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.
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Proving filter to be an ultrafilter
I'm trying to prove the following result:
Let $r:X\longrightarrow Y$ be a relation and $\mathcal{U}$ and $\mathcal{V}$ be ultrafilters such that $r^{-1}V \in\mathcal{U}$ for all $V\in\mathcal{V}$. ...
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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 := ...
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Equivalence of categories of coalgebras
I'm studying monadicity and comonadicity and I´m stuck with the following:
Let $L\dashv R:X\rightarrow Y$ be an adjunction with unit $\eta$ and counit $\varepsilon$. The induced monad on $Y$ is ...
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What is a monad in a $2$-category?
The wikipedia article on monads somewhat mysteriously notes that
Monads can be defined in any 2-category ${\mathfrak C}$. The monads defined above are for ${\mathfrak C}$ = Cat.
where Cat is the ...
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Monad = Reflective Subcategory?
After answering this question here about Kleisli triples, I realized that this whole Kleisli triple construction:
$T:{\rm Ob}\mathcal C\to{\rm Ob}\mathcal C$, $\ \eta_A:A\to TA$ for all $A\in ...
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Does a Kleisli triple need naturality conditions?
I'm reading a paper by Eugenio Moggi entitled "Notions of Computation and Monads". It introduces the concept of a “Kleisli triple” on a category $\mathcal C$, which is $(T, \eta, -^*)$, where:
$T$ ...
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Examples of Monads and their Algebras
I'd like to get some examples of monads; specifically, I'd love a big list of different monads and a description of what their algebras are. Alternatively, online resources and especially exercices ...
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The free abelian group monad
The forgetful functor $U : \mathsf{Ab} \to \mathsf{Set}$ is monadic, this follows from Beck's monadicity theorem or some other general result. Anyway, I would like to prove this directly, thereby ...
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Lax algebras as lax morphisms
Wondering for the ncatlab I've encountered the following pages: one about lax-morphisms and the other about lax-algebras for $2$-monads.
For what I could get it seems that lax algebras can be ...
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Category Theory for programmers
I'm a developer and have become fixated on functional programming due to its expressivity. I have begun learning Haskell but have reached a very significant wall when trying to comprehend functors, ...
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Do probability distributions form a comonad?
$\def\unit{{\rm unit}}\def\join{{\rm join}}$It's well known that (discrete) probability distributions form a monad. Specifically, if we let $PX$ be the set of discrete probability distributions on ...
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Applications of monads in general topology?
What are applications of monads in general topology?
For example, for GT is important the notion of products, products are adjoints, so adjoints may be important for GT, but what's about monads?
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Godement resolution using monads
Given a topological space $X$ the Godement construction for a sheaf $F$ returns a sheaf $G^0(F)$ constructed as follows. For each point $x\in X$, define
$$ G^0(F)(U):=\prod_{x\in U} F_x.$$
This ...
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What are the algebras of the double powerset monad?
Let $\mathscr{P} : \textbf{Set} \to \textbf{Set}^\textrm{op}$ be the (contravariant) powerset functor, taking a set $X$ to its powerset $\mathscr{P}(X)$ and a map $f : X \to Y$ to the inverse image ...
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Is there a way to make tangent bundle a monad?
The tangent bundle functor $T: \mathbf{Diff} \to \mathbf{Diff}$ together with the bundle projection $\pi: T \Rightarrow 1_\mathbf{Diff}$ basically screams 'monad' at me, especially because both $\pi ...
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Do comonads always induce cosimplicial objects? Vice-Versa?
So Charles Weibel, in his book "Introduction to Homological Algebra" discusses the idea of a cotriple or a comonad. I believe he say that, given a comonad $\bot$ and an object $X$ which is ...
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Simple explanation of a monad
I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...
