Monads and their Algebras — Index

Monads and their Algebras

Routing Summary

• Need the big picture / where to start? → Monads - Overview
• Need the definition of a monad (endofunctor $T$, unit $\eta$, multiplication $\mu$, coherence laws)? → Monads and the Monad Laws
• Need why every adjunction gives a monad ($T=UF=GF$, $\mu =UϵF$)? → Adjunctions Induce Monads
• Need algebras for a monad — Eilenberg–Moore ${\mathsf{C}}^T$, Kleisli ${\mathsf{C}}_T$, the comparison functor? → Algebras for a Monad - Eilenberg-Moore and Kleisli
• Need when is a functor monadic — Beck’s theorem, split coequalizers, monadicity over $\mathsf{Set}$, limits/colimits of algebras? → Beck’s Monadicity Theorem
• Need a comonad, closure operator, or the monoid-in-endofunctors view? → Monads and the Monad Laws
• Need the list / maybe / power-set / state monad examples? → Monads and the Monad Laws and Algebras for a Monad - Eilenberg-Moore and Kleisli

Concept Map

Concept	Note	Type	Depends On	Key Result
Orientation & motivation	Monads - Overview	overview	Adjoint Functors, Units and Counits	Monad = shadow of an adjunction; monoid in endofunctors
Monad definition & laws	Monads and the Monad Laws	definition	Functors, Natural Transformations	Def 5.1.1: $\mu \cdot T\mu =\mu \cdot \mu T$, $\mu \cdot \eta T=\mathrm{id}=\mu \cdot T\eta$
Adjunction → monad	Adjunctions Induce Monads	concept	Monads and the Monad Laws, Adjoint Functors	Lemma 5.1.3: $T=UF$, $\mu =UϵF$
Algebras: EM & Kleisli	Algebras for a Monad - Eilenberg-Moore and Kleisli	definition	Adjunctions Induce Monads	Def 5.2.4/5.2.10; ${\mathsf{C}}_T$ initial, ${\mathsf{C}}^T$ terminal (5.2.13); comparison full & faithful (5.2.14)
Monadicity	Beck’s Monadicity Theorem	theorem	Algebras for a Monad - Eilenberg-Moore and Kleisli	Thm 5.5.1: $U$ monadic ⟺ creates coequalizers of $U$-split pairs

Notes

• Monads - Overview — CONTAINS: the chapter’s organizing slogan (“monad = shadow of an adjunction”); the six-section roadmap; the list of categories monadic over $\mathsf{Set}$; how the five notes interrelate.
• Monads and the Monad Laws — CONTAINS: Definition 5.1.1 (monad: $T$, $\eta$, $\mu$, associativity + unit diagrams); Remark 5.1.2 (monoid in the monoidal category of endofunctors); Definition 5.1.6 (comonad); Example 5.1.7 (closure/interior operators on a preorder); the list, maybe, power-set, double power-set, free-module, free-group, state, Giry monads.
• Adjunctions Induce Monads — CONTAINS: Lemma 5.1.3 ($T=UF$, $\mu =UϵF$) with proof via the triangle identities and naturality of $ϵ$; the $\mathsf{Set}\rightleftarrows \mathsf{Ab}$ worked example; free $⊣$ forgetful sources of monads; reflective subcategories → idempotent monads.
• Algebras for a Monad - Eilenberg-Moore and Kleisli — CONTAINS: Definition 5.2.4 ($T$-algebra, EM category ${\mathsf{C}}^T$, algebra homomorphism); Definition 5.2.8 (free $T$-algebra, free functor $F^T$); Lemma 5.2.9 (EM adjunction $F^T⊣U^T$, counit = structure map); Definition 5.2.10 (Kleisli category, Kleisli composite); Lemma 5.2.12 (Kleisli adjunction); Proposition 5.2.13 (Kleisli initial, EM terminal); Lemma 5.2.14 (comparison functor full & faithful, image = free algebras); algebra/Kleisli examples.
• Beck’s Monadicity Theorem — CONTAINS: Definition 5.3.1 (monadic / strictly monadic); Definition 5.4.4 (split coequalizer) + Lemma 5.4.6 (absolute colimit); Proposition 5.4.2 (canonical presentation of an algebra); Definition 5.4.8 ($U$-split, creating coequalizers); Theorem 5.5.1 (Beck / PTT); Proposition 5.5.8 (reflexive tripleability) and CTT/VTT variants; Theorem 5.5.9 (Paré: ${\mathsf{Set}}^{op}$ monadic); §5.6 limits/colimits results (Lemma 5.6.1, Theorem 5.6.5, Corollaries 5.6.6/5.6.7/5.6.9, Prop 5.6.11, Thm 5.6.12); monadic and non-monadic examples over $\mathsf{Set}$.

Sources

• Riehl - Category Theory in Context.pdf — Chapter 5, “Monads and their Algebras” (pp. 179–216): §5.1 Monads from adjunctions, §5.2 Adjunctions from monads, §5.3 Monadic functors, §5.4 Canonical presentations via free algebras, §5.5 Recognizing categories of algebras, §5.6 Limits and colimits in categories of algebras.