nLab
measure coalgebra

Idea

Measure coalgebras (or measuring coalgebras) are an enrichment of the category of commutative rings (or commutative $\mathbb{Z}$-algebras) in the cartesian closed category $k$ Cocomm Coalg of cocommutative coalgebras (which we will write simply as $\mathrm{Coalg}$), given a ground field $k$.

The starting point is the observation that the category $\mathrm{Coalg}$ acts on the category Alg of commutative algebras: there is a functor

$$\{-,-\}:{\mathrm{Coalg}}^{\mathrm{op}}\times \mathrm{Alg}\to \mathrm{Alg}$$\{-, -\}: Coalg^{op} \times Alg \to Alg

where, given a coalgebra $C$ and an algebra $A$, $\{C,A\}$ is the abelian-group hom of additive homomorphisms $f:C\to A$, made into an algebra whose multiplication $f\cdot g$ is given by

$$C\stackrel{d}{\to }C\otimes C\stackrel{f\otimes g}{\to }A\otimes A\stackrel{m}{\to }A$$C \overset{d}{\to} C \otimes C \overset{f \otimes g}{\to} A \otimes A \overset{m}{\to} A

where $d$ is the coalgebra comultiplication and $m$ is the algebra multiplication. That this is an “action” here means that there is a natural isomorphism

$$\{C\otimes D,A\}\cong \{C,\{D,A\}\}$$\{C \otimes D, A\} \cong \{C, \{D, A\}\}

of algebras; here $\mathrm{Alg}$ is sometimes described as an actegory over $\mathrm{Coalg}$.

Definition

Definition

Given two algebras $A,B$, the measure coalgebra $\mu (A,B)$ is by definition the representing object of the functor

$$\mathrm{Alg}(A,\{-,B\}):{\mathrm{Coalg}}^{\mathrm{op}}\to \mathrm{Set}$$Alg(A, \{-, B\}): Coalg^{op} \to Set

so that there is an isomorphism, natural for coalgebras $C$, of the form

$$\mathrm{Coalg}(C,\mu (A,B))\cong \mathrm{Alg}(A,\{C,B\})$$Coalg(C, \mu(A, B)) \cong Alg(A, \{C, B\})

Assume the existence of equalizers in $\mathrm{Coalg}$, and of a right adjoint

$$\mathrm{Cof}:\mathrm{Vect}\to \mathrm{Coalg}$$Cof: Vect \to Coalg

to the forgetful functor $U:\mathrm{Coalg}\to \mathrm{Vect}$ (the cofree cocommutative coalgebra construction). We let

$$\pi :U\circ \mathrm{Cof}\to 1_{\mathrm{Vect}}$$\pi: U \circ Cof \to 1_{Vect}

denote the counit of the adjunction $U⊣\mathrm{Cof}$.

We construct $\mu (A,B)$ explicitly as the equalizer in $\mathrm{Coalg}$ of a pair of maps of the form

$$\mathrm{Cof}(B^A)\overrightarrow{\to }\mathrm{Cof}(B^{A\otimes A})\times \mathrm{Cof}(B^k)$$Cof(B^A) \overset{\to}{\to} Cof(B^{A \otimes A}) \times Cof(B^k)

where we denote the internal hom in Vect by exponentiation (and we recall here that the cartesian product in $\mathrm{Coalg}$ is given by tensor product at the level of $\mathrm{Vect}$). The first of these maps is

$$⟨\mathrm{Cof}(B^{m_A}),\mathrm{Cof}(B^{u_A})⟩:\mathrm{Cof}(B^A)\to \mathrm{Cof}(B^{A\otimes A})\times \mathrm{Cof}(B^k)$$\langle Cof(B^{m_A}), Cof(B^{u_A}) \rangle: Cof(B^A) \to Cof(B^{A \otimes A}) \times Cof(B^k)

where $m_A:A\otimes A\to A$ is the multiplication on $A$ and $u_A:k\to A$ is the unit. The second is given by a pair of maps

$$⟨\Phi ,\Psi ⟩$$\langle \Phi, \Psi \rangle

which we now describe separately.

The map $\Phi :\mathrm{Cof}(B^A)\to \mathrm{Cof}(B^{A\otimes A})$ is the unique coalgebra map such that $U\Phi$ lifts the map

$$U\mathrm{Cof}(B^A)\stackrel{\delta }{\to }U\mathrm{Cof}(B^A)\otimes U\mathrm{Cof}(B^A)\stackrel{\pi \otimes \pi }{\to }{B}^A\otimes B^A\stackrel{{\otimes }_1}{\to }(B\otimes B{)}^{A\otimes A}\stackrel{m_B^{A\otimes A}}{\to }{B}^{A\otimes A}$$U Cof(B^A) \overset{\delta}{\to} U Cof(B^A) \otimes U Cof(B^A) \overset{\pi \otimes \pi}{\to} B^A \otimes B^A \overset{\otimes_1}{\to} (B \otimes B)^{A \otimes A} \overset{m_{B}^{A \otimes A}}{\to} B^{A \otimes A}

through $\pi :U\mathrm{Cof}(B^{A\otimes A})\to B^{A\otimes A}$. Here $\delta$ denotes the comultiplication (same as the diagonal map as seen in $\mathrm{Coalg}$), and ${\otimes }_1$ indicates the structure of enriched functoriality for $\otimes$.

The map $\Psi :\mathrm{Cof}(B^A)\to \mathrm{Cof}(B^k)$ is the unique coalgebra map such that $U\Psi$ lifts the map

$$U\mathrm{Cof}(B^A)\stackrel{\epsilon }{\to }k\stackrel{u_B}{\to }B\cong B^k$$U Cof(B^A) \overset{\varepsilon}{\to} k \overset{u_B}{\to} B \cong B^k

through $\pi :U\mathrm{Cof}(B^A)\to B^A$. Here $\epsilon$ denotes the counit (same as the unique map to the terminal object as seen in $\mathrm{Coalg}$).

Enrichment of algebras in coalgebras

Proposition

The measure coalgebra $\mu (A,B)$ indeed gives an enrichment

$$\mu (-,-):{\mathrm{Alg}}^{\mathrm{op}}\times \mathrm{Alg}\to \mathrm{Coalg}.$$\mu(-, -): Alg^{op} \times Alg \to Coalg \,.

Here the composition law in $\mathrm{Coalg}$

$$\mu (A_0,A_1)\times \mu (A_1,A_2)\to \mu (A_0,A_2)$$\mu(A_0, A_1) \times \mu(A_1, A_2) \to \mu(A_0, A_2)

(recalling that the product in $\mathrm{Coalg}$ is the tensor product of the underlying additive groups) is derived by universality from a composition of maps:

$$\begin{array}{cccc}\mathrm{Coalg}(C,\mu (A_0,A_1)\times \mu (A_1,A_2))& \cong & \mathrm{Coalg}(C,\mu (A_0,A_1))\times \mathrm{Coalg}(C,\mu (A_1,A_2))& (\mathrm{Coalg}(C,-)\mathrm{preserves}\mathrm{products})\\ & \cong & \mathrm{Alg}(A_0,\{C,A_1\})\times \mathrm{Alg}(A_1,\{C,A_2\})& (\mathrm{definition}\mathrm{of}\mu )\\ & \to & \mathrm{Alg}(A_0,\{C,A_1\})\times \mathrm{Alg}(\{C,A_1\},\{C,\{C,A_2\}\})& (\mathrm{functoriality}\mathrm{of}\{C,-\})\\ & \to & \mathrm{Alg}(A_0,\{C,\{C,A_2\}\})& (\mathrm{composition}\mathrm{law})\\ & \cong & \mathrm{Alg}(A_0,\{C\otimes C,A_2\})& (\mathrm{actegory}\mathrm{constraint})\\ & \to & \mathrm{Alg}(A_0,\{C,A_2\})& (\mathrm{using}d:C\to C\otimes C)\\ & \cong & \mathrm{Coalg}(C,\mu (A_0,A_2))& (\mathrm{definition}\mathrm{of}\mu )\end{array}$$\array{ Coalg(C, \mu(A_0, A_1) \times \mu(A_1, A_2)) & \cong & Coalg(C, \mu(A_0, A_1)) \times Coalg(C, \mu(A_1, A_2)) & (Coalg(C, -) preserves products)\\ & \cong & Alg(A_0, \{C, A_1\}) \times Alg(A_1, \{C, A_2\}) & (definition of \mu) \\ & \to & Alg(A_0, \{C, A_1\}) \times Alg(\{C, A_1\}, \{C, \{C, A_2\}\}) & (functoriality of \{C, -\})\\ & \to & Alg(A_0, \{C, \{C, A_2\}\}) & (composition law)\\ & \cong & Alg(A_0, \{C \otimes C, A_2\}) & (actegory constraint)\\ & \to & Alg(A_0, \{C, A_2\}) & (using d: C \to C \otimes C)\\ & \cong & Coalg(C, \mu(A_0, A_2)) & (definition of \mu) }