple? Or: How should I define fusion categories?

I'm confused about the notion of simple objects. Now ncatlab says that an object is simple in an abelian category if it only has itself and 0 as subobjects. On another page, it says that the simple ...
2
votes
1answer
65 views

If the cohomology of two objects in the derived category are equal, are the objects isomorphic?

Let $\mathcal{A}$ be an abelian category. Given objects $A^\bullet,B^\bullet$ in the derived category $D(\mathcal{A})$. Assume that $H^n(A^\bullet)=H^n(B^\bullet)$ for all $n\in\mathbb{Z}$. Can we ...
2
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1answer
38 views

What does the notation $U(\frak{g})[[\hbar]]$ mean?

I'm reading the following motivation for studying quantum group but I'm unfamiliar with the double bracket notation in $$U(\frak{g})[[\hbar]].$$ Is this a special set of polynomials with coefficient ...
5
votes
2answers
190 views

Applications of Eckmann–Hilton argument

I am looking for applications of the Eckmann–Hilton argument. I found one application in Algebraic Topology where we show that the fundamental group is abelian in case of a topological group. Thank ...
0
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1answer
52 views

Universal Properties and Isomorphisms

If two objects satisfy the same universal property, we know that they are isomorphic in that category. Is the converse true? That is, if two objects are isomorphic in some category, can we construct ...
3
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1answer
42 views

Equivalence of category of cones

If $ E \colon I \rightarrow J $ is an equivalence of categories and $ D \colon J \rightarrow C $ is a diagram of shape $ J $ in $ C $, is the category of cones over $ D $ equivalent to the category of ...
4
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0answers
59 views

Spectral sequence $\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{H}^j,\mathcal{H}^k)\Rightarrow \mathrm{Hom}^{p+q}(P,P)$

Reading the proof in Bondal-Orlov reconstruction theorem (http://arxiv.org/pdf/alg-geom/9712029v1.pdf), I found the spectral sequence in the title ...
0
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0answers
43 views

How to prove that a functor factors through another functor under some certain conditions

Let $\mathcal C$ be an additive category, $B$ and $L$ two objects in $\mathcal C$, and $G$ a functor $\mathcal C^{op} \to \mathsf{Ab}$. Suppose there are morphisms $$\alpha \colon \hom_{\mathcal ...
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0answers
27 views

Subobject classifier in $Set^{C^{op}}$

I'm reading "Sheaves in geometry and logic" and I'm not sure if i'm understanding some definitions. We have our functor $\Omega$ defined on objects by $\Omega(C)$$=\{$$S|$ $S$ is a sieve on C in ...
2
votes
2answers
54 views

Equivalent categories are elementarily equivalent: Formalization?

Equivalent categories should be elementarily equivalent in the sense of mathematical logic. How to make this precise? Here is an attempt: Let $F : \mathcal{C} \to \mathcal{D}$ be an equivalence of ...
0
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0answers
22 views

Terminal object in $Set^{C^{op}}$ and subobject classifier.

This is from Sheaves in Geometry and Logic pg 38. I'm not sure if I understood it correctly but the subobject classifier in $Set^{C^{op}}$ when $C$ is a small category is a map (natural ...
5
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0answers
64 views

Does the vector space of compactly-supported continuous functions $X \rightarrow \mathbb{R}$ satisfy an interesting universal property?

Let $S$ denote a set. Then the vector space $FS$ freely generated by $S$ can be identified with the set of all finitely-supported functions $S \rightarrow \mathbb{R}$. This gave me the following idea; ...
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0answers
13 views

strong epis in the category of banach spaces with linear contractions

In Borceux's Handbook volume 1, page 145, the strong epis in the category of Banach spaces with bounded linear maps of norm <= 1 is characterized as the maps whose restriction on the unit balls is ...
5
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0answers
60 views

When does Sheafification commute with direct image?

Given a presheaf $\mathcal{F}$ on a space $X$ and a map $f: X \rightarrow Y$, when does $f_* A(\mathcal{F}) = A(f_* \mathcal{F})$, where $A$ is the associated sheaf/sheafification functor? Since ...
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0answers
21 views
+50

Projective family of probability spaces

I'm confused about the definition of a projective family of probability spaces $(S_t,\mathscr S _t,\mu_t,f_{ts})$. Let our indexing set be a poset $T$. The conditions $f_{tt}=1_{S_t}$ ...
3
votes
1answer
77 views

Two categories sharing the same objects and morphisms

Is there a natural example of two categories $\mathcal{C}$, $\mathcal{C}'$ which have the same class of objects and the same class of morphisms, including source and target maps, but different ...
1
vote
2answers
58 views

Identity about a Functor

I'm moving my first steps in CT and suddenly after reading about functors, this question came up in my mind: Let $F \colon A \to B$ a functor between two categories $A$ and $B$, is it true that for ...
3
votes
1answer
112 views

Does internalization loses informations everywhere?

It is well known that a group object in Grp is necessarily abelian. This can be understood as "internalization loses information". Indeed, if one was to study group theory by looking at group objects ...
1
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1answer
35 views

Action of $Aut(X)$ on $Coh(X)$

I was reading about Bondal and Orlov reconstruction theorem. In particular that for a smooth variety with ample or anti-ample canonical bundle $\mathrm{Aut}(D^b(X)) \cong \mathbb{Z} \times ...
2
votes
2answers
54 views

Multiple categorifications of structures

I recently read about how the category of finite sets and the category of finite-dimensional vector spaces are both categorifications of the natural numbers. I was wondering if there are any other ...
8
votes
1answer
108 views

Are there any non-obvious colimits of finite abelian groups?

Does the forgetful functor $U : \mathsf{FinAb} \to \mathsf{Ab}$ from finite abelian groups to abelian groups preserve colimits? Morally this should be true, but it is not so easy (for me) to come up ...
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0answers
24 views

Are functors that are right-cancellable full, or do they have other characterizations? [duplicate]

In a former question it has become clear to me that a functor is left-cancellable if and only if it is injective on morphisms. This provides a nice characterization of monomorphisms in category ...
1
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1answer
46 views

Are functors that are left-cancellable necessarily injective on morphisms?

Let it be that $\mathcal C$ and $\mathcal D$ are categories and that $F:\mathcal C\rightarrow\mathcal D$ is a functor. If $F$ is injective on morphisms then it is easy to verify that it will be ...
2
votes
1answer
39 views

Set notation for generalized elements

I am currently reading Steve Awodey's book on category theory. On pg. 101 he uses set notation for generalized elements, namely $\{ a \mid f(a) = g(a) \}$ What does it mean in an arbitrary category?
0
votes
1answer
50 views

The image of an object under a subfunctor

Let $C$ be an additive category and $X$ is an object in $C$, $G$ is a functor in $(C^{op},Ab)$. $H$ is a subgroup of $G(X)$. Define $G_{H}(C)$ to be the set of all $a$ in $G(C)$ such that $G(f)(a)$ ...
0
votes
0answers
58 views

direct proof of the dual statement of the Yoneda lemma

The dual statement of the Yoneda lemma should read: Given any object $A$ in a locally small category $\mathsf{C}$ and any functor $F: \mathsf{C} \to \mathsf{Sets}$, we have an isomorphism ...
2
votes
1answer
32 views

If a finitely complete category has system of factorization (E,M) then the class M is stable under pullbacks.

We have to show that in this diagram ( see below) if $f \in$$\mathcal M$ then $g \in$$\mathcal M$. So $\forall e \in$ $\mathcal E$ we have to show $e \perp g$. I consider this diagram: But ...
2
votes
0answers
56 views

T-Algebras for a monad

Suppose $R$ is a ring with identity, $G:R$-$Mod\rightarrow Set$ is the forgetful functor and $F:Set\rightarrow R$-$Mod$ its left adjoint. I want to prove that the structure maps for the T-Algebras of ...
1
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0answers
65 views

Can we say anything about the relationship between these functors?

I am working with a category $\mathcal{C}$ and two functors $F:\mathcal{C}\rightarrow \mathbb{R}$-$\mathbf{Vect}$ and $G:\mathcal{C}^{\operatorname{op}}\rightarrow \mathbb{R}$-$\mathbf{Vect}$ where ...
1
vote
1answer
38 views

Does this property characterize monomorphisms?

Is requiring that $f:\mathrm{Hom}(A,B)$ is mono the same as requiring that the pullback $A\times_BA$ of $f$ along itself is isomorphic to $A$? In Sheaves in Geometry and Logic, I read the following ...
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0answers
22 views

Can two smooth categories be equivalent if their object manifolds aren't diffeomorphic or homotopy equivalent?

There is a category of "smooth categories", where the objects and the morphisms don't form sets, but manifolds (and there are some other conditions that I won't repeat here). Important examples are ...
1
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1answer
32 views

Modules in Morita Equivalence

In Method of Homological Algebra by Gelfand and Manin (Exercise 2.2.3). How are $\mathrm{Hom}_A(P,X)$ and $\mathrm{Hom}_B(P^*,Y)\,$ regarded as a $B$-module and $A$-module respectively?
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0answers
23 views

Regular representation, representability of the fiber functor, and hom-distributivity for Hilbert spaces

I've culled together a slick proof of $\Bbb C[G]\cong\bigoplus_{V\in\widehat{G}}{\rm End}(V)$ (Peter-Weyl decomposition) for finite groups using the fact that the fiber functor (that is, the forgetful ...
6
votes
1answer
55 views

Can a general version of the covariant powerset monad be derived from the universal property of power objects?

As the title asks, I'm wondering if one can generally squeeze a "covariant power object monad" out of a topos (following the usual example in $\mathcal{Set}$ with functor part the direct image ...
1
vote
1answer
47 views

Multiplication with category theory

Using category theory why is 3*2=6?. In book conceptual mathematics this is explained as : So there are 3 object 6,3 & 2 with two maps , level & shadow. There are 6 mapping from object 6 ...
0
votes
1answer
29 views

How to prove that homology is a functor?

Is a homology operator $H_k:(cKom) \rightarrow (Ab)$ a functor? I know this is a really simple question, but I'm not familiar with category theory and do not know how to prove this... (I'm even not ...
6
votes
1answer
83 views

An equivalence of categories which looks like Voevodsky's Univalence Axiom

Let $\mathcal{C}$ be a category. Consider the full subcategory $\mathrm{Isom}(\mathcal{C})$ of $\mathrm{Mor}(\mathcal{C})$ whose objects are isomorphisms $A \xrightarrow{\cong} B$. It has a full ...
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0answers
36 views

Characterization of Projective Objects

In which categories is an object $P$ projective if and only if every short exact sequence ending with it splits? $$0\longrightarrow A\longrightarrow B\longrightarrow P \longrightarrow 0$$