All Questions
Tagged with monoidal-categories monads
12 questions
3
votes
1
answer
133
views
Is there a non-symmetric monoidal monad?
Recall that a monoidal monad on a monoidal category $(\mathcal{C}, \otimes, I)$ is a monad $(M, \eta, \mu)$ on $\mathcal{C}$ such that $M$ is also equipped with the structure of a lax monoidal functor ...
- 3,550
3
votes
1
answer
108
views
Cocommutative bimonads: Why does this diagram commute?
1. Definitions
Let $(C, \otimes,I, a, l,r,c)$ be a monoidal category with braiding $c:\otimes \rightarrow\otimes ^{op}$. Let $(S,\mu,\eta,\tau,\theta)$ be a bimonad on $C$.
Following Turaev and ...
- 3,158
1
vote
1
answer
419
views
What makes every strong monad on a certain category be a monoidal functor?
A concept named Monad is used a lot in functional programming. And in spite their definition is not completely same with the definition of monad in category theory, as I know, Monad on a programming ...
- 13
2
votes
0
answers
83
views
Effectus theory and the Giry monad
Is the Kleisli category of the Giry monad a monoidal effectus with copiers, in the sense of Definition 70 from An Introduction to Effectus Theory ? The fact that this category is an effectus is ...
- 1,823
5
votes
2
answers
180
views
Recover the monoidal structure on $\mathbb{Ab}$ from the monad over Set
The category of abelian groups $\mathbb{Ab}$ has a monoidal (closed) structure $(\otimes, \mathbb{Z})$. Moreover, it is monadic over the category of sets via the free abelian group monad $$\mathbb{Z}[\...
- 4,929
0
votes
0
answers
171
views
Elements of the Monoid in the category of endofunctors
Quoting from Categories for the Working Mathematician by Saunders Mac Lane:
All told, a monad in X is just a monoid in the category of
endofunctors of X, with product × replaced by composition of
...
- 337
7
votes
1
answer
361
views
Monoid in the category of endofunctors and Monoid as a category with one object
Quoting from Categories for the Working Mathematician by Saunders Mac Lane:
All told, a monad in X is just a monoid in the category of
endofunctors of X, with product × replaced by composition of
...
- 337
5
votes
0
answers
237
views
Monads, monoidal categories
I'm interested in learning about monads and their relations to algebraic structures (as a generalization of universal algebra, if I understand well -correct me if not) .
In the process of learning ...
- 44.4k
3
votes
1
answer
132
views
2-monad and 2-operad of monoidal categories, explicit construction
It is well-known that monoids are algebras for the free monoid monad, and can be seen as well as algebras for the associative operad.
Less known is the categorified statement: for example, monoidal ...
- 8,307
15
votes
1
answer
2k
views
Details in applying the Barr-Beck monadicity theorem to Tannakian reconstruction
The Barr-Beck monadicity theorem gives necessary and sufficient conditions for a category $\mathcal{C}$ to be equivalent to a category of (co)algebras over a (co)monad. A functor $F:\mathcal{C}\to\...
- 3,619
5
votes
1
answer
129
views
Algebras for monads in Cat and 2-categories
An algebra for a monad $(T, \mu, \eta)$ on a category $\mathbb{C}$ is defined as a morphism $T X \to X$ for some object $X$ such that the obvious diagrams commute.
If I look at the monad as a monoid ...
- 270
17
votes
3
answers
1k
views
Theory of promonads
I'm led to define a promonad in $\bf D$ as a monoid in the category of endo-profunctors of a category $\bf D$, where the product of two profunctors is their composition as profunctors:
$$
F\odot G := \...
- 12.2k