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measuring coalgebra

Measuring coalgebras

Idea

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

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

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

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

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

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

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

of algebras; here AlgAlg is sometimes described as an actegory over CoalgCoalg.

Definition

Definition

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

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

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

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

Assume the existence of equalizers in CoalgCoalg, and of a right adjoint

Cof:VectCoalgCof: Vect \to Coalg

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

π:UCof1 Vect\pi: U \circ Cof \to 1_{Vect}

denote the counit of the adjunction UCofU \dashv Cof.

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

Cof(B A)Cof(B AA)×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 CoalgCoalg is given by tensor product at the level of VectVect). The first of these maps is

Cof(B m A),Cof(B u A):Cof(B A)Cof(B AA)×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:AAAm_A: A \otimes A \to A is the multiplication on AA and u A:kAu_A: k \to A is the unit. The second is given by a pair of maps

Φ,Ψ\langle \Phi, \Psi \rangle

which we now describe separately.

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

UCof(B A)δUCof(B A)UCof(B A)ππB AB A 1(BB) AAm B AAB AAU 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 π:UCof(B AA)B AA\pi: U Cof(B^{A \otimes A}) \to B^{A \otimes A}. Here δ\delta denotes the comultiplication (same as the diagonal map as seen in CoalgCoalg), and 1\otimes_1 indicates the structure of enriched functoriality for \otimes.

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

UCof(B A)εku BBB kU Cof(B^A) \overset{\varepsilon}{\to} k \overset{u_B}{\to} B \cong B^k

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

Enrichment of algebras in coalgebras

Proposition

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

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

Here the composition law in CoalgCoalg

μ(A 0,A 1)×μ(A 1,A 2)μ(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 CoalgCoalg is the tensor product of the underlying additive groups) is derived by universality from a composition of maps:

Coalg(C,μ(A 0,A 1)×μ(A 1,A 2)) Coalg(C,μ(A 0,A 1))×Coalg(C,μ(A 1,A 2)) (Coalg(C,)preservesproducts) Alg(A 0,{C,A 1})×Alg(A 1,{C,A 2}) (definitionofμ) Alg(A 0,{C,A 1})×Alg({C,A 1},{C,{C,A 2}}) (functorialityof{C,}) Alg(A 0,{C,{C,A 2}}) (compositionlaw) Alg(A 0,{CC,A 2}) (actegoryconstraint) Alg(A 0,{C,A 2}) (usingd:CCC) Coalg(C,μ(A 0,A 2)) (definitionofμ)\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)