Monads, Algebras, and the Barr–Beck Theorem¶
Job: develop monads from adjunctions, the Eilenberg–Moore / Kleisli categories, the Barr–Beck monadicity theorem, and distributive laws — with the free \(R\)-module monad / Barr–Beck as the worked example.
Core¶
Monads and adjunctions¶
The monad \(T=\langle T,\eta,\mu\rangle\) (the \(T:\mathcal{C}\to\mathcal{C}\), the unit \(\eta:1\to T\), the multiplication \(\mu:T^2\to T\)). The adjunction \(\to\) monad: every adjunction \(F\dashv G\) gives a monad \(T=GF\) with \(\eta\) the unit, \(\mu=G\varepsilon F\) (the counit \(\varepsilon\)).
| Monad / adjunction / EM | Description |
|---|---|
| Monad \(T=\langle T,\eta,\mu\rangle\) | \(T:\mathcal{C}\to\mathcal{C}\), \(\eta:1\to T\), \(\mu:T^2\to T\) (the monad laws \(\mu\circ T\mu=\mu\circ\mu T\), \(\mu\circ T\eta=\mu\circ\eta T=1\)) |
| Adjunction \(\to\) monad | \(F\dashv G\) gives \(T=GF\), \(\eta\) the unit, \(\mu=G\varepsilon F\) |
| Eilenberg–Moore \(\mathcal{C}^T\) | the \(T\)-algebras \((A,\alpha:TA\to A)\), the comparison \(K:\mathcal{D}\to\mathcal{C}^T\) |
| Comparison | \(K:\mathcal{D}\to\mathcal{C}^T\) the comparison functor (the EM the universal) |
The Kleisli and the free algebras¶
| Kleisli / free / EM vs Kleisli | Description |
|---|---|
| Kleisli \(\mathcal{C}_T\) | the free \(T\)-algebras (the Kleisli, the objects \(=\) objects of \(\mathcal{C}\), the morphisms \(X\to Y\) the \(X\to TY\) in \(\mathcal{C}\)) |
| EM vs Kleisli | the EM the all \(T\)-algebras, the Kleisli the free (the Kleisli \(\subset\) EM, the full subcategory of the free) |
The Barr–Beck monadicity¶
Theorem (Barr–Beck). \(U:\mathcal{D}\to\mathcal{C}\) is monadic (i.e. \(\mathcal{D}\simeq\mathcal{C}^T\) for \(T=UF\)) iff \(U\) has a left adjoint, \(U\) reflects isos, and \(\mathcal{D}\) has + \(U\) preserves coequalizers of \(U\)-split pairs.
| Barr–Beck / monadicity / descent | Description |
|---|---|
| Barr–Beck | \(U\) monadic iff \(U\) has left adjoint, reflects isos, coequalizers of \(U\)-split pairs exist and are preserved |
| Monadic descent | the Barr–Beck the descent (the higher categories, the derived AG) |
The distributive laws and the composite¶
| Distributive / composite | Description |
|---|---|
| Distributive law | \(\lambda:TS\to ST\) (the distributive, the composite monad \(ST\)) |
| Composite / iterated | the Beck (the composite monad from the distributive law, the iterated) |
See it¶
Monad of the free \(R\)-module and the Barr–Beck for modules. The monad: \(T(S)=\) the free \(R\)-module on \(S\) (the \(F\dashv U\), the \(T=UF\), the \(\eta:S\to UF(S)\) the inclusion of generators, the \(\mu:T^2(S)\to T(S)\) the \(R\)-linear extension). The EM: the \(T\)-algebras \(=\) the \(R\)-modules (the \(\mathcal{C}^T\simeq R\text{-}\mathbf{Mod}\), the \(T\)-algebra \(\alpha:TA\to A\) the \(R\)-action). The Barr–Beck: the \(U:R\text{-}\mathbf{Mod}\to\mathbf{Set}\) monadic (the \(U\) has the left adjoint \(F\), reflects isos, the coequalizers of \(U\)-split pairs exist and are preserved). The key point: the free \(R\)-module the monad, the EM the modules, the Barr–Beck the monadic.
Check¶
Show that every adjunction \(F\dashv G\) gives a monad \(T=GF\), and verify the monad laws.
Given \(F:\mathcal{C}\to\mathcal{D}\), \(G:\mathcal{D}\to\mathcal{C}\) with unit \(\eta:1_\mathcal{C}\to GF\) and counit \(\varepsilon:FG\to1_\mathcal{D}\). Define \(T=GF:\mathcal{C}\to\mathcal{C}\), the unit \(\eta:1\to T\) (the adjunction unit), the multiplication \(\mu=G\varepsilon F:T^2=GFGF\to GF=T\) (the \(G\varepsilon F\)). The monad laws: \(\mu\circ T\eta=G\varepsilon F\circ GF\eta=G(\varepsilon\circ F\eta)F=G\cdot1_F=1_{GF}\) (the triangle \(\varepsilon\circ F\eta=1_F\)), \(\mu\circ\eta T=G\varepsilon F\circ\eta GF=G(\varepsilon F\circ\eta F)F...\) the other triangle, \(\mu\circ T\mu=G\varepsilon F\circ GF G\varepsilon F=G(\varepsilon\circ FG\varepsilon)F=G(\varepsilon\varepsilon)F\)... the associativity from the counit law. So \(T=GF\) the monad with \(\eta\) the unit, \(\mu=G\varepsilon F\).
Explain the Barr–Beck conditions and why each is needed, and the monadic descent connection.
The Barr–Beck: \(U:\mathcal{D}\to\mathcal{C}\) monadic iff (1) \(U\) has a left adjoint \(F\) (the monad \(T=UF\)), (2) \(U\) reflects isos (the \(U(f)\) iso \(\Rightarrow\) \(f\) iso — the conservative, the descent: the objects glued from the local), (3) \(\mathcal{D}\) has + \(U\) preserves coequalizers of \(U\)-split pairs (the split coequalizer the descent: the coequalizer the gluing, the \(U\)-split the the fork splits after \(U\) — the descent data). Each needed: (1) the monad, (2) the reflection (the conservative, without it the comparison not an equivalence), (3) the coequalizers (the descent the gluing, without it the EM objects not the \(\mathcal{D}\) objects). The monadic descent: the Barr–Beck the categorical descent (the \(U\)-split the descent data, the coequalizer the gluing, the higher categories the \(\infty\)-categorical version, the derived AG the derived descent).
Pitfalls¶
- The monad laws (\(\mu\circ T\mu=\mu\circ\mu T\), \(\mu\circ T\eta=\mu\circ\eta T=1\)) the associativity / the unit — the monad the categorical, the laws the algebraic.
- The EM vs the Kleisli (the EM the all \(T\)-algebras, the Kleisli the free — the Kleisli \(\subset\) EM, the full subcategory of the free) — the two the different (the EM the universal, the Kleisli the free).
- The Barr–Beck the \(U\)-split pairs (not the arbitrary coequalizers — the \(U\)-split the specific, the split after \(U\), the descent data).
- The monadic descent the higher categories / the derived AG (the \(\infty\)-categorical Barr–Beck, the derived descent — the Lurie the \(\infty\)-version).
Refs¶
- Mac Lane, Categories for the Working Mathematician (Ch. VI: monads); Eilenberg–Moore (1965)
- Beck, "Distributive laws" (1969); the Barr–Beck monadicity theorem
- Lurie, Higher Algebra (the \(\infty\)-categorical monadicity / descent); the derived AG
Next¶
CT index · Foundations · Monoidal & Enriched · Accessible & Presentable · Categorical Logic & Topoi · Higher categories · Derived AG