The monoidal-categories tag has no usage guidance.

**23**

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**4**answers

4k views

### Tannakian Formalism

The Tannakian formalism says you can recover a complex algebraic group from its category of finite dimensional representations, the tensor structure, and the forgetful functor to Vect. Intuitively, ...

**3**

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**0**answers

124 views

### symmetric monoidal dagger endofunctor categories

Take an arbitrary category $C$, and consider $End(C)$, any endofunctor category consisting of some or all endofunctors of $C$.
I have several related questions:
What restrictions must we impose on ...

**5**

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**2**answers

447 views

### A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category

I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea.
Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...

**4**

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**1**answer

481 views

### When is homotopy orbit space weakly equivalent to orbit space, other than situation of free action?

Let $M$ be a closed symmetric monoidal model category. Let $X$ be a cofibrant object (it can also be fibrant if you like) and let $\Sigma_n$ act on $X^{\otimes n}$ by permuting the factors (note that ...

**14**

votes

**1**answer

602 views

### Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the
category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...

**8**

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**1**answer

953 views

### Exterior powers in tensor categories

Let $\mathcal{C}$ be a cocomplete $R$-linear tensor category. Many notions of commutative algebra can be internalized to $\mathcal{C}$. For example a commutative algebra is an object $A$ in $\mathcal{...

**11**

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**1**answer

629 views

### Reference for “lax monoidal functors” = “monoids under Day convolution”

Suppose $A$ and $C$ two symmetric monoidal categories. Let's say that $A$ is small and $C$ is locally presentable, and let's assume also that the tensor product on $C$ preserves colimits separately in ...

**9**

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**2**answers

3k views

### Is there a meaningful difference between biased and unbiased composition?

In higher category theory, there are notions of biased and unbiased definitions of composition of $n$-morphisms (or, as a special case, tensor products of objects). In the biased framework, we define ...

**8**

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**1**answer

294 views

### Reference for a generalization of Γ-spaces to monoidal model categories

Γ-spaces were introduced by Segal in 1969 as models for what can be now described
as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...

**8**

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**1**answer

641 views

### Saavedra's Definition of Tannakian Category

I have been reading Deligne-Milne's Tannakian Categories and got to the point at the end of part 3 where they discuss what went wrong with Saavedra's definition. Motivated by their counterexample, ...

**8**

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**1**answer

538 views

### Hovey's unit axiom in monoidal model categories

Let $\mathcal{C}$ be a monoidal model category in the sense of Hovey's book. He assumes the following unit axiom not considered in other references (e.g. Schwede-Shipley): given a cofibrant ...

**7**

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**4**answers

772 views

### 180˚ vs 360˚ Twists in String Diagrams for Ribbon Categories

Ribbon categories are braided monoidal categories with a twist or balance, $\theta_B:B\to B$, which is a natural transformation from the identity functor to itself. In the string diagram calculus for ...

**4**

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**2**answers

264 views

### Coherence in pseudo.monoids

In the article of A. Joyal and R. Street:
Braided tensor categories, (Adv. Math. N.102 (1993), no. 1, 20-78)
they define a tensor object (or pseudo.monoids) in a monoidal 2-category $\mathcal{C}$ (...

**4**

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**1**answer

531 views

### Category of modules over commutative monoid in symmetric monoidal category

Let $\left(\cal{C},\otimes ,I\right)$ be a symmetric monoidal category (not necessarily closed) and $A$ a commutative monoid in $\cal{C}$. In his DAG III (page 95), Lurie writes:
In many cases, ...

**12**

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**1**answer

190 views

### Is there a finite test for isomorphisms of rigid monoidal abelian categories?

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors $...

**9**

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**0**answers

113 views

### t-structures on the tensor product of stable $\infty$-categories

It is a matter of checking definitions that the tensor product of presentable $\infty$-categories restrict to a tensor product between stable presentable $\infty$-categories; I was wondering if there ...

**6**

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**1**answer

336 views

### Generalization of analytic functors

It's been a long time since I posted the following question on stackexchange.
Now I think it's better to adress it to you, in the hope I will reach the right audience: Martin's comment, albeit useful,...

**5**

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**1**answer

713 views

### Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects

Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$.
Questions: are there definitions of image and kernel for a ...

**6**

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**0**answers

119 views

### Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.)
Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G ...

**4**

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**0**answers

280 views

### When is the cofibrant replacement of a product the product of the cofibrant replacements?

I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE \...

**2**

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**0**answers

148 views

### Pseudomodules, “general coherence theorem”

A pseudomonoid is defined within a monoidal bicategory. It is like a monoid in a monoidal category except that the usual axioms hold up to coherent invertible 2-cells. Pseudomonoid is like a monoidal ...