# nLab measuring coalgebra

### Context

#### Algebra

higher algebra

universal algebra

# Measuring coalgebras

## Idea

Measuring coalgebras are an enrichment of the category of commutative rings (or commutative $ℤ$-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

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

where, given a coalgebra $C$ and an algebra $A$, $\left\{C,A\right\}$ 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

$\left\{C\otimes D,A\right\}\cong \left\{C,\left\{D,A\right\}\right\}$\{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 measuring coalgebra $\mu \left(A,B\right)$ is by definition the representing object of the functor

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

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

$\mathrm{Coalg}\left(C,\mu \left(A,B\right)\right)\cong \mathrm{Alg}\left(A,\left\{C,B\right\}\right)$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 $\mu \left(A,B\right)$ explicitly as the equalizer in $\mathrm{Coalg}$ of a pair of maps of the form

$\mathrm{Cof}\left({B}^{A}\right)\stackrel{\to }{\to }\mathrm{Cof}\left({B}^{A\otimes A}\right)×\mathrm{Cof}\left({B}^{k}\right)$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

$⟨\mathrm{Cof}\left({B}^{{m}_{A}}\right),\mathrm{Cof}\left({B}^{{u}_{A}}\right)⟩:\mathrm{Cof}\left({B}^{A}\right)\to \mathrm{Cof}\left({B}^{A\otimes A}\right)×\mathrm{Cof}\left({B}^{k}\right)$\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 $\Phi :\mathrm{Cof}\left({B}^{A}\right)\to \mathrm{Cof}\left({B}^{A\otimes A}\right)$ is the unique coalgebra map such that $U\Phi$ lifts the map

$U\mathrm{Cof}\left({B}^{A}\right)\stackrel{\delta }{\to }U\mathrm{Cof}\left({B}^{A}\right)\otimes U\mathrm{Cof}\left({B}^{A}\right)\stackrel{\pi \otimes \pi }{\to }{B}^{A}\otimes {B}^{A}\stackrel{{\otimes }_{1}}{\to }\left(B\otimes B{\right)}^{A\otimes A}\stackrel{{m}_{B}^{A\otimes A}}{\to }{B}^{A\otimes A}$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 $\pi :U\mathrm{Cof}\left({B}^{A\otimes A}\right)\to {B}^{A\otimes A}$. 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 $\Psi :\mathrm{Cof}\left({B}^{A}\right)\to \mathrm{Cof}\left({B}^{k}\right)$ is the unique coalgebra map such that $U\Psi$ lifts the map

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

through $\pi :U\mathrm{Cof}\left({B}^{A}\right)\to {B}^{A}$. 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 $\mu \left(A,B\right)$ indeed gives an enrichment

$\mu \left(-,-\right):{\mathrm{Alg}}^{\mathrm{op}}×\mathrm{Alg}\to \mathrm{Coalg}\phantom{\rule{thinmathspace}{0ex}}.$\mu(-, -): Alg^{op} \times Alg \to Coalg \,.

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

$\mu \left({A}_{0},{A}_{1}\right)×\mu \left({A}_{1},{A}_{2}\right)\to \mu \left({A}_{0},{A}_{2}\right)$\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:

$\begin{array}{cccc}\mathrm{Coalg}\left(C,\mu \left({A}_{0},{A}_{1}\right)×\mu \left({A}_{1},{A}_{2}\right)\right)& \cong & \mathrm{Coalg}\left(C,\mu \left({A}_{0},{A}_{1}\right)\right)×\mathrm{Coalg}\left(C,\mu \left({A}_{1},{A}_{2}\right)\right)& \left(\mathrm{Coalg}\left(C,-\right)\mathrm{preserves}\mathrm{products}\right)\\ & \cong & \mathrm{Alg}\left({A}_{0},\left\{C,{A}_{1}\right\}\right)×\mathrm{Alg}\left({A}_{1},\left\{C,{A}_{2}\right\}\right)& \left(\mathrm{definition}\mathrm{of}\mu \right)\\ & \to & \mathrm{Alg}\left({A}_{0},\left\{C,{A}_{1}\right\}\right)×\mathrm{Alg}\left(\left\{C,{A}_{1}\right\},\left\{C,\left\{C,{A}_{2}\right\}\right\}\right)& \left(\mathrm{functoriality}\mathrm{of}\left\{C,-\right\}\right)\\ & \to & \mathrm{Alg}\left({A}_{0},\left\{C,\left\{C,{A}_{2}\right\}\right\}\right)& \left(\mathrm{composition}\mathrm{law}\right)\\ & \cong & \mathrm{Alg}\left({A}_{0},\left\{C\otimes C,{A}_{2}\right\}\right)& \left(\mathrm{actegory}\mathrm{constraint}\right)\\ & \to & \mathrm{Alg}\left({A}_{0},\left\{C,{A}_{2}\right\}\right)& \left(\mathrm{using}d:C\to C\otimes C\right)\\ & \cong & \mathrm{Coalg}\left(C,\mu \left({A}_{0},{A}_{2}\right)\right)& \left(\mathrm{definition}\mathrm{of}\mu \right)\end{array}$\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) }

Revised on March 5, 2012 03:30:42 by Todd Trimble (67.80.8.47)