symmetric monoidal (∞,1)-category of spectra
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
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
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
of algebras; here $Alg$ is sometimes described as an actegory over $Coalg$.
Given two algebras $A, B$, the measuring coalgebra $\mu(A, B)$ is by definition the representing object of the functor
so that there is an isomorphism, natural for coalgebras $C$, of the form
Assume the existence of equalizers in $Coalg$, and of a right adjoint
to the forgetful functor $U: Coalg \to Vect$ (the cofree cocommutative coalgebra construction). We let
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
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
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
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
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
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 $Coalg$).
The measure coalgebra $\mu(A, B)$ indeed gives an enrichment
Here the composition law in $Coalg$
(recalling that the product in $Coalg$ is the tensor product of the underlying additive groups) is derived by universality from a composition of maps: