All Questions
Tagged with monads monoidal-categories
12 questions
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Monoidal categories with canonical left-strengths of monads
It is well-known that every monad on Set is left-strong and that left-strength on a Set-monad is unique.
Is there an abstract characterization of monoidal categories $C$ for which every monad on $C$ ...
- 1,347
3
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98
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Yetter-Drinfeld modules for Hopf monads
1. Context.
1.1. Classical Yetter-Drinfeld modules.
Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ ...
- 764
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Separable monads do not induce separable monoids
Let us first recall the categorical notion of monad: if we have a category $\mathcal{C}$ then a monad on it consists in an endofunctor $\mathbb{A}\colon \mathcal{C}\rightarrow \mathcal{C}$ together ...
- 767
2
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283
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The Kleisli category of a monoidal monad
Let $C$ be a symmetric monoidal category equipped with diagonals $\triangle_x: x \to x \otimes x$, that is, equipped with natural transformations $e_x: x \to 1$ and $\triangle_x : x \to x \otimes x $ ...
- 123
3
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354
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The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors
In 6.5 of the book by Kelly,
Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, No. 10, 2005.
the author claims that the $2$-category $\mathsf{Cat}_{\...
- 9,111
1
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1
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164
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Internal commutative monoid gives commutative monad
Let $(C,\otimes,1)$ be a symmetric monoidal category. Let $(M,\mu,\eta)$ be an internal commutative monoid object.
The functor $X\mapsto M\otimes X$ has a canonical monad structure, with unit and ...
- 2,129
10
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What are the (co)algebras for the $(\operatorname{Hom}(A,-), A\otimes-)$ adjunction (co)monad?
A module $A$ over a commutative ring $k$ gives a pair of adjoint endofunctors, $(A\otimes_k-)$ left adjoint to $\operatorname{Hom}_k(A,-)$. They produce a monad $T_A$ and a comonad $C_A$. Is there any ...
- 18.4k
2
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1
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339
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Monad induced by actegory
It seems to be folklore that if we have an actegory, i.e. a monoidal functor from a monoidal category $C$ to an endofunctor category $Cat(D,D)$, we can obtain from it a monad on $D$. This appears for ...
- 2,129
5
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Monads which are monoidal and opmonoidal
Do monads which are monoidal and opmonoidal have a name? (Bimonoidal?) In case they have already been studied, who can point me to a reference?
More in detail. Let $(C,\otimes)$ be a symmetric (or ...
- 2,129
2
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1
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When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?
I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is ...
- 5,565
6
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340
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Compatibility between strength and costrength of a monoidal monad
Let $C$ be a closed monoidal category, and let $T : C \to C$ be a monad on the underlying category. Let $\sigma$ be a tensorial strength of $T$ and let $\sigma'$ be a cotensorial strength of $T$. A ...
- 63.1k
2
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Seems like Reader monad composed with a strong monad produces a monad, am I right?
Take a Cartesian (or monoidal) closed category; define Reader monad for a given object $E$ as
$X \mapsto X^E$; and take a strong monad $M$ (strong means preserves product or tensor product).
Now the ...
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