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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About the definition of Day's convolution

I'm struggling with the definition of Day's convolution. Given a monoidal category $(\mathcal C,\otimes, I)$, there is a way to define a monoidal product on the category of presheaves ...
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2answers
12 views

Kernel of biproduct projection is the other biproduct injection

For some reason I'm unable to figure out what should be a trivial step in a proof.. Let $A\oplus B$ be a biproduct with injections $i_1,i_2$ and projections $p_1,p_2$. I aim to prove $i_1=\ker p_2$. ...
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33 views

Do functors that send direct sums to tensor products have a name?

Suppose a category $C$ is given with direct sums and tensor products, and let $F:C\to C$ be a functor with the property that $F(A\oplus B)\cong F(A)\otimes F(B)$. It would be tempting to call $F$ an ...
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66 views

A construction with homotopy colimits and homotopy pullbacks for descent.

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a ...
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43 views

Inverse limit of isomorphic objects

Why the inverse limit of isomorphic objects is isomorphic to each one of them? I know there is this question: inverse limit of isomorphic vector spaces but it seems that there they have a total ...
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3answers
68 views

Generators and relations for $\mathbb{R}_{\geq 0} \cup \{\infty\}$ involving infinite sums

A countably-complete semiring is basically a semiring with some additional structure making infinite sums possible. Formally, it is a tuple $R=(|R|,\Sigma,\cdot,1)$, where $|R|$ is a set, ...
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1answer
21 views

Constructing a $G$-equivariant map from given set map.

In my category theory text I've come across a lot of questions where my own background is lacking in some topics. I have the following question I'm trying to work through: Describe nontrivial ...
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1answer
29 views

Given A : $\mathcal{C} \rightarrow \mathcal{E} $ when does Colim(A)=A(1)?

Suppose $\mathcal{C}$ a small category and $A$ : $\mathcal{C} \rightarrow \mathcal{E} $ where $\mathcal{E}$ is a cocomplete category and $\mathcal{C}$ has a terminal object $1$ When is the colimit ...
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70 views

Yoneda-type lemma for compositions on the hom-functor

The Yoneda lemma basically rephrases a rigidity property of natural transformations out of a covariant hom functor: A natural transformation $\psi : \mathsf{Hom}(Z,-)\Rightarrow F$ is determined by ...
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38 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 ...
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6answers
295 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 ...
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1answer
22 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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36 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 ...
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1answer
41 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
35 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: $V\mapsto V^G$ ...
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1answer
58 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 ...
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2answers
67 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. ...
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36 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 ...
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1answer
44 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 ...
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1answer
26 views

Homotopy split monomorphisms [closed]

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 ...
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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 ...
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1answer
79 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 ...
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1answer
39 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 ...
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194 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 ...
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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 ...
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1answer
44 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 ...
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0answers
64 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 ...
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44 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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28 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 ...
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2answers
56 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 ...
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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 ...
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84 views
+50

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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14 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 ...
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64 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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51 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
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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 ...
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2answers
59 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
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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 ...
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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
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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 ...
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1answer
110 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 ...
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1answer
47 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 ...
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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?
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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 $G(f)(a)$ ...
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60 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
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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
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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 ...
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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 ...