Movatterモバイル変換

Question 1

2votes

0answers

39views

On the definition of a Monoidal Category

In all sources I have read, a monoidal category is defined essentially as follows. A monoidal category is a 6-tuple $(\mathcal{C}, I, \otimes, \alpha, \lambda, \rho)$ where$\mathcal{C}$ is a ...

Luna Elliott

233

asked14 hours ago

Question 2

2votes

1answer

45views

Duality and projectively stable category

Let $A$ be a finite‑dimensional algebra over an algebraically closed field $k$. I work with right modules.We have a duality given by $F=\operatorname{Hom}(_,A):\mathrm{mod}(A)\to\mathrm{mod}(A^{op})$....

Theo

41

asked21 hours ago

Question 3

0votes

1answer

61views

Morphisms in $\mathfrak{R}^3$ and $\mathfrak{R}^6$ with $\mathcal{mod}\,A$

I am working with representation theory of finite-dimensional algebras.While studying powers of the radical of the module category, I found the followingdefinition (for indecomposable modules):$$\...

the topological beast

171

askedyesterday

Question 4

2votes

1answer

78views

The role of strictness in the equivalence between $ 2 $-dimensional topological quantum field theories and Frobenius algebras

Monoidal categories and monoidal functors come in many flavors. The former can be weak or strict, while the latter can be lax, strong or even strict themself.In it's Frobenius Algebras and 2D ...

GeometriaDifferenziale

1,005

asked2 days ago

Question 5

0votes

1answer

50views

Additive Right Adjoint functor F preserves left exact sequence? Question about the proof

$f:A\to B$ and $g: B\to C$ are two morphisms of R-Mod. $0\to A \to B\to C\to 0$ is exact and F is an additive right adjoint functor.I want to show $0\to F(A) \to F(B)\to F(C)\to 0$.I know right ...

AmazingBBoy

11

asked2 days ago

Question 6

2votes

1answer

136views

Why are the projection functors in the definition of the product of categories explicitly defined?

Consider the usual definition of a product in a fixed category $\mathcal C$:we have a family of objects (with projection maps) $\left\{\left( X_i, \, \pi_i \colon X \to X_i \right) \right\}_{i \in \...

Markus Klyver

2,582

askedNov 25 at 19:44

Question 7

0votes

0answers

41views

Supremum of Subsheaves

I'm reading MacLane and Moerdijk's "Sheaves in Geometry and Logic," and I'm having trouble understanding a description given in section 8 of chapter III of the supremum of a family of ...

Itserpol

543

askedNov 25 at 15:55

Question 8

0votes

0answers

50views

Doubt on how much is unique a certain homomorphism into the modules of differential

Let $k$ be a ring and let $A$ be a $k$-algebra. Denote by $\Omega^1_{A/k}$ the module of derivations on $A$ over $k$.Let $I\subset A$ be an ideal. The composition of $k$-linear homomorphisms $$I\...

Stanis LaRochelle

333

askedNov 24 at 17:30

Question 9

0votes

0answers

23views

split coequalisers which are not split epi [closed]

Any split epi is a split coequaliser. I assume the converse is not true, otherwise there will be no reason to have both concepts. However, I do not have a counterexample for this.

diacon

27

askedNov 23 at 20:34

Question 10

0votes

0answers

148views

$a \cong a$ using arrows-only axioms [closed]

This is an exercise in Topoi by Goldblatt:$a \cong a$,which means:every object in a category $\mathscr{C}$ is isomorphic to itself.A very nice way to think about this, in my experience, is to ...

Attila Vajda

113

askedNov 23 at 6:44

Question 11

1vote

1answer

99views

degeneracy maps when computing limits

I am reading page 15 of this note. It is stated thatMay I ask, in the case of 2-limit, are the degeneracy maps needed to form the correct 2-limit?(Perhaps a related question: In general, how do one ...

user14411

673

askedNov 21 at 5:17

Question 12

0votes

0answers

34views

Extending a mapping on objects to a functor in the category of Abelian Groups

The motivation for this problem is that it is a component in proving the Freyd-Mitchell embedding theorem.Let $F$ be a functor from a small abelian category $C$ to the category of abelian groups $\...

Nicholas White

11

askedNov 20 at 20:22

Question 13

3votes

1answer

50views

Characters of the multiplicative group

Fix a commutative, unital, associative ring $R$, and let $C$ be the category of commutative, unital, associative $R$-algebras.Let $\mathbb{G}_m$ be the functor from $C$ to groups, which sends $A \in \...

Joshua Ruiter

3,242

askedNov 19 at 18:25

Question 14

0votes

0answers

54views

Is a retract a projective object in the category of TOP?

A retract of a top space $X$ is a subset $A\subset X$ s.t. there is a cts surjection $r: X\to A$ satisfying $r\circ \iota_A=Id_A$. The map $r$ is called a retraction and satisfies $r^2=r$.Is every ...

user760

2,982

askedNov 19 at 12:27

Question 15

2votes

1answer

63views

Class coincidence in sSet

Let $S$ be a class of simplicial sets such thata) $S$ is closed under coproducts;b) if in the pushout$\begin{array}{ccc}A & \xrightarrow{f} & A' \\\downarrow{g} & & \downarrow{h} ...

Michal Lolis

168

askedNov 18 at 23:24

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Stack Exchange Network

Questions tagged [category-theory]

On the definition of a Monoidal Category

Duality and projectively stable category

Morphisms in $\mathfrak{R}^3$ and $\mathfrak{R}^6$ with $\mathcal{mod}\,A$

The role of strictness in the equivalence between $ 2 $-dimensional topological quantum field theories and Frobenius algebras

Additive Right Adjoint functor F preserves left exact sequence? Question about the proof

Why are the projection functors in the definition of the product of categories explicitly defined?

Supremum of Subsheaves

Doubt on how much is unique a certain homomorphism into the modules of differential

split coequalisers which are not split epi [closed]

$a \cong a$ using arrows-only axioms [closed]

degeneracy maps when computing limits

Extending a mapping on objects to a functor in the category of Abelian Groups

Characters of the multiplicative group

Is a retract a projective object in the category of TOP?

Class coincidence in sSet

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