Questions tagged [monads]
A monad is a functor from a category to itself together with two natural transformations, commonly called μ (the "multiplication") and η (the "unit"), satisfying conditions that make μ monoidal and η an identity for it.
253
questions
1
vote
1
answer
64
views
Cosimplicial resolution associated to a monad
Let $\mathcal{C}$ be a category and $\mathbf{T}$ a monad on $\mathcal{C}$ with functor part $T : \mathcal{C} \to \mathcal{C}$ (I would actually like to consider the case where $\mathcal{C}$ is an $(\...
- 804
3
votes
1
answer
80
views
When is $\text{Ab}(\mathcal{C}) \to \mathcal{C}$ monadic?
Consider the free abelian group monad $T: \text{Sets} \to \text{Sets}$. Then the category $\text{Ab}$ of abelian groups is equivalent to the category of $T$-algebras, and thus we have a monadic (...
- 254
5
votes
1
answer
138
views
The monad induced by the group multiplication
Let $G$ be any group. We can define its delooping, $\mathbf{B}G$, to be the groupoid with a single object $\bullet$ and $\mathrm{Hom}_{\mathbf BG}(\bullet, \bullet) = G$ with composition given by the ...
- 3,320
3
votes
1
answer
67
views
Notion of commutativity for monads
I have read that in functional programming a monad is commutative if
a >>= \x -> b >>= \y -> f x y is equivalent to ...
- 83
1
vote
0
answers
21
views
How does the EM category for the multiset monad work?
I am working with the standard multiset monad defined on sets and functions and I want to start looking at algebras for the monad, the EM-category of the multiset monad. I understand that this is ...
- 1,281
0
votes
0
answers
26
views
Majority vote EM category
I am working with the multiset monad. I want the EM category to have a structure map that is majority vote. That is to say that the map $f: X \rightarrow M(X)$ takes the set element with the highest ...
- 1,281
0
votes
0
answers
29
views
Concrete example on the commutative diagram of comparison functor
I am trying to familiarize myself with the properties of monad. I discovered in nlab the following convoluted picture about the whole story.
https://ncatlab.org/nlab/show/comparison+functor
It is ...
- 4,223
0
votes
0
answers
23
views
Forks of algebras over monads in proof of Beck's criterion
Consider a monad $T=(T,\mu,\eta)$ and a $T$-algebra $(X,\xi).$ Then lots of expositions of Beck's monadicity criterion refer to the following fact: there is a parallel pair of morphisms of $T$-...
- 4,123
1
vote
0
answers
33
views
Free monads without fixing the base category
Usually, free monads are defined with respect to a fixed base category.
By this I mean that given an endofunctor $F: \mathcal{C} \to \mathcal{C}$, the free monad $\bar{F}$ on $F$ is defined using only ...
- 440
6
votes
0
answers
143
views
Algebraic Structures involving 𝙽𝚊𝙽 (absorbing element).
IEEE 754 floating point numbers contain the concept of 𝙽𝚊𝙽 (not a number), which "dominates" arithmetical operations ($+,-,⋅,÷$ will return ...
- 11.8k
3
votes
1
answer
115
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 ...
- 1,579
0
votes
0
answers
34
views
Proving that a monad $(D,\eta,\mu)$ induces a monad $(T,\lambda, \rho)$.
I'm trying to formalize some of my code within Category Theory, and I ended up having to prove that a certain construction $(T,\lambda,\rho)$ was a monad. Here is the setup.
Let $(D,\eta,\mu)$ be a ...
- 3,309
0
votes
1
answer
45
views
Can a (co)monad composed with itself be a (co)monad again?
I have very little categorical knowledge, my question comes mostly from programming, but I'm interested in the categorical solution of this problem.
Let's assume we have a monad $(T, \eta, \mu)$
Is $T^...
1
vote
0
answers
39
views
If $\mathsf{C}$ is a cocomplete category and $\mathsf{I}\to\mathsf{J}$ is a functor, when is $\mathsf{C}^\mathsf{J}\to\mathsf{C}^\mathsf{I}$ monadic?
$\def\C{\mathsf{C}}
\def\res{\operatorname{res}}
\def\I{\mathsf{I}}
\def\J{\mathsf{J}}
\def\A{\mathsf{A}}
\def\colim{\mathop{\operatorname{colim}}}$In Riehl's Category Theory in Context, we find:
...
0
votes
1
answer
33
views
Reflexive Tripleability Theorem in Riehl's Category Theory in Context
Exercise 5.5.iii in Riehl's Category Theory in Context consists in proving the following result:
Proposition 5.5.8 (RTT). If $U: \mathsf{D} \rightarrow \mathsf{C}$ has a left adjoint and if
$\mathsf{...