# 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 $\mathbb{Z}$-algebras) in the cartesian closed category $k$ Cocomm Coalg of cocommutative coalgebras (which we will write simply as $Coalg$), given a ground field $k$.

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

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

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

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

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

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

of algebras; here $Alg$ is sometimes described as an actegory over $Coalg$.

## Definition

###### Definition

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

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

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

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

Assume the existence of equalizers in $Coalg$, and of a right adjoint

$Cof: Vect \to Coalg$

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

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

denote the counit of the adjunction $U \dashv Cof$.

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

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

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

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

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

$\langle \Phi, \Psi \rangle$

which we now describe separately.

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

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

through $\pi: U Cof(B^{A \otimes A}) \to B^{A \otimes A}$. Here $\delta$ denotes the comultiplication (same as the diagonal map as seen in $Coalg$), and $\otimes_1$ indicates the structure of enriched functoriality for $\otimes$.

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

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

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

## Enrichment of algebras in coalgebras

###### Proposition

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

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

Here the composition law in $Coalg$

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

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

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

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