Contents

### Context

category theory

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## Idea

A functor $U \,\colon\, D\to C$ is monadic iff it has a left adjoint $F \,\colon\, C\to D$ such that – under the relation between adjunctions and monads – the adjunction $F\dashv U$ is that induced by the monad which it induces – in which case it is called a monadic adjunction.

In this situation $U$ is identified with the forgetful functor from the Eilenberg-Moore category $EM(U \circ F)$ of the monad $(U \circ F, \eta, U\epsilon_F)$ on $C$, and hence shares the properties of these forgetful functors.

The monadicity theorem characterizes monadic functors and makes these ‘nice properties’ precise.

Monadic functors are sometimes called functors of effective descent (type). See the page on monadic descent for more on this aspect.

## Definition

Given a pair of adjoint functors $F \colon C \to D :U$, $F \dashv U$, with unit $\eta: Id_C \to U \circ F$ and counit $\epsilon: F \circ U \to Id_D$, one constructs a monad $\mathbf{T}=(T,\mu,\eta)$ setting $T = U \circ F: C \to C$, $\mu = U \epsilon F: T T = U F U F \to U F = T$.

Consider the Eilenberg-Moore category $C^{\mathbf{T}}$ of $T$-algebras ($T$-modules) in $C$. Clearly $U (\epsilon_M): T U M = U F U M \to U M$ is a $T$-action. In fact there is a canonical comparison functor $K^{\mathbf{T}} \colon D \to C^{\mathbf{T}}$ given on objects by $K(M) \coloneqq \big(U M, U (\epsilon_M) \big)$. We then say that we have a (resp. strictly) monadic adjunction iff $K$ is an equivalence (resp. isomorphism) of categories.

###### Definition

A functor $U \colon D \to C$ is monadic (resp. strictly monadic) if it has a left adjoint $F \colon C\to D$ and the comparison functor $K^{\mathbf{T}} \colon D \to C^{\mathbf{T}}$ is an equivalence of categories (resp. an isomorphism of strict categories).

In other words, up to equivalence, monadic functors are precisely the forgetful functors defined on Eilenberg-Moore categories for monads, and strictly monadic functors are the same as these forgetful functors up to isomorphism of strict categories.

###### Definition

A category $D$ is called monadic over a category $C$ if there is any functor $U \colon D \to C$ which is monadic (Def. ).

## Properties

### Basic properties

1. faithful (by the definition of Eilenberg-Moore category)

Moreover:

###### Proposition
1. creates all limits that exist in its codomain;

2. creates all colimits that exist in its codomain and are preserved by the corresponding monad (or, equivalently, by the monadic functor itself).

(e.g. MacLane 71, Exercise IV.2.2 (p. 138))

###### Remark

Beware that the class of monadic functors is not generally closed under composition.

For a specific counter-example: the category of reflexive graphs is monadic over Set via the functor $RefGph \to Set$ sending a graph to its set of edges, and the category of categories is monadic over reflexive graphs via the forgetful functor $Cat \to RefGph$, but $Cat$ is not monadic over $Set$ (via any functor whatsoever, since monadic categories over Set are regular categories which Cat is not).

This is an instance of a general phenomenon: Let $\mathcal{C}$ be a reflective subcategory of a presheaf category $\widehat{A}$ (e.g. any locally presentable category is of this form). Then the adjunction between $\mathcal{C}$ and $\widehat{A}$ is monadic, and the adjunction between $\widehat{A}$ and $\mathrm{Set}^{\mathrm{Ob} A}$ is also monadic. But the composite adjunction between $\mathcal{C}$ and $\mathrm{Set}^{\mathrm{Ob} A}$ is often not monadic. For instance, if it is monadic, then $\mathcal{C}$ must be a Barr-exact category.

Monadic functors have the following cancellation property:

###### Proposition

Consider a pair of adjunctions: If here $U' U$ is monadic, then $U$ is of descent type and the comparison functor has a left adjoint. If $U'$ is furthermore conservative (and in particular if it is monadic), then $U$ is monadic.

This is Propositions 4 and 5 of Bourn.

###### Proposition

A monadic functor is strictly monadic if and only if it is also an amnestic isofibration.

###### Proof

Clearly, a strictly monadic functor is an amnestic isofibration; and if a monadic functor $U$ is amnestic, then the comparison functor $K$ is also amnestic, and if $U$ is a monadic isofibration, so is $K$; therefore in this case $K$ must be an isomorphism of categories.

Various versions of Beck’s monadicity theorem (also: “tripleability theorem” in older literature) give sufficient, and sometimes necessary, conditions for a given functor to be monadic. There are also dual, comonadic versions.

• Every monadic functor $U \,\colon\, D \to \mathrm{Set}$ is a solid functor.

• A category is monadic over $\mathrm{Set}$ (i.e. it admits a monadic functor to $\mathrm{Set}$) if and only if is Barr exact, cocomplete, and has a projective generator.