📐 Category Theory IV: Adjoint Functor Theorems and Monads

Table of Contents

• #1. Motivation: When Does a Functor Have an Adjoint|1. Motivation: When Does a Functor Have an Adjoint
• #2. Adjoints and Infima in Posets|2. Adjoints and Infima in Posets
• #3. Monads: Definition and First Examples|3. Monads: Definition and First Examples
  • #3.1 The Definition|3.1 The Definition
  • #3.2 Every Adjunction Gives a Monad|3.2 Every Adjunction Gives a Monad
  • #3.3 Every Monad Arises from an Adjunction|3.3 Every Monad Arises from an Adjunction
• #4. Eilenberg-Moore Algebras|4. Eilenberg-Moore Algebras
  • #4.1 T-Algebras and Their Morphisms|4.1 T-Algebras and Their Morphisms
  • #4.2 The Eilenberg-Moore Category|4.2 The Eilenberg-Moore Category
  • #4.3 The Comparison Functor|4.3 The Comparison Functor
• #5. Absolute Coequalizers and Split Pairs|5. Absolute Coequalizers and Split Pairs
  • #5.1 Reflexive Pairs|5.1 Reflexive Pairs
  • #5.2 Split Coequalizers|5.2 Split Coequalizers
• #6. Set-Adjoint, Representable, and Limit-Preserving Functors|6. Set-Adjoint, Representable, and Limit-Preserving Functors
• #7. The General Adjoint Functor Theorem|7. The General Adjoint Functor Theorem
  • #7.1 The Solution Set Condition|7.1 The Solution Set Condition
  • #7.2 Proof Sketch|7.2 Proof Sketch
• #8. Well-Powered Categories and the Subobject Functor|8. Well-Powered Categories and the Subobject Functor
• #9. The Special Adjoint Functor Theorem|9. The Special Adjoint Functor Theorem
  • #9.1 Coseparating Sets|9.1 Coseparating Sets
  • #9.2 Proof Sketch via GAFT|9.2 Proof Sketch via GAFT
• #10. Applications of GAFT and SAFT|10. Applications of GAFT and SAFT
• #11. The Eilenberg-Moore Category: Creation Theorems|11. The Eilenberg-Moore Category: Creation Theorems
  • #11.1 Creation of Limits|11.1 Creation of Limits
  • #11.2 Creation of Coequalizers for Absolute Pairs|11.2 Creation of Coequalizers for Absolute Pairs
• #12. Idempotent Monads|12. Idempotent Monads
• #13. Presheaves Are Monadic|13. Presheaves Are Monadic
• #14. The Beck Monadicity Theorem|14. The Beck Monadicity Theorem
  • #14.1 The Huge Monadicity Theorem|14.1 The Huge Monadicity Theorem
  • #14.2 Proof Strategy|14.2 Proof Strategy
• #15. The Comparison Functor Revisited|15. The Comparison Functor Revisited
• #16. Geometric Realization via SAFT|16. Geometric Realization via SAFT
• #17. Comonads from Adjunctions|17. Comonads from Adjunctions
• #18. Monadic Categories over Set and Top|18. Monadic Categories over Set and Top
• #19. Canonical Presentations and the Bar Construction|19. Canonical Presentations and the Bar Construction 🔩
• #References|References

1. Motivation: When Does a Functor Have an Adjoint 🔍

A central result proved in concepts/category-theory/03-limits-colimits|File 3 is that right adjoints preserve limits. More precisely, if \(F \dashv G\) with \(G \colon \mathcal{D} \to \mathcal{C}\), then \(G\) preserves all limits that exist in \(\mathcal{D}\).

This raises the converse question: when does limit-preservation suffice to guarantee the existence of a left adjoint? The answer is the content of the Adjoint Functor Theorems, the main subject of this file.

The plan is: 1. Understand the poset case, where the theorem is transparent. 2. State and prove the General Adjoint Functor Theorem (GAFT) using the solution set condition. 3. Derive the Special Adjoint Functor Theorem (SAFT) using well-poweredness. 4. Develop the theory of monads and prove Beck’s Monadicity Theorem.

2. Adjoints and Infima in Posets 📊

A poset \((P, \leq)\) is a category whose objects are elements of \(P\) and whose hom-sets satisfy \(|P(p, q)| \leq 1\), with a morphism \(p \to q\) iff \(p \leq q\). A functor between posets is exactly a monotone map.

Definition (Adjunction in a Poset). Let \(P\) and \(Q\) be posets. Monotone maps \(f \colon P \to Q\) and \(g \colon Q \to P\) form an adjunction \(f \dashv g\) if and only if: \[f(p) \leq q \iff p \leq g(q) \quad \text{for all } p \in P,\, q \in Q.\] The map \(f\) is called a Galois connection lower adjoint and \(g\) the upper adjoint.

Proposition. Let \(g \colon Q \to P\) be a monotone map where \(Q\) has all infima. Then \(g\) has a left adjoint if and only if \(g\) preserves all infima. In that case, the left adjoint is: \[f(p) = \inf\{q \in Q : p \leq g(q)\}.\]

Proof sketch. If \(f \dashv g\), then for any \(p \in P\) the set \(\{q \in Q : p \leq g(q)\} = \{q : f(p) \leq q\}\), and its infimum is \(f(p)\). This shows \(g\) must preserve infima. Conversely, given that \(g\) preserves infima, define \(f(p) = \inf\{q : p \leq g(q)\}\). One checks \(p \leq g(f(p))\) (using infimum closure and preservation) and \(f(g(q)) \leq q\), establishing \(f \dashv g\).

3. Monads: Definition and First Examples 🔑

3.1 The Definition

A monad is the categorical abstraction of a monoid acting on itself — or, as the slogan goes, “a monad is just a monoid in the category of endofunctors.”

Definition (Monad). A monad on a category \(\mathcal{C}\) is a triple \((T, \eta, \mu)\) consisting of: - an endofunctor \(T \colon \mathcal{C} \to \mathcal{C}\), - a natural transformation \(\eta \colon \mathrm{id}_{\mathcal{C}} \Rightarrow T\) called the unit, - a natural transformation \(\mu \colon T^2 \Rightarrow T\) (where \(T^2 = T \circ T\)) called the multiplication,

subject to the following two axioms, expressed as commutative diagrams:

Associativity:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
T^3 \arrow[r, "T\mu"] \arrow[d, "\mu T"'] & T^2 \arrow[d, "\mu"] \\
T^2 \arrow[r, "\mu"'] & T
\end{tikzcd}
\end{document}

That is: \(\mu \circ T\mu = \mu \circ \mu T\) as natural transformations \(T^3 \Rightarrow T\).

Unit laws:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
T \arrow[r, "\eta T"] \arrow[dr, "\mathrm{id}_T"'] & T^2 \arrow[d, "\mu"] & T \arrow[l, "T\eta"'] \arrow[dl, "\mathrm{id}_T"] \\
& T &
\end{tikzcd}
\end{document}

That is: \(\mu \circ \eta T = \mathrm{id}_T = \mu \circ T\eta\).

Examples:

3.2 Every Adjunction Gives a Monad

Proposition. Let \(F \dashv G\) with \(F \colon \mathcal{C} \to \mathcal{D}\) and \(G \colon \mathcal{D} \to \mathcal{C}\), unit \(\eta \colon \mathrm{id}_{\mathcal{C}} \Rightarrow GF\), and counit \(\varepsilon \colon FG \Rightarrow \mathrm{id}_{\mathcal{D}}\). Then \((T, \eta, \mu)\) where \(T = GF\) and \(\mu = G\varepsilon F \colon GFGF \Rightarrow GF\) is a monad on \(\mathcal{C}\).

Proof sketch. The unit law \(\mu \circ \eta T = \mathrm{id}_T\) follows from one of the triangle identities \(G\varepsilon \circ \eta G = \mathrm{id}_G\). The law \(\mu \circ T\eta = \mathrm{id}_T\) follows from \(\varepsilon F \circ F\eta = \mathrm{id}_F\) and applying \(G\). Associativity \(\mu \circ T\mu = \mu \circ \mu T\) reduces to naturality of \(\varepsilon\).

3.3 Every Monad Arises from an Adjunction

Theorem. Every monad \((T, \eta, \mu)\) on \(\mathcal{C}\) arises from at least one adjunction. In fact, it arises from (at least) two canonical ones: 1. The Kleisli adjunction \(F_T \dashv U_T \colon \mathcal{C}_T \to \mathcal{C}\) via the Kleisli category \(\mathcal{C}_T\) (objects of \(\mathcal{C}\), morphisms \(A \to B\) in \(\mathcal{C}_T\) are morphisms \(A \to TB\) in \(\mathcal{C}\)). 2. The Eilenberg-Moore adjunction \(F^T \dashv U^T \colon \mathcal{C}^T \to \mathcal{C}\) via the Eilenberg-Moore category \(\mathcal{C}^T\) of \(T\)-algebras (defined in §4).

The Kleisli adjunction is the initial adjunction giving rise to \((T, \eta, \mu)\); the Eilenberg-Moore adjunction is the terminal one.

4. Eilenberg-Moore Algebras 🏗️

4.1 T-Algebras and Their Morphisms

Definition (T-Algebra). Let \((T, \eta, \mu)\) be a monad on \(\mathcal{C}\). A \(T\)-algebra (or Eilenberg-Moore algebra) is a pair \((A, a)\) where \(A \in \mathcal{C}\) and \(a \colon TA \to A\) is a morphism in \(\mathcal{C}\), called the structure map, satisfying:

• Unit law: \(a \circ \eta_A = \mathrm{id}_A\) (the unit is a section of \(a\)),
• Multiplication law: \(a \circ \mu_A = a \circ Ta\) (the two ways of collapsing \(T^2A\) to \(A\) agree).

As commutative diagrams:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A \arrow[r, "\eta_A"] \arrow[dr, "\mathrm{id}_A"'] & TA \arrow[d, "a"] \\
& A
\end{tikzcd}
\end{document}

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
T^2A \arrow[r, "\mu_A"] \arrow[d, "Ta"'] & TA \arrow[d, "a"] \\
TA \arrow[r, "a"'] & A
\end{tikzcd}
\end{document}

Definition (T-Algebra Morphism). A morphism of \(T\)-algebras \(f \colon (A, a) \to (B, b)\) is a morphism \(f \colon A \to B\) in \(\mathcal{C}\) such that \(f \circ a = b \circ Tf\).

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
TA \arrow[r, "Tf"] \arrow[d, "a"'] & TB \arrow[d, "b"] \\
A \arrow[r, "f"'] & B
\end{tikzcd}
\end{document}

4.2 The Eilenberg-Moore Category

Definition (Eilenberg-Moore Category). The Eilenberg-Moore category \(\mathcal{C}^T\) has: - Objects: \(T\)-algebras \((A, a)\), - Morphisms: \(T\)-algebra morphisms, - Composition and identities inherited from \(\mathcal{C}\).

There is a canonical adjunction \(F^T \dashv U^T \colon \mathcal{C}^T \rightleftharpoons \mathcal{C}\): - The forgetful functor \(U^T \colon \mathcal{C}^T \to \mathcal{C}\) sends \((A, a) \mapsto A\). - The free \(T\)-algebra functor \(F^T \colon \mathcal{C} \to \mathcal{C}^T\) sends \(A \mapsto (TA, \mu_A)\).

Verification that \((TA, \mu_A)\) is a \(T\)-algebra: unit law \(\mu_A \circ \eta_{TA} = \mathrm{id}_{TA}\) is the monad unit law, and multiplication law \(\mu_A \circ \mu_{TA} = \mu_A \circ T\mu_A\) is monad associativity.

The monad induced by \(F^T \dashv U^T\) is \(U^T F^T = T\) with unit \(\eta\) and multiplication \(\mu\), recovering the original monad.

4.3 The Comparison Functor

For any adjunction \(F \dashv G \colon \mathcal{C} \rightleftharpoons \mathcal{D}\) with counit \(\varepsilon\) and induced monad \(T = GF\) on \(\mathcal{C}\), the comparison functor \(K \colon \mathcal{D} \to \mathcal{C}^T\) is defined by: \[K(d) = (G(d),\, G(\varepsilon_d)) \quad \text{and} \quad K(f) = G(f).\]

One checks that \((G(d), G(\varepsilon_d))\) is indeed a \(T\)-algebra: the unit law uses the triangle identity \(G\varepsilon \circ \eta G = \mathrm{id}_G\), and the multiplication law follows from naturality of \(\varepsilon\).

Definition (Monadic Functor). \(G \colon \mathcal{D} \to \mathcal{C}\) is monadic if it has a left adjoint and the comparison functor \(K \colon \mathcal{D} \to \mathcal{C}^T\) is an equivalence of categories.

5. Absolute Coequalizers and Split Pairs ⚙️

5.1 Reflexive Pairs

Definition (Reflexive Pair). A parallel pair \(f, g \colon A \rightrightarrows B\) is reflexive if there exists a common section \(s \colon B \to A\) with \(fs = \mathrm{id}_B = gs\).

5.2 Split Coequalizers

Definition (Split Coequalizer). A diagram \[A \overset{f}{\underset{g}{\rightrightarrows}} B \xrightarrow{q} C\] is a split coequalizer (or absolute coequalizer) if there exist morphisms \(s \colon B \to A\) and \(t \colon C \to B\) such that: 1. \(q \circ f = q \circ g\), 2. \(f \circ s = \mathrm{id}_B\), 3. \(q \circ t = \mathrm{id}_C\), 4. \(g \circ s = t \circ q\).

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A \arrow[r, "f", shift left=2] \arrow[r, "g"', shift right=2] & B \arrow[r, "q"] \arrow[l, "s", bend right=40] & C \arrow[l, "t", bend right=40]
\end{tikzcd}
\end{document}

Conditions 1–4 together imply \(q\) is a coequalizer of \(f\) and \(g\).

Proposition (Absoluteness). Every split coequalizer is preserved by every functor.

Proof. The conditions 1–4 are purely equational. Applying any functor \(H\) preserves all equations, so \(H(A) \rightrightarrows H(B) \to H(C)\) with \(H(s)\) and \(H(t)\) satisfies the same conditions, making \(H(q)\) a split coequalizer.

6. Set-Adjoint, Representable, and Limit-Preserving Functors 🧩

Three conditions on a functor \(U \colon \mathcal{C} \to \mathbf{Set}\) are naturally ordered by strength:

• (A) Set-adjoint: \(U\) has a left adjoint \(F \dashv U\) (equivalently, for every set \(S\) there is a “free object” \(F(S) \in \mathcal{C}\) with \(\mathcal{C}(F(S), C) \cong \mathbf{Set}(S, UC)\) naturally).
• (R) Representable: \(U \cong \mathcal{C}(A, -)\) for some \(A \in \mathcal{C}\) (naturally in the \(\mathcal{C}\) variable).
• (L) Limit-preserving: \(U\) preserves all small limits.

Proposition. \((A) \Rightarrow (R) \Rightarrow (L)\).

Proof. \((A) \Rightarrow (R)\): If \(F \dashv U\) then \(U(C) \cong \mathbf{Set}(1, UC) \cong \mathcal{C}(F(1), C)\) naturally, so \(U\) is represented by \(F(1)\).

\((R) \Rightarrow (L)\): Representable functors are right adjoints to \(\mathcal{C}(-,A) \colon \mathcal{C}^{op} \to \mathbf{Set}\)… more directly, \(\mathcal{C}(A,-)\) preserves limits since limits in \(\mathbf{Set}\) of hom-sets compute correctly (this is the standard fact that hom-functors preserve limits).

Proposition. If \(\mathcal{C}\) has all small coproducts, then \((R) \Rightarrow (A)\).

Proof sketch. Given \(U \cong \mathcal{C}(A,-)\), define \(F(S) = \coprod_{s \in S} A\) (coproduct of \(|S|\) copies of \(A\)). Then \(\mathcal{C}(F(S), C) \cong \prod_{s \in S} \mathcal{C}(A, C) \cong \mathbf{Set}(S, \mathcal{C}(A,C)) \cong \mathbf{Set}(S, UC)\).

7. The General Adjoint Functor Theorem 📐

7.1 The Solution Set Condition

Definition (Solution Set Condition). A functor \(G \colon \mathcal{D} \to \mathcal{C}\) satisfies the solution set condition if for every object \(A \in \mathcal{C}\) there exists a small set \(I\) and a family of morphisms \(\{f_i \colon A \to G(D_i)\}_{i \in I}\) such that every morphism \(h \colon A \to G(D)\) in \(\mathcal{C}\) factors as \[A \xrightarrow{f_i} G(D_i) \xrightarrow{G(k)} G(D)\] for some \(i \in I\) and some \(k \colon D_i \to D\) in \(\mathcal{D}\).

In categorical language: the full subcategory of \((A \downarrow G)\) on \(\{(D_i, f_i)\}_{i \in I}\) is a weakly initial set for the comma category.

7.2 Proof Sketch

Theorem (General Adjoint Functor Theorem, Freyd). Let \(G \colon \mathcal{D} \to \mathcal{C}\) where \(\mathcal{D}\) is locally small and complete. Then \(G\) has a left adjoint if and only if: 1. \(G\) preserves all small limits, and 2. \(G\) satisfies the solution set condition.

Proof sketch (sufficiency). Fix \(A \in \mathcal{C}\). We must construct an initial object of \((A \downarrow G)\).

Step 1: Weakly initial object. By the SSC, there is a small set \(\{(D_i, f_i \colon A \to GD_i)\}_{i \in I}\). Let \(W = \prod_{i \in I} D_i\) (existing by completeness). The projections \(W \to D_i\) give a morphism \(A \to GW\) (using limit-preservation of \(G\) and the \(f_i\)). This makes \((W, A \to GW)\) a weakly initial object in \((A \downarrow G)\): every \((D, h)\) is hit via the relevant \(D_i \to D\) composed with \(W \to D_i\).

Step 2: Reflexive equalizer to get an initial object. The comma category \((A \downarrow G)\) is locally small (since \(\mathcal{D}\) is locally small). Consider the set of all endomorphisms of \(W\) in \((A \downarrow G)\). Take the equalizer of all of them simultaneously (a small diagram since \(\mathcal{D}(W,W)\) is a set). The resulting object is an initial object of \((A \downarrow G)\), which is the required universal arrow \(A \to G(FA)\).

Necessity. Right adjoints preserve limits (File 3). For the SSC: take \(I = \{\ast\}\) and \(f_\ast = \eta_A \colon A \to GF(A)\); every \(h \colon A \to GD\) factors through \(\eta_A\) by universality.

Key conclusion: \(G\) has a left adjoint if and only if \(G\) preserves limits and satisfies the SSC. This is the most general sufficient condition for existence of a left adjoint.

8. Well-Powered Categories and the Subobject Functor 🔬

Definition (Subobject). A subobject of \(A \in \mathcal{C}\) is an isomorphism class of monomorphisms \(m \colon S \hookrightarrow A\). (Two monics \(m \colon S \to A\) and \(m' \colon S' \to A\) are isomorphic as subobjects if there is an isomorphism \(\phi \colon S \xrightarrow{\sim} S'\) with \(m' \phi = m\).)

Definition (Well-Powered). A category \(\mathcal{C}\) is well-powered if for every object \(A\), the collection \(\mathrm{Sub}(A)\) of subobjects of \(A\) is a set (not a proper class).

Proposition. In a well-powered category \(\mathcal{C}\) with pullbacks, there is a well-defined functor \[\mathrm{Sub} \colon \mathcal{C}^{op} \to \mathbf{Set}\] sending each object \(A\) to \(\mathrm{Sub}(A)\), and each morphism \(f \colon A \to B\) to the pullback-along-\(f\) map \(f^* \colon \mathrm{Sub}(B) \to \mathrm{Sub}(A)\).

Proof sketch. Given a monic \(m \colon S \hookrightarrow B\) representing a subobject of \(B\), form the pullback: \[\begin{array}{ccc} f^*S & \to & S \\ \downarrow & & \downarrow m \\ A & \xrightarrow{f} & B \end{array}\] The pullback of a monic is monic (a standard lemma), so \(f^*S \hookrightarrow A\) is a subobject of \(A\). Well-poweredness ensures \(f^*\) is a function between sets. Functoriality follows from the universal property of pullbacks.

9. The Special Adjoint Functor Theorem ✨

9.1 Coseparating Sets

Definition (Coseparating Set). A set \(\mathcal{S}\) of objects of \(\mathcal{C}\) is a coseparating set (or cogenerating set) if for every pair of distinct parallel morphisms \(f, g \colon A \rightrightarrows B\) in \(\mathcal{C}\) with \(f \neq g\), there exists \(S \in \mathcal{S}\) and a morphism \(h \colon B \to S\) with \(h \circ f \neq h \circ g\).

Equivalently: the family of functors \(\{\mathcal{C}(-, S)\}_{S \in \mathcal{S}}\) is jointly faithful.

9.2 Proof Sketch via GAFT

Theorem (Special Adjoint Functor Theorem, Freyd). Let \(\mathcal{C}\) be locally small, complete, well-powered, and with a small coseparating set \(\mathcal{S}\). Then every limit-preserving functor \(G \colon \mathcal{C} \to \mathcal{D}\) has a left adjoint.

Proof sketch. By the GAFT, it suffices to verify the SSC. Fix \(A \in \mathcal{C}\) and let \(f \colon A \to G(D)\) be any morphism.

Constructing the solution set. Consider the equalizer \(E_D = \mathrm{Eq}(G \circ \pi_S)_{S \in \mathcal{S}}\)… more concretely:

For each object \(D \in \mathcal{D}\), the morphism \(f \colon A \to G(D)\) factors through the intersection of all subobjects of \(G(D)\) through which \(f\) factors. Call this \(\mathrm{Im}_{\mathcal{S}}(f)\). Well-poweredness ensures \(\mathrm{Sub}(G(D))\) is a set, and the coseparating set ensures the image can be computed using \(\mathcal{S}\).

Precisely: define \(\mathcal{I}\) to be the set of all pairs $(D’, f’) $ where \(D'\) is a subobject of some \(\prod_{S \in \mathcal{S}} S^{n}\) for some finite \(n\) and \(f' \colon A \to G(D')\). Well-poweredness (applied to the product \(\prod_{S \in \mathcal{S}} S^n\), which exists by completeness) makes \(\mathcal{I}\) a set. The coseparating property ensures every \(f \colon A \to G(D)\) factors through some element of \(\mathcal{I}\). Hence \(\mathcal{I}\) is the required solution set.

This is the key: well-poweredness controls the size of \(\mathrm{Sub}\), and the coseparating set ensures every object can be separated into this controlled size. Together they imply SSC without needing to know the specific functor \(G\).

10. Applications of GAFT and SAFT 🌐

Free algebraic structures (GAFT). For any finitary algebraic theory \(\mathbb{T}\) (groups, rings, \(R\)-modules, lattices, etc.), the forgetful functor \(U \colon \mathbf{T}\text{-}\mathbf{Alg} \to \mathbf{Set}\) preserves limits and satisfies the SSC (the solution set for a set \(S\) is the single object \(F(S)\) = the \(\mathbb{T}\)-algebra presented by generators \(S\) and no relations). Hence \(U\) has a left adjoint, the free-algebra functor.

Stone-Čech compactification (SAFT). Let \(i \colon \mathbf{KHaus} \hookrightarrow \mathbf{Top}\) be the inclusion of compact Hausdorff spaces into all topological spaces. \(\mathbf{KHaus}\) is complete (products and equalizers of compact Hausdorff spaces are compact Hausdorff), well-powered (subspaces of compact Hausdorff spaces form a set), and has a small coseparating set (\(\{[0,1]\}\) works by Urysohn’s theorem). The inclusion \(i\) trivially preserves limits. Hence by the SAFT, \(i\) has a left adjoint — the Stone-Čech compactification \(\beta \colon \mathbf{Top} \to \mathbf{KHaus}\).

Geometric realization (SAFT). Covered in §16.

11. The Eilenberg-Moore Category: Creation Theorems 🏗️

11.1 Creation of Limits

Theorem. The forgetful functor \(U^T \colon \mathcal{C}^T \to \mathcal{C}\) creates all limits.

Proof sketch. Let \(D \colon \mathcal{J} \to \mathcal{C}^T\) be a diagram of \(T\)-algebras \(\{(A_j, a_j)\}\). Let \(A = \lim_j U^T(A_j) = \lim_j A_j\) in \(\mathcal{C}\), with projections \(\pi_j \colon A \to A_j\).

Since \(T\) is a functor, \(TA = T(\lim_j A_j)\) with projections \(T\pi_j \colon TA \to TA_j\). Define \(a \colon TA \to A\) by the universal property of \(A\) (as a limit): the maps \(a_j \circ T\pi_j \colon TA \to A_j\) are compatible (using the \(T\)-algebra morphisms in the diagram), so there is a unique \(a \colon TA \to A\) with \(\pi_j \circ a = a_j \circ T\pi_j\) for all \(j\).

Checking \((A, a)\) is a \(T\)-algebra: both the unit law and multiplication law hold componentwise (at each \(\pi_j\)) and are inherited from the \(A_j\) by uniqueness of the limit morphism.

Corollary. \(\mathcal{C}^T\) is complete whenever \(\mathcal{C}\) is complete.

11.2 Creation of Coequalizers for Absolute Pairs

Theorem. \(U^T\) creates coequalizers of \(U^T\)-absolute pairs.

Proof sketch. Let \(f, g \colon (A, a) \rightrightarrows (B, b)\) be a pair of \(T\)-algebra morphisms such that \(U^T f, U^T g \colon A \rightrightarrows B\) has a split coequalizer \(q \colon B \to C\) with splitting \(s \colon B \to A\), \(t \colon C \to B\) in \(\mathcal{C}\). We must produce a \(T\)-algebra structure on \(C\) making \(q\) a coequalizer in \(\mathcal{C}^T\).

Define \(c \colon TC \to C\) by: since \(q \colon B \to C\) has a section \(t\), we set \(c = q \circ b \circ Tt\). One verifies: - \(c\) is well-defined (independent of the choice of factorization, using that \(q\) coequalizes \(f\) and \(g\)). - The unit law for \(c\): \(c \circ \eta_C = q \circ b \circ Tt \circ \eta_C = q \circ b \circ \eta_B \circ t = q \circ \mathrm{id}_B \circ t = q \circ t = \mathrm{id}_C\). - The multiplication law: a similar computation using associativity of \(b\) and naturality.

Moreover, every \(T\)-algebra is a coequalizer of free algebras:

Proposition. For any \(T\)-algebra \((A, a)\), the diagram \[F^T(TA) \overset{F^T(a)}{\underset{\mu_A^{\sharp}}{\rightrightarrows}} F^T(A) \xrightarrow{a^{\sharp}} (A, a)\] is a coequalizer in \(\mathcal{C}^T\), where \(a^{\sharp}\) is the \(T\)-algebra map \((TA, \mu_A) \to (A, a)\) corresponding to \(a\) under the adjunction, and the parallel pair is identified by Exercise 4.

12. Idempotent Monads 🔄

Definition (Idempotent Monad). A monad \((T, \eta, \mu)\) on \(\mathcal{C}\) is idempotent if the multiplication \(\mu \colon T^2 \Rightarrow T\) is a natural isomorphism.

Proposition. For an idempotent monad \((T, \eta, \mu)\): 1. Every \(T\)-algebra \((A, a)\) is isomorphic to the free \(T\)-algebra \((TA, \mu_A)\). 2. The Eilenberg-Moore category \(\mathcal{C}^T\) is equivalent to the full subcategory of \(T\)-objects in \(\mathcal{C}\): objects \(A\) with \(\eta_A \colon A \to TA\) an isomorphism.

Proof. For (1): given \((A, a \colon TA \to A)\), the unit law gives \(a \circ \eta_A = \mathrm{id}_A\), so \(\eta_A\) is a split monomorphism. The multiplication law \(a \circ \mu_A = a \circ Ta\) with \(\mu_A\) an iso gives \(a \circ \mu_A \circ (\mu_A)^{-1} = a \circ Ta \circ (\mu_A)^{-1}\), which together with naturality shows \(\eta_A\) is also an epimorphism (using \(T\eta_A = \eta_{TA}\) and the unit law). Hence \(\eta_A \colon A \to TA\) is an isomorphism, and \((A, a) \cong (TA, \mu_A)\).

Idempotent monads and reflective subcategories. Idempotent monads on \(\mathcal{C}\) correspond bijectively to reflective subcategories of \(\mathcal{C}\): the Eilenberg-Moore category \(\mathcal{C}^T\) is the reflective subcategory, and the reflector is \(T\) itself.

13. Presheaves Are Monadic 📦

Let \(\mathcal{C}\) be a small category with object set \(\mathrm{ob}(\mathcal{C})\). Write \(\mathbf{Set}^{\mathrm{ob}(\mathcal{C})}\) for the product category \(\prod_{C \in \mathrm{ob}(\mathcal{C})} \mathbf{Set}\) (families of sets indexed by objects of \(\mathcal{C}\)). There is a forgetful functor \[V \colon [\mathcal{C}^{op}, \mathbf{Set}] \to \mathbf{Set}^{\mathrm{ob}(\mathcal{C})}\] sending a presheaf \(F\) to the family \((F(C))_{C \in \mathrm{ob}(\mathcal{C})}\) of its values (forgetting the restriction maps).

Theorem. \(V\) is monadic.

Proof sketch. The left adjoint \(L \colon \mathbf{Set}^{\mathrm{ob}(\mathcal{C})} \to [\mathcal{C}^{op}, \mathbf{Set}]\) sends a family \((S_C)_C\) to the presheaf \(L(S)(D) = \bigsqcup_{C \in \mathrm{ob}(\mathcal{C})} S_C \times \mathcal{C}(D, C)\) (the “free presheaf” on the family). The induced monad \(T = VL\) on \(\mathbf{Set}^{\mathrm{ob}(\mathcal{C})}\) sends \((S_C)_C\) to \(\left(\bigsqcup_{C'} S_{C'} \times \mathcal{C}(C, C')\right)_C\). A \(T\)-algebra structure precisely encodes the action of all hom-sets \(\mathcal{C}(C, C')\) on the family, which is exactly a presheaf on \(\mathcal{C}\). The comparison functor is an isomorphism (not just equivalence) of categories.

Corollaries: 1. \([\mathcal{C}^{op}, \mathbf{Set}]\) creates limits from \(\mathbf{Set}^{\mathrm{ob}(\mathcal{C})}\), hence is complete (limits computed objectwise). 2. \(V\) reflects isomorphisms: a natural transformation is an isomorphism iff all components are isomorphisms. 3. Every presheaf is a colimit of representable presheaves (recoverable as a corollary, since the free presheaf \(L(S)\) is a coproduct of representables \(\mathcal{C}(-,C)\), and every presheaf is a coequalizer of free presheaves by §11).

14. The Beck Monadicity Theorem 🏆

14.1 The Huge Monadicity Theorem

Definition (U-Split Pair / U-Absolute Pair). Let \(U \colon \mathcal{D} \to \mathcal{C}\). A parallel pair \(f, g \colon A \rightrightarrows B\) in \(\mathcal{D}\) is \(U\)-split (or \(U\)-absolute) if \(Uf, Ug \colon UA \rightrightarrows UB\) has a split coequalizer in \(\mathcal{C}\).

Theorem (Beck Monadicity / Huge Monadicity). Let \(U \colon \mathcal{D} \to \mathcal{C}\) have a left adjoint \(F \dashv U\) with induced monad \(T = UF\) on \(\mathcal{C}\). The following five conditions are equivalent:

Corollary (Crude Monadicity, Beck). \(U\) is monadic iff \(U\) has a left adjoint, \(U\) reflects isomorphisms, and \(\mathcal{D}\) has and \(U\) preserves coequalizers of \(U\)-split pairs.

14.2 Proof Strategy

The proof of the equivalence of (i)–(v) proceeds as follows.

\((v) \Rightarrow (i)\): If \(K\) is an equivalence, then \(\mathcal{D} \simeq \mathcal{C}^T\). By §11, \(U^T\) creates coequalizers of \(U^T\)-absolute pairs. Transporting along \(K\) gives the same for \(U\).

\((i) \Rightarrow (ii) \Rightarrow (iii)\): Creation implies preservation and reflection; preservation and reflection implies preservation. These are formal implications from the definitions.

\((iii) \Rightarrow (iv)\): Suppose \(U\) preserves coequalizers of \(U\)-absolute pairs. For any isomorphism-reflecting: if \(Uf\) is an isomorphism, then \(f, \mathrm{id}\) is a \(U\)-split pair (with \(U\mathrm{id}\) as splitting), so their coequalizer in \(\mathcal{D}\) exists and \(U\) preserves it — a short diagram chase shows \(f\) is an isomorphism.

\((iv) \Rightarrow (v)\) (key direction). This is the hardest step. We must show the comparison functor \(K \colon \mathcal{D} \to \mathcal{C}^T\) is an equivalence. For every \(T\)-algebra \((A, a) \in \mathcal{C}^T\), we need an object \(d \in \mathcal{D}\) with \(K(d) \cong (A, a)\).

By §11, \((A, a)\) is a coequalizer of free \(T\)-algebras. The corresponding diagram in \(\mathcal{D}\) (via \(F\)) is \(F(TA) \rightrightarrows F(A)\), which is a \(U\)-split pair (since applying \(U\) gives the split coequalizer of Exercise 4). By hypothesis (iv), \(\mathcal{D}\) has a coequalizer \(d = \mathrm{coeq}(F(TA) \rightrightarrows F(A))\), and \(U\) preserves it. One checks \(K(d) \cong (A, a)\) using that \(U\) reflects isomorphisms.

15. The Comparison Functor Revisited 🔭

Recall the comparison functor \(K \colon \mathcal{D} \to \mathcal{C}^T\) for an adjunction \(F \dashv U\) with monad \(T = UF\).

Proposition. \(K\) has a left adjoint if \(\mathcal{D}\) has coequalizers of \(U\)-absolute pairs and \(U\) preserves them.

Proof sketch. For each \(T\)-algebra \((A, a)\), we saw in §14 that \((A, a)\) is a coequalizer of free algebras \(K(FTA) \rightrightarrows K(FA)\) in \(\mathcal{C}^T\). The \(U\)-preservation hypothesis ensures \(\mathcal{D}\) has the corresponding coequalizer, giving a functor \(L \colon \mathcal{C}^T \to \mathcal{D}\) with \(L(A, a) = \mathrm{coeq}(FTA \rightrightarrows FA)\). One verifies \(L \dashv K\) by a universal property argument.

Proposition. The unit \(\eta^{LK} \colon \mathrm{id}_{\mathcal{D}} \Rightarrow KL\) of \(L \dashv K\) is a natural isomorphism iff \(U\) reflects isomorphisms.

Corollary. Both the unit and counit of \(L \dashv K\) are natural isomorphisms (i.e., \(K\) is an equivalence of categories) iff \(U\) is monadic.

16. Geometric Realization via SAFT 🌐

The simplex category \(\Delta\) has: - Objects: finite non-empty ordinals \([n] = \{0 < 1 < \cdots < n\}\) for \(n \geq 0\), - Morphisms: order-preserving (non-strictly) functions \([m] \to [n]\).

A simplicial set is a functor \(X \colon \Delta^{op} \to \mathbf{Set}\). The category of simplicial sets is \(\mathbf{sSet} = [\Delta^{op}, \mathbf{Set}]\).

The standard geometric simplices are \(|\Delta^n| = \{(t_0, \ldots, t_n) \in \mathbb{R}^{n+1}_{\geq 0} : \sum t_i = 1\}\). These assemble into a functor \(|\Delta^{\bullet}| \colon \Delta \to \mathbf{Top}\).

The singular complex functor \(\mathrm{Sing} \colon \mathbf{Top} \to \mathbf{sSet}\) is defined by: \[\mathrm{Sing}(X)_n = \mathbf{Top}(|\Delta^n|, X).\] Its restriction and degeneracy maps are precomposition with the face and degeneracy maps of \(\Delta\).

Theorem. \(\mathrm{Sing}\) has a left adjoint \(|-| \colon \mathbf{sSet} \to \mathbf{Top}\), called geometric realization.

Proof via SAFT. Apply the SAFT to \(G = \mathrm{Sing} \colon \mathbf{Top} \to \mathbf{sSet}\)… but wait: we want the left adjoint to \(\mathrm{Sing}\), so we apply SAFT to the opposite category situation, or more directly apply GAFT:

\(\mathrm{Sing}\) preserves limits (since \(\mathbf{Top}(|\Delta^n|, -)\) preserves limits for each \(n\), limits in \(\mathbf{sSet}\) are computed pointwise). For the SSC: for any simplicial set \(X\), the solution set consists of a single object — the geometric realization \(|X|\) if we already know it exists, or more carefully: the set of all quotients of coproducts of simplices \(\bigsqcup_\sigma |\Delta^{n_\sigma}|\) satisfying the \(n\)-skeleta conditions. Since \(X\) has only a set of simplices, this set is small.

Alternatively, one constructs \(|X|\) directly as the coend: \[|X| = \int^{[n] \in \Delta} X_n \cdot |\Delta^n| = \left(\bigsqcup_{n \geq 0} X_n \times |\Delta^n|\right) \big/ \sim\] where \((x, \phi^*(t)) \sim (\phi_*(x), t)\) for \(\phi \colon [m] \to [n]\) order-preserving. This formula shows \(|-|\) is well-defined and one verifies the adjunction bijection \(\mathbf{sSet}(X, \mathrm{Sing}(Y)) \cong \mathbf{Top}(|X|, Y)\) by the universal property of the coend.

17. Comonads from Adjunctions ⬇️

Definition (Comonad). A comonad on \(\mathcal{C}\) is a triple \((G, \varepsilon, \delta)\) where \(G \colon \mathcal{C} \to \mathcal{C}\) is an endofunctor, \(\varepsilon \colon G \Rightarrow \mathrm{id}_{\mathcal{C}}\) is the counit, and \(\delta \colon G \Rightarrow G^2\) is the comultiplication, satisfying coassociativity \(\delta G \circ \delta = G\delta \circ \delta\) and counit laws \(\varepsilon G \circ \delta = \mathrm{id}_G = G\varepsilon \circ \delta\).

Dually, a \(G\)-coalgebra is a pair \((A, \alpha \colon A \to GA)\) satisfying \(\varepsilon_A \circ \alpha = \mathrm{id}_A\) and \(\delta_A \circ \alpha = G\alpha \circ \alpha\).

Theorem. Let \((T, \eta, \mu)\) be a monad on \(\mathcal{C}\) and suppose \(T\) has a right adjoint \(G \colon \mathcal{C} \to \mathcal{C}\) (i.e., \(T \dashv G\)). Then: 1. \(G\) carries a canonical comonad structure \((G, \varepsilon^G, \delta^G)\). 2. The category of \(G\)-coalgebras is isomorphic to the category of \(T\)-algebras \(\mathcal{C}^T\).

Proof sketch. Let \(\phi_{A,B} \colon \mathcal{C}(TA, B) \xrightarrow{\sim} \mathcal{C}(A, GB)\) be the adjunction bijection for \(T \dashv G\).

Counit of \(G\): Define \(\varepsilon^G_A \colon GA \to A\) by \(\varepsilon^G_A = \phi^{-1}_{A,A}(\mathrm{id}_{GA})\)…

More precisely: The comonad structure on \(G\) is obtained by dualizing the monad structure on \(T\) via the adjunction. Specifically: - The counit \(\varepsilon^G \colon G \Rightarrow \mathrm{id}\) corresponds under \(\phi\) to the unit \(\eta \colon \mathrm{id} \Rightarrow T\) (via the bijection \(\mathcal{C}(T \cdot \mathrm{id}, \mathrm{id}) \cong \mathcal{C}(\mathrm{id}, G \cdot \mathrm{id})\)). - The comultiplication \(\delta^G \colon G \Rightarrow G^2\) corresponds to the multiplication \(\mu \colon T^2 \Rightarrow T\) (via \(\mathcal{C}(T^2, \mathrm{id}) \cong \mathcal{C}(\mathrm{id}, G^2)\)).

Isomorphism of algebra categories: Given a \(T\)-algebra \((A, a \colon TA \to A)\), define a \(G\)-coalgebra structure \(\bar{a} \colon A \to GA\) by \(\bar{a} = \phi_{A,A}(a)\). The unit law \(a \circ \eta_A = \mathrm{id}_A\) translates to the coalgebra counit law \(\varepsilon^G_A \circ \bar{a} = \mathrm{id}_A\), and the multiplication law translates to the comultiplication law. This correspondence is functorial and gives an isomorphism \(\mathcal{C}^T \cong \mathcal{C}_G\) (Eilenberg-Moore algebras for \(T\) vs. coalgebras for \(G\)).

18. Monadic Categories over Set and Top 📋

We apply Beck’s theorem to determine monadicity for various concrete categories.

Key justifications:

Fields are not monadic over \(\mathbf{Set}\): The forgetful functor \(U \colon \mathbf{Field} \to \mathbf{Set}\) has no left adjoint — there is no free field on a set (the “free field on \(\{x, y\}\)” would need to include \(x/y\) but \(y\) might be zero; this leads to contradiction). Formally: in any monad on \(\mathbf{Set}\), the algebra over the empty set is nonempty (it must contain the unit), but there is no field with 0 elements and the field with 1 element does not exist (fields require \(0 \neq 1\)). So the initial algebra, which would have to be the “field on 0 generators”, does not exist.

Totally ordered sets are not monadic over \(\mathbf{Set}\): The same failure of reflection of isomorphisms applies — a bijective order-preserving map between total orders need not be an isomorphism (think of the map \(\{0 < 1 < 2\} \to \{0 < 1 < 2\}\) given by \(0 \mapsto 1, 1 \mapsto 2, 2 \mapsto 0\); this is a bijection but not order-preserving in that direction… actually, let us note that any bijection that is order-preserving between total orders IS an isomorphism — a bijective order-preserving map between finite total orders is an isomorphism. For infinite total orders: \(\mathbb{Z} \to \mathbb{N}\) via a bijection can be order-preserving with non-order-preserving inverse. The key issue is that there is no free totally ordered set on more than 1 element, since all orderings on a finite set are a choice and none is canonical.)

Alternatively: \(\mathbf{TOrd}\) does not have coequalizers of all \(U\)-split pairs — certain coequalizer constructions are not totally ordered.

Topological groups are not monadic over \(\mathbf{Top}\): The forgetful functor \(U \colon \mathbf{TopGp} \to \mathbf{Top}\) does not have a left adjoint in general (there is no free topological group on a topological space with the expected universal property in the category of all topological groups, without further conditions on the space).

19. Canonical Presentations and the Bar Construction 🔩

Every T-algebra arises from a coequalizer of free T-algebras — this was established in the coequalizer characterization underlying Beck’s theorem (Exercise 10(iii)). But there is something more: there is a canonical and functorial way to present any algebra as such a coequalizer, built from the monad structure alone. This is the bar construction.

19.1 Motivation

Definition (Presentation of a T-algebra). A presentation of a T-algebra \((A, a)\) by free algebras is a coequalizer diagram in \(\mathcal{C}^T\): \[T^2 A \underset{d_1}{\overset{d_0}{\rightrightarrows}} TA \xrightarrow{q} A\] where \(d_0 = \mu_A \colon T^2 A \to TA\) (apply multiplication) and \(d_1 = Ta \colon T^2 A \to TA\) (apply \(T\) to the algebra map). Here \((TA, \mu_A)\) and \((T^2 A, \mu_{TA})\) are free T-algebras.

The coequalizer exists and equals \((A, a)\) because the pair \((d_0, d_1)\) is a reflexive split coequalizer in \(\mathcal{C}^T\): the section \(\eta_{TA} \colon TA \to T^2 A\) satisfies \(d_0 \circ \eta_{TA} = \mathrm{id}_{TA} = d_1 \circ \eta_{TA}\) (after unfolding), and the underlying pair in \(\mathcal{C}\) is split by the maps \(\eta_A \colon A \to TA\) and \(\eta_{TA}\).

Key question: Is this presentation functorial in \((A, a)\), and does it extend to an entire resolution rather than just a single coequalizer step? The bar construction answers both questions affirmatively.

19.2 The Bar Resolution

Definition (Bar Resolution). Let \((T, \eta, \mu)\) be a monad on \(\mathcal{C}\) and let \((A, a \colon TA \to A)\) be a T-algebra. The bar resolution of \((A, a)\) is the augmented simplicial object in \(\mathcal{C}\):

\[\cdots \underset{}{\overset{}{\underset{}{\overset{}{\rightrightarrows}}}} T^3 A \underset{}{\overset{}{\rightrightarrows}} T^2 A \underset{d_0}{\overset{d_1}{\rightrightarrows}} TA \xrightarrow{a} A\]

where at simplicial level \(n\) the object is \(T^{n+1} A\), the face maps are: \[d_i = T^i \mu_{T^{n-i-1} A} \colon T^{n+1} A \to T^n A, \quad 0 \leq i \leq n,\] and the degeneracy maps are: \[s_i = T^i \eta_{T^{n-i} A} \colon T^n A \to T^{n+1} A, \quad 0 \leq i \leq n-1.\] The augmentation map at level \(-1\) is \(a \colon TA \to A\).

📐 Concretely: At level \(n = 0\), the face maps are \(d_0 = \mu_A \colon T^2 A \to TA\) and \(d_1 = Ta \colon T^2 A \to TA\). At level \(n = 1\), the three face maps are \(d_0 = \mu_{TA}\), \(d_1 = T\mu_A\), and \(d_2 = T^2 a\).

[!EXAMPLE]- Bar resolution for groups Take \(T\) = the free group monad on Set. For a group \(G\) with underlying set \(|G|\), we have \(T^n|G|\) = the free group on the free group on \(\cdots\) on \(|G|\) (\(n\) times). The face map \(d_0 = \mu_{|G|}\) evaluates the outer free group using \(G\)’s multiplication, while \(d_1 = T(a) = T(\text{evaluation})\) evaluates one level inside. The degeneracy \(s_0 = \eta_{T|G|}\) inserts the unit into the free group.

Verification of simplicial identities. The face and degeneracy maps satisfy the standard simplicial identities (\(d_i d_j = d_{j-1} d_i\) for \(i < j\), etc.) by the monad axioms: associativity of \(\mu\) and the unit laws for \(\eta\).

19.3 The Simplicial Bar Construction in C^T

The bar resolution lives in the underlying category \(\mathcal{C}\), but it lifts to a simplicial object in the category of T-algebras \(\mathcal{C}^T\).

Definition (Simplicial Bar Construction). The simplicial bar construction \(B_\bullet(T, A)\) is the simplicial object in \(\mathcal{C}^T\) defined by: - Level \(n\): the free T-algebra \((T^{n+1} A,\, \mu_{T^n A})\) - Face maps: \(d_i = T^i(\mu_{T^{n-i-1} A}) \colon (T^{n+1} A, \mu_{T^n A}) \to (T^n A, \mu_{T^{n-1} A})\) (these are T-algebra maps because \(\mu\) is natural) - Degeneracy maps: \(s_i = T^i(\eta_{T^{n-i} A}) \colon (T^n A, \mu_{T^{n-1} A}) \to (T^{n+1} A, \mu_{T^n A})\) - Augmentation: the algebra map \(a \colon (TA, \mu_A) \to (A, a)\)

Each level \(B_n(T, A) = (T^{n+1} A, \mu_{T^n A})\) is the free T-algebra on \(T^n A\). The bar construction resolves every T-algebra by free algebras.

19.4 Geometric Realization and the Coequalizer

The geometric realization of the bar construction (i.e., the colimit of the simplicial diagram) recovers the original algebra \((A, a)\).

Theorem (Realization of B_bullet). The coequalizer of \(d_0, d_1 \colon T^2 A \rightrightarrows TA\) in \(\mathcal{C}^T\) is \((A, a)\). Moreover, if \(\mathcal{C}^T\) has all coequalizers of reflexive pairs (which it does when \(\mathcal{C}\) does, by the creation theorem from §11), then the geometric realization (homotopy colimit) of the entire simplicial object \(B_\bullet(T, A)\) is again \((A, a)\).

Proof sketch for the coequalizer step. We need: 1. \(a \circ d_0 = a \circ d_1\) (i.e., \(a \circ \mu_A = a \circ Ta\)): this is the T-algebra associativity law. 2. \((A, a)\) is the universal such coequalizer: given any T-algebra map \(h \colon (TA, \mu_A) \to (B, b)\) with \(h \circ d_0 = h \circ d_1\), there is a unique T-algebra map \(\bar{h} \colon (A, a) \to (B, b)\) with \(\bar{h} \circ a = h\). Setting \(\bar{h} = h \circ \eta_A\) works: one checks \(\bar{h}\) is a T-algebra map using the split structure.

The key observation is that the pair \((d_0, d_1)\) is split (not merely reflexive): the maps \(s_{-1} = \eta_A \colon A \to TA\) and \(s_0 = \eta_{TA} \colon TA \to T^2 A\) provide an explicit splitting. This makes the coequalizer absolute — preserved by every functor.

19.5 Contractibility and Splitting

🔑 The splitting maps are the heart of why the bar construction works. Define the extra degeneracy maps: \[s_{-1}^{(n)} \colon T^n A \to T^{n+1} A, \quad s_{-1}^{(n)} = \eta_{T^n A}.\] These are not natural transformations of the simplicial structure (they go in the “wrong” direction with respect to the augmentation), but they provide a contraction of the augmented simplicial object \(A \leftarrow TA \leftarrow T^2 A \leftarrow \cdots\) in \(\mathcal{C}\).

Definition (Contractible augmented simplicial object). An augmented simplicial object \(X_{-1} \leftarrow X_0 \leftarrow X_1 \leftarrow \cdots\) is contractible (or split) if there exist maps \(s_{-1}^{(n)} \colon X_n \to X_{n+1}\) (not required to be natural) such that: \[d_0 \circ s_{-1}^{(n)} = \mathrm{id}_{X_n}, \quad d_{i+1} \circ s_{-1}^{(n)} = s_{-1}^{(n-1)} \circ d_i \text{ for } i \geq 0.\]

Proposition. The augmented bar resolution \(A \leftarrow TA \leftarrow T^2 A \leftarrow \cdots\) (in \(\mathcal{C}\)) is contractible, with contraction given by \(s_{-1}^{(n)} = \eta_{T^n A}\).

Proof. Check: \(d_0 \circ s_{-1}^{(n)} = \mu_{T^{n-1} A} \circ \eta_{T^n A} = \mathrm{id}_{T^n A}\) by the right unit law for \(\mu\).

19.6 Universal Property: Canonical Presentation

Theorem (Canonical Presentation). For any T-algebra \((A, a)\), the diagram \[\bigl(T^2 A,\, \mu_{TA},\, d_0 = \mu_A,\, d_1 = Ta,\, s_0 = \eta_{TA}\bigr) \underset{d_1}{\overset{d_0}{\rightrightarrows}} \bigl(TA,\, \mu_A\bigr) \xrightarrow{a} (A, a)\] is a reflexive split coequalizer diagram in \(\mathcal{C}^T\), functorial in \((A, a)\).

Corollary. The functor \(F^T U^T \colon \mathcal{C}^T \to \mathcal{C}^T\) (where \(F^T\) is the free algebra functor and \(U^T\) the forgetful functor) comes with a canonical augmented simplicial resolution: for every \((A, a)\), the bar construction provides a functorial simplicial object \(B_\bullet(T, A)\) in \(\mathcal{C}^T\) whose geometric realization is \((A, a)\).

This is the sense in which every algebra is canonically presented by free algebras.

19.7 Applications

The bar construction appears throughout mathematics:

1. Homological algebra. 💡 For the monad \(T = A \otimes_k (-)\) on \(k\)-Mod (for a \(k\)-algebra \(A\)), the bar resolution of an \(A\)-module \(M\) is the bar resolution: \[\cdots \to A^{\otimes 3} \otimes M \to A^{\otimes 2} \otimes M \to A \otimes M \to M \to 0\] with face maps given by the multiplication in \(A\). This is precisely the standard resolution used to compute \(\mathrm{Tor}_*^A(-, M)\) and \(\mathrm{Ext}_A^*(-, M)\).

2. Algebraic topology. For the monad \(T = \Sigma^\infty \Omega^\infty\) on spectra, the bar construction produces the Adams resolution, used to compute stable homotopy groups via the Adams spectral sequence.

3. Classifying spaces. For a discrete group \(G\) viewed as a one-object groupoid, the bar construction on the monad associated to the free-forgetful adjunction \(\mathbf{Set} \rightleftharpoons \mathbf{Grp}\) produces \(B_n G = G^n\). The geometric realization \(|B_\bullet G|\) is the classifying space \(BG\) of \(G\).

4. \(\infty\)-category theory. The bar construction is a fundamental tool in \((\infty, 1)\)-category theory, where it provides the two-sided bar construction \(B(M, A, N)\) computing derived tensor products in the \(\infty\)-categorical sense.

19.8 The Cobar Construction

The cobar construction is the dual of the bar construction, defined for comonads rather than monads.

Definition (Cobar Construction). Let \((G, \varepsilon, \delta)\) be a comonad on \(\mathcal{C}\) and let \((A, \alpha \colon A \to GA)\) be a G-coalgebra. The cobar construction \(\Omega^\bullet(G, A)\) is the cosimplicial object in \(\mathcal{C}_G\) (the Eilenberg-Moore category of G-coalgebras) defined by: - Level \(n\): the cofree G-coalgebra \((G^{n+1} A,\, \delta_{G^n A})\) - Coface maps: \(\delta^i = G^i(\delta_{G^{n-i} A}) \colon G^n A \to G^{n+1} A\) for \(0 \leq i \leq n\) - Codegeneracy maps: \(\varepsilon^i = G^i(\varepsilon_{G^{n-i} A}) \colon G^{n+1} A \to G^n A\) for \(0 \leq i \leq n\) - Augmentation: the coalgebra map \(\alpha \colon A \to GA\)

Theorem (Dual Splitting). The augmented cosimplicial object \(A \to GA \to G^2 A \to \cdots\) is cosplit by the extra codegeneracy maps \(\varepsilon_{G^n A} \colon G^{n+1} A \to G^n A\). The equalizer of \(\varepsilon^0, \varepsilon^1 \colon GA \rightrightarrows A\) (in \(\mathcal{C}_G\)) is \((A, \alpha)\).

References