The monads tag has no usage guidance.

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### For which category (if any) are Lie algebras the algebras of a monad? [migrated]

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...

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### List is a monad, but is it a comonad with these natural transformations?

List is known to be a monad. It takes a set and maps it to lists of elements of that set. The natural transformations are, singleton and flatten, whereby we map a set to a set of singleton lists ...

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### Intuition for density comonad in relation to lifting problems

In Emily Riehl's Categorical Homotopy Theory, there is a section on Garner's Small Object Argument which I'm trying and failing to understand. Originally I followed most of Garner's paper, using the ...

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### “Why” are monadicity and descent related?

This question is probably too vague for experts, but I really don't know how to avoid it.
I've read in several places that under mild conditions, a morphism is an effective descent morphism iff the ...

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### Algebras for probability monad

What is the Eilenberg-Moore category for the non-finitary probability distribution monad is, that is, the monad $D \colon \mathbf{Set} \to \mathbf{Set}$ defined by
$$
DX = \left\{ p \in [0,1]^X \ \...

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### What is known about the category of monads on Set?

Monads on the category Set of sets and functions are somehow fundamental objects of category theory, and moreover they have important applications to computer science. We know of a good number of ...

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### iterated loop spaces and configuration spaces [closed]

In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map
$$
\phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y)
$$
is defined. And a map
$$
\eta_n=\phi^{-...

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### Correspondence between operads and monads requires tensor distribute over coproduct?

In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without ...

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### tree derived from monad is itself a monad

I have constructed a functor from a monad that appears (based on computer experiments to test the monad laws) to also have monad properties but I am having trouble proving it.
Here is the idea: M[A] ...

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### Distributive law between Kleisli triples

A distributive law of a monad $S$ over a monad $T$ is a natural transformation $l : T S \to S T$ such that:
$l \circ T \eta^S = \eta^S T$
$l \circ \eta^T S = S \eta^T$
$\mu^S T \circ S l \circ l S = ...

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### Whiskering a monad

In "The Geometry of Iterated Loop Spaces", May shows that any monad that is coming from an operad may be "whiskered", so that the unit map becomes a closed cofibration. The ability to do this is vital ...

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### opposite category

In the 2-category Cat of small categories, for each category $C$ (an object of Cat) there is also the dual category (I dare not write "dual object") $C^{op}$.
Is ${op}$ the instance in Cat of a more ...

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### Comonads from monoids

The following construction is probably known. I think it should work in any closed symmetric monoidal category, but I will play it safe and formulate the question in the concrete, cartesian closed ...

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### When is an Eilenberg-Moore category or Kleisli category braided monoidal? When semisimple?

I have a braided monoidal, semisimple linear category $\mathcal{C}$. (Imagine representations of a semisimple quasitriangular Hopf algebra.) I also have a monad $(T,\mu,\eta)$ on it, however, $T$ is ...

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### Further relation between monads and theories

This question want to be a follow up of the following question.
In that thread I was interested in understanding relation between various presentation of algebraic theories. In particular in Eduardo ...

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### Free cocommutative commutative Hopf monoids

I have some questions about generalizations of abelian groups, relative to symmetric monoidal categories.
1) Let $C$ be a cocomplete cartesian monoidal category with equalizers. I can show that the ...

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### Universal property of module categories over monads

Let $T$ be a monad on a cocomplete category $\mathcal{C}$. Let's assume that $T$ preserves reflexive coequalizers (or something weaker?). Then the category of $T$-modules $\mathsf{Mod}(T)$ is ...

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### Transporting algebraic structure along adjoint equivalences

I have two questions, one general and the other particular to the case I am interested in.
The 'homotopically correct' notion of equivalence of categories is an adjoint equivalence (from one point of ...

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### comparison between two monadic definitions for an operad

According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$
According to Leinster, an operad is ...

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### Monadicity theorem in homotopy theory.

Let $\mathbf{C}$ be a cofibrantly generated model category (assume for simplicity that all objects are fibrant) and $\mathbf{C}^{\mathrm{T}}$ the category of $\mathrm{T}$-algebras with the induced ...

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### Higher Descent Cohomology

Descent cohomology for a comonad is defined at degrees 0 and 1 by Mesablishvili in his paper "On Descent Cohomology" (as well as by many other authors in many other contexts). For a comonad $\bot$ on ...

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### Regarding a difficulty in the Fakir article about associated idempotent triple

I just had post this question in SE: http://math.stackexchange.com/questions/518054/about-details-of-the-fakir-theorem-proof-associated-idempotent-triple but dont get any answer.
I understand that at ...

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### Coequalizers in an Eilenberg-Moore category

Last month I proved that some category $\mathbf C$ that I happen to care about is isomorphic to the Eilenberg-Moore category for a monad on the category of bounded posets $\mathbf{BPos}$.
I know from ...

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### Iterating Monad-Comonads structures

Let $(T, \mu , \eta )$ a monad on the category $\mathscr{C}$ , with the usual EM (Eilenberg-Moore) adjunction $\langle F_T, U_T, \eta_Y, \epsilon_T \rangle: \mathscr{C}^T \to \mathscr{C}$ where ...

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### Is there a monad on Set whose algebras are Tychonoff spaces?

Compact Hausdorff spaces are algebras of the ultrafilter monad on Set.
Is the category of Tychonoff spaces also monadic over Set?

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### When do reflexive coequalizers preserve weak equivalences?

In my work I've run into the following situation. In a model category, I have two reflexive coequalizers $A_i \stackrel{\to}{\to} B_i \to C_i$ and a map of diagrams which is levelwise a weak ...

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### Reference for my monads?

I'm looking for a reference for a certain pair of monads on $Cat$. One problem is that I don't know the modern way of thinking about some basic things, so excuse me if my presentation is naive.
First ...

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### Free monad or monad defined from an adjunction.

My first question here.
Accordingly to M. Barr "Coequalizers and free triples" by a free triple (or free monad) generated by an endofunctor $R: X\rightarrow{X}$ we mean a
triple $T=(T,\eta,\nu)$ and ...

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### Compatibility between strength and costrength of a monoidal monad

Let $C$ be a closed monoidal category, and let $T : C \to C$ be a monad on the underlying category. Let $\sigma$ be a tensorial strength of $T$ and let $\sigma'$ be a cotensorial strength of $T$. A ...

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### The crude monadicity theorem

In order to test the monadicity of a functor, there is a precise monadicity theorem (PM) as well as a crude monadicity theorem (CM), see the nlab. In CM, the forgetful functor should create reflexive ...

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### Applications of the Giry monad in probability and statistics

In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$.
Will Sawin described the ...

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### Seems like Reader monad composed with a strong monad produces a monad, am I right?

Take a Cartesian (or monoidal) closed category; define Reader monad for a given object $E$ as
$X \mapsto X^E$; and take a strong monad $M$ (strong means preserves product or tensor product).
Now the ...

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### Can we describe equivariant vector bundles of free group action in terms of descent theory (Barr-Beck theorem)?

It is well known that for a compact topological group $G$ acts (say, from the right) freely on a compact space $X$. Then the category of equivariant complex vector bundles on $X$, $\text{Vect}_G(X)$, ...

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### “Functors between monads”: what are these really called?

Let $(S,\eta,\mu)$ be a monad on a category $C$, and $(T,\eta,\mu)$ a monad on a category $D$. The following kind of gadget is ubiquitous: a functor $F:D\to C$, together with a natural map $\sigma: ...

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### Where is there a treatment of “exponential monads”?

I have a category $C$, which is equipped with a symmetric monoidal structure (tensor product $\otimes$, unit object $1$). My category also has finite coproducts (I'll write them using $\oplus$, and ...

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### When do functors induce monadic adjunctions of presheaf categories

For a category $C$, let $C-Set$ denote the category of set-valued functors $\delta\colon C\to Set$. Given categories $C$ and $D$, and a functor $F\colon C\to D$, composition with $F$ yields a functor ...

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### Coproducts of modules over an algebraic monad

Coproducts of modules over an algebraic monad $\Sigma$ are described in Section 4.16.14/15 in Durov's thesis. It is claimed there that for $\Sigma$-modules $M,N$, the set $M \coprod N$ generates $M \...

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### The reflexive free-category comonad-resolution is a cofibrant replacement of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?

Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to its underlying ...

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### Re-seating a monad

Let $\mathcal C$ and $\mathcal D$ be categories with suitable limits and colimits for the following discussion. Is it possible to re-interpret, or "re-seat" a monad $T : \mathcal C \to \mathcal C$ as ...

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### Relation between monads, operads and algebraic theories

I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any ...

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### Monad arising from operad

It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way ...

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### What are the algebras over $\Omega^k\Sigma^k$ ?

Let $Ho(Spc)$ be the homotopy category of spaces. There is an adjoint pair
$$
\Sigma^k \colon Ho(Spc) \leftrightarrows Ho(Spc)\colon \Omega^k,
$$
where $\Sigma^k$ is the $k$-th supension functor and $\...

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### Reference request: 2-Monads and 2-Adjunctions

Given a category $\mathcal C$ together with a monad $T$ on $\mathcal C$, we get an adjunction $$\mathcal C^T(T-,-)\cong \mathcal C(-,\mathrm{For}-).$$
Is the same true for 2-monads on a 2-category?

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### Characterization of Kleisli adjunctions

There's a well known theorem due to Beck that characterizes when an adjunction is monadic, that is, if $F$ is left adjoint to $G$, $G:D \to C$, $GF:=T$ is always a monad on $C$, and the adjunction is ...

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### Adjunctions: Algebras of the induced monad VS. Coalgebras of the induced comonad.

Given an adjunction, we get a monad on one side and a comonad on the other side. What is the connection between their algebra and coalgebra categories? Are they allways equivalent?
The example i have ...

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### An elementary question about adjunctions between presheaf categories preserving pullbacks.

A functor $C \to D$ between categories induces a morphism of presheaf categories $Pre(D) \to Pre(C)$. This functor has a left adjoint given by left Kan extension and I am interested in knowing when ...

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### Cartesian-closed categories of algebras

If the Kleisli-category of a monad is Cartesian-closed, can we say when the category of Eilenberg-Moore algebras is?

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### Eilenberg–Moore algebras in terms of Kleisli ones

Suppose I know what the category of free algebras for a particular monad look like. Can I then describe what the category of Eilenberg–Moore algebras look like? E.g. Suppose that I have a good handle ...

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### Right actions of operads and monads

Given an operad $A$, there is an associated monad $M_A$ given by $M_A(X) = \coprod_n A(n) \otimes X^{\otimes n}$ such that being an $A$-algebra and being an algebra over the monad $M_A$ is the same ...

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### Lax and Colax Monads

Is there much known about the theory of lax and colax monads on a bicategory? Here, I really mean lax or colax, not weak. I'm aware of some literature about weak monads. I'm interested in distributive ...