nLab measuring coalgebra

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:

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

Literature

• M. E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, 1969
• Mathieu Anel, André Joyal, Sweedler Theory for (co)algebras and the bar-cobar constructions, arXiv:1309.6952

A dual notion to measuring coalgebra is Tambara's universal coacting bialgebra from

• Daisuke Tambara, The coendomorphism bialgebra of an algebra, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 37, 425-456, 1990 pdf
• Martin Hyland, Ignacio López Franco?, Christina Vasilakopoulou, Hopf measuring comonoids and enrichment, Proc. London Math. Soc. 115:5 (2017) 1118-1148, doi
• Christina Vasilakopoulou, Enrichment of categories of algebras and modules, arXiv:1205.6450
• Marjorie Batchelor, Measuring coalgebras, quantum group-like objects, and non-commutative geometry, In: Bartocci, C., Bruzzo, U., Cianci, R. (eds) Differential Geometric Methods in Theoretical Physics. Lecture Notes in Physics 375 doi; Measuring comodules - their applications, J. Geom. Physics 36:3-4 (2009) 251-269 (doi); Difference operators, measuring coalgebras and quantum group like objects, Adv. Math. 105, 190-218 (1994) pdf
• Maximilien Péroux, The coalgebraic enrichment of algebras in higher categories, J. Pure Appl. Alg. 226:3 (2022) 106849 doi
• Paige Randall North, Maximilien Péroux, Coinductive control of inductive data types, arXiv:2303.16793