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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Vector bundles on Hirzebruch surface $\mathbb{F}_2$

I would like to know a classification for all holomorphic vector bundles on the second Hirzebruch surface $\mathbb{F}_2$. Is this known? What is known? In particular, I'm looking for holomorphic ...
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What is a Monad in the two category $Rel$?

The 2-category $Rel$ is a category with sets as $0$-cells, relations as $1$-cells (with relation composition as composition), and inclusions as $2$-cells (with vertical composition being the fact that ...
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“Lossy” v.s. “Lossless” Monads

Let us look at two monads on $\bf Set$. The first will be the finite sequence monad (from the free forgetful adjunction with $\bf Mon$.) $$\eta(x)=[x]$$ $$\mu([[a,b, \dots, z],[\alpha,\beta, \dots, ...
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Extending actions of Monads on Endofunctors.

Let $X^{X}$ be the category of endofunctors on a category $X$. Then if we define $\otimes :X^{X}\times X^{X}\rightarrow X^{X}$ by $R\otimes S=RS$ on objects and $\tau \otimes \sigma $ to be horizontal ...
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Why are morphisms of monads lax and not oplax natural transformations

Monads in a bicategory $\mathscr B$ correspond to lax functors $* → \mathscr B$, so one expects morphisms of monads should correspond to nautral transformations between them. A natural ...
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What is the Eilenberg-Moore category of this diagonal-like monad?

The Eilenberg-Moore category of a monad $(T:C \to C, \eta, \mu)$ has as objects pairs $(x \in Ob(C), h:Tx \to x)$ such that $h \circ \mu_x = h \circ Th:T(Tx) \to x$ and $h \circ \eta_x = id_{x}$. A ...
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Is there a “properly categorical” description of Eilenberg-Moore algebras on a relative monad?

A relative monad is defined in Altenkirch et al. essentially as a pair of functors $J,F:\mathcal{C}\to\mathcal{D}$, a natural transformation $\eta:J\to F$, and an operation ...
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Topological insight on an algebraic theory.

I recently read about the marvelous bar construction. As far as I understand, it is something of a free resolution in an abstract setting. I'm wondering if it can be used to get topological insights ...
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The tensor product of monads.

It is known that the tensor product of endofunctors End(C) over a given category C is given by composition and the category of monads Mon(C) over a given category is cartesian. That cartesian product ...
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Mathematical notation for monad bind

Background I'm defining the semantics of a programming language, and I'd like to use a monad to help me. My specific monad, not that it much matters, is $ M(\tau) = \mathcal P(\tau \times ...
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Is the continuation monad terminal?

For each $R$ an object of a cartesian closed category, there is a monad $\mathrm{Cont}_R(A) = [[A,R],R]$, the continuation monad. If $M$ is a strong monad, we can find for each pair of objects $A$ ...
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Flabby sheaf comonad

If one Googles sufficiently hard one finds the statement that Roger Godement gives the first example of a comonad, used to compute flabby resolutions of sheaves, in his monograph "Topologie algébrique ...
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Describing where a Kleisli Triple fits into a Monad ontology

I'm trying to map a Kleisli triple onto my existing understanding of Monads. I can represent my understanding of Monads like this: (courtesy of Jim Duey's slides at 13) Could you please point to ...
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Do adjoint functors really define monads?

It is often claimed as "obvious" that a pair of adjoint functors: $L\colon{\cal V}\to {\cal M}$ and $R\colon{\cal M}\to {\cal V}$ defines a cotriple $(\bot, \epsilon, \delta)$ and a monad. What is ...
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Regarding a difficulty in the Fakir article about associated idempotent triple

I just had post this question here but in a less exact form . I understand that at the first sight it seems a silly/trivial verifications (to me too) but I discussed the matter with some known person ...
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About details of the Fakir theorem proof (associated idempotent triple)

On the ncatlab work http://ncatlab.org/toddtrimble/published/Associated+idempotent+monad+of+a+monad Todd Trimbe quote the Fakir theorem about the associated idempotent triple, and this is based on ...
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Examples of a monad in a monoid (i.e. category with one object)?

I've been trying to figure out what having a monad in a monoid (i.e. a category with one object) would mean. As far as I can tell it would be a homomorphism (functor) $T : M → M$, with two elements ...
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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 ...