symmetric monoidal (∞,1)-category of spectra
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A functor $U \,\colon\, D\to C$ is monadic iff it has a left adjoint $F \,\colon\, C\to D$ and the adjunction $F\dashv U$ “comes from: the induced monad on $C$” – that is, $U$ monadic iff $F\dashv U$ is a monadic adjunction.
In this situation $U$ “looks like” the forgetful functor from the Eilenberg-Moore category of the monad $(U\circ F, \eta, U\epsilon_F)$ on $C$, and has ‘nice properties’ similar to 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.
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.
(monadic functor)
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.
A category $D$ is called monadic over a category $C$ if there is any functor $U \colon D \to C$ which is monadic (Def. ).
A monadic functor is strictly monadic if and only if it is also an amnestic isofibration.
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.
Beware that the class of monadic functors is not closed under composition.
For a specific counter-example: the category of reflexive quivers 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 such categories are regular categories but $Cat$ 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. every 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.
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.
Francis Borceux, Def. 4.4.1 in: Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]
Emily Riehl, §5.3 in: Category Theory in Context, Dover Publications (2017) [pdf]
Last revised on November 6, 2022 at 10:50:32. See the history of this page for a list of all contributions to it.