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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Multisets are containers, also called bags. A multiset is a set that can have repeats: $$\{ a, a, a, c, b, c \}$$ Usually when researchers talk about multisets, they use this kind of presentation: $$... 3 votes 0 answers 37 views ### Kleisli categories and Eilenberg-Moore categories induced by a pair of subcategories Let \mathfrak{A} be a full subcategory of \mathfrak{B}, where \mathfrak{A} and \mathfrak{B} are both concrete over a given category \mathfrak{C}. This means that we can consider the two ... 6 votes 1 answer 139 views ### What is the categorical construction for a list of nested lists? In Category Theory, the List functor is the Free Monoid over a given type (i.e. object) T. One can then consider the category of ... 1 vote 0 answers 43 views ### Induced Comonad of a Monad on the Eilenberg-Moore and the Kleisli category It is well known that for every monad T, we can consider the Eilenberg-Moore-Category and get an adjunction which induces T, similar for the Kleisli-Category. But since every adjunction induces a ... 1 vote 0 answers 32 views ### Construction of 2-category of monoidal categories and (lax) monoidal functors as strict algebra category of a 2-monad As motivation for 2-monads, I would like to understand an explicit construction of the 2-monad T of which derived 2-category T-\operatorname{Alg}_l of algebras as described in Lack's 2-categories ... 3 votes 0 answers 82 views ### Fundamental Description Of Algebras Over The Monad PP^{op}:\text{Set}\to\text{Set}  \newcommand{\eps}{\epsilon} \newcommand{\op}{^\text{op}} \newcommand{\id}{\text{id}} \newcommand{\set}{\text{Set}} \newcommand{\xra}{\xrightarrow} Can someone help me to understand how algebras ... 0 votes 1 answer 33 views ### Understading usage of functor composed with natural transformation in CT definitions I'm reading about Monads on Wiki, I'm confused about these two defining diagrams. Where T is the endofunctors \mu: T^2 \to T and \eta: 1_C \to T. So, my question is, why is the two arrow in ... 0 votes 0 answers 26 views ### Does the monad on V associated to an V-operad preserve reflexive coequalizers for a closed monoidal category V? This might be a very basic question, but a non-symmetric operad P in the symmetric closed monoidal category V induces a monad T_P on V which has the same category of algebras as P and which ... 3 votes 1 answer 74 views ### How to internalize the extension operator of a monad in a Cartesian closed category? Let \mathbb C be a Cartesian closed category and (T, \eta, \mu) be a monad on \mathbb C. The extension operator \_^{\sharp} : Hom(X, TY) → Hom(TX, TY) is defined by:$$f^{\sharp} = \mu_{Y}\...
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Question: I will first state the question, then do explanation of terminology below. All categories should be considered as cocomplete stable $\infty$-categories, and colimits, limits etc are the ...
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I'll motivate my question with a neat observation: let $G$ be a group, and let $\mathcal{P}(G)$ denote the power set of $G$, regarded as a category in the usual way (i.e., as a poset converted into a ...
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### Coproducts in Eilenberg–Moore categories

In Category Theory In Context, Proposition 5.6.11 Riehl says that for objects $(A_1,\alpha_1)$ and $(A_2,\alpha_2)$ in the Eilenberg–Moore category of monad $\mathcal{C} \xrightarrow{T} \mathcal{C}$ (...
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We know that two monads compose if there is a distributive law. What is the law that is necessary to compose a monad and a co-monad? When you combine a monad and co-monad, you get a (co)monad that ...
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### Algebras for the continuation monad?

Given a monad $(T,\mu,\eta)$, a map $\alpha : TA\to A$ commuting with $\mu$ and $\eta$ is a $T$-algebra. Given a set $D$, the continuation monad is given by the functor $C:X\mapsto D^{(D^X)}$ (see ...
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### What do you need to define a map of monads?

Suppose I have a monad $M_S = \langle S , \eta_S, \mu_S \rangle$ and a monad map given by a natural transformation $\phi: T \rightarrow S$. What are the natural transformations of the resulting monad?...
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I'm learning about monads in Riehl's Category Theory in Context, and after reading an example about the free monoid monad (also known as the list monad to computer scientists) on the monoidal category ...
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### Internal monomorphism in the category of functors

The codensity monad can be seen as a sort of endomorphism monoid. The internal hom or residual with respect to functor composition as the monoid being basically kan extension. In Set (G/F)(A) = \...
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Let $T:\mathcal{C}\to\mathcal{C}$ be a monad with unit $\eta:1_\mathcal{C}\Rightarrow T$ and multiplication $\mu:T\circ T\Rightarrow T$, and let $(A,\alpha)$ be a $T$-algebra. When is $\big(T(A),T(\... 3 votes 1 answer 97 views ### Cocommutative bimonads: Why does this diagram commute? 1. Definitions Let$(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$. Following Turaev and ... 1 vote 0 answers 44 views ### On the multiplication of the symmetric algebra monad In their paper "Generators and bases for algebras over a monad" (https://arxiv.org/pdf/2010.10223.pdf) Zetzsche, Silva and Sammartino introduce the Symmetric algebra monad. It is an ... 2 votes 2 answers 147 views ### The functor that maps a (co-)monad to its (co-)Eilenberg-Moore category I have noticed that the function that maps a monad$T : C \to C$to the Eilenberg-Moore category$C^T$can easily be extended into a functor$E_C$from the category of monads$\textbf{Mnd}_C$to$\...
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If a comonad $D$ is left adjoint to an endofunctor $T$, then $T$ can be made into a monad: its unit and multiplication are given respectively by the mates of the counit and comultiplication of $D$. ...
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For a topological space $X$ and the lattice of open sets $LX$, there is an extension of the "inclusion" (don't know what to call it) functor $F:LX\to\text{Top}/X$ along the Yoneda embedding $... 1 vote 0 answers 76 views ### What is an algebra for an algebraic theory? Definition 1.1 of Algebraic Theories says an algebraic theory is a small category$\mathcal{T}$with finite products, and an algebra for theory$\mathcal{T}$is a product preserving functor$A$from$\...
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Basically can you generalize adjunction to "graded adjunction?" I have a hunch you would end up with something similar to some linear logic stuff. My thoughts are you can use monads for ...
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### When does transporting a monad along an adjunction "preserve" its category of modules?

If I have an adjunction $F \dashv G$ where $F \colon C \to D$, and $(N, m, u)$ is a monad on $D$, then I can define a monad $\widetilde{N}$ on $C$ via \begin{align} (GNF, \widetilde{m} = G m_F \...
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### What are the functors that can be approximated by a monadic functor?

Let $\newcommand{\Fc}{\mathbf{Func}}\Fc$ be the category whose objects are functors between small categories and whose morphisms are commutative squares (the commutativity is given by a natural ...
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### Modules over monoids vs algebra over monads

I read somewhere that the construction of algebras over monads is motivated by/ similar to the construction of modules over monoids, but I'm having difficulty seeing this. I see that a monad "...
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### 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 ...
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In reference to Atiyah's book - "Geometry of Yang-Mills Fields" (1979). In chapter 5, section 3, he describes how the monad construction for $Sp(n)$ potentials can be interpreted in terms of ...
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### On the monadicity of Cat

I heard that the category of small categories is monadic over the category of small graphs, the monad being the free-graph functor. I interpreted that statement as: the Eilenberg-Moore category of the ...
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### Definition of strict 2-monad on Cat

The category $\text{Cat}$ can be thought of as a $2$-category. I was hoping somebody could help by telling me the explicit definition of a strict 2-monad $(T, \eta, \mu)$ on $\text{Cat}.$ In ...
### Category of Monads on $PreOrd$
In a recent paper by Adamek ("Finitary Monads on the Category of Posets", see it on Arxiv) the author introduces the category of finitary monads on $Pos$ and analyzes it in great detail. ...