Todd Trimble
Notes on group objects

Let C\mathbf{C} be a complete cartesian closed category. A running example will be the category of cocommutative coalgebras over a field kk (which is cartesian closed and locally finitely presentable, hence complete; see the next section).

We will be studying group objects in such categories. For example, group objects in the category of cocommutative coalgebras over kk are precisely cocommutative Hopf algebras.

Notation: if f:XYf: X \to Y and g:XZg: X \to Z are morphisms in a category of products, then f,g\langle f, g \rangle denotes the unique map h:XY×Zh: X \to Y \times Z such that π 1h=f\pi_1 \circ h = f and π 2h=g\pi_2 \circ h = g; a similar notation extends to more general products (not just binary products).

Properties of the category of cocommutative coalgebras

Proposition

Let V\mathbf{V} be a symmetric monoidal category with tensor product \otimes and monoidal unit II. For any two cocommutative comonoids A,BA, B with counits ϵ A:AI\epsilon_A: A \to I, ϵ B:BI\epsilon_B: B \to I, the maps

AB1 Aϵ BAIA,ABϵ A1 BIBBA \otimes B \stackrel{1_A \otimes \epsilon_B}{\to} A \otimes I \cong A, \qquad A \otimes B \stackrel{\epsilon_A \otimes 1_B}{\to} I \otimes B \cong B

provide projection maps that exhibit ABA \otimes B as the cartesian product of AA and BB in the category CoCom(V)CoCom(\mathbf{V}) of cocommutative comonoids in V\mathbf{V}. Moreover, for any cartesian monoidal category M\mathbf{M} there is an equivalence between the category of symmetric monoidal functors MV\mathbf{M} \to \mathbf{V} and product-preserving functors MCoCom(V)\mathbf{M} \to CoCom(\mathbf{V}).

In the case of V=Vect k\mathbf{V} = Vect_k, cocommutative comonoids are the same as cocommutative kk-coalgebras. The forgetful functor CoCom(Vect k)Vect kCoCom(Vect_k) \to Vect_k creates colimits in Cocom(Vect k)Cocom(Vect_k). Since A:Vect kVect kA \otimes - : Vect_k \to Vect_k preserves colimits for any vector space AA, it follows that C:CoCom(Vect k)CoCom(Vect k)C \otimes - : CoCom(Vect_k) \to CoCom(Vect_k) preserves colimits, i.e., cartesian products in Cocom(Vect k)Cocom(Vect_k) distribute over colimits.

Theorem

The category of cocommutative kk-coalgebras is the IndInd-completion of the category of finite-dimensional cocommutative kk-coalgebras, and is cocomplete, hence locally finitely presentable. The finitely presentable objects are precisely the finite-dimensional objects.

According to Gabriel-Ulmer duality, this means there is an equivalence

CoCom(Vect k)Lex(CoCom fd op,Set)CoCom(Vect_k) \simeq Lex(CoCom_{fd}^{op}, Set)

and moreover, by taking linear duals, there is an equivalence

CoCom fd opCAlg fd.CoCom_{fd}^{op} \simeq CAlg_{fd}.

where CAlg fdCAlg_{fd} is the category of finite-dimensional commutative kk-algebras.

Corollary

The category CoCom(Vect k)CoCom(Vect_k) is complete, cocomplete, and cartesian closed.

Proof

Locally presentable categories C,D\mathbf{C}, \mathbf{D} are complete, and enjoy a strong form of an adjoint functor theorem, where a functor F:CDF: \mathbf{C} \to \mathbf{D} has a right adjoint iff it is cocontinuous. Since CoCom(Vect k)CoCom(Vect_k) is locally presentable, the product functor CC \otimes - has a right adjoint for any cocommutative coalgebra CC; therefore CoCom(Vect k)CoCom(Vect_k) is cartesian closed.

Exponentials C DC^D in Cocom(Vect k)Cocom(Vect_k) are often called measuring coalgebras.

Exponentials of group objects

Now let C\mathbf{C} be a complete cartesian closed category. If XX is an arbitrary object of C\mathbf{C}, the right adjoint () X(-)^X (with left adjoint ×X- \times X) preserves arbitrary limits and in particular finite products. This means that the canonical map

c Y 1,,Y n=π 1 X,,π n X:(Y 1××Y n) XY 1 X××Y n Xc_{Y_1, \ldots, Y_n} = \langle \pi_1^X, \ldots, \pi_n^X \rangle : (Y_1 \times \ldots \times Y_n)^X \to Y_1^X \times \ldots \times Y_n^X

is invertible.

It follows that if GG is a group object with multiplication m:G×GGm: G \times G \to G, identity e:1Ge: 1 \to G, and inverse i:GGi: G \to G, then G XG^X is a group object. The multiplication is defined to be the composite

G X×G X(c G,G) 1(G×G) XmG XG^X \times G^X \stackrel{(c_{G, G})^{-1}}{\to} (G \times G)^X \stackrel{m}{\to} G^X

and the identity and inverse are defined similarly. Indeed, for any model MM of a Lawvere theory T\mathbf{T} in C\mathbf{C}, the same principle shows that M XM^X carries a T\mathbf{T}-model structure canonically induced from the structure on MM. (Proof: a TT-model is given precisely by a product-preserving functor M:TCM: \mathbf{T} \to \mathbf{C}, and the composite

TMC() XC\mathbf{T} \stackrel{M}{\to} \mathbf{C} \stackrel{(-)^X}{\to} \mathbf{C}

is also product-preserving.)

Automorphism groups

If N,GN, G are ordinary groups, a homomorphism NGN \to G may be defined as a function ff that preserves multiplication (it may be shown that such functions also preserve the identity and inverse):

f(nn)=f(n)f(n)f(n \cdot n') = f(n) \cdot f(n')

for all n,nNn, n' \in N. The left side represents the function N×Nm NNfGN \times N \stackrel{m_N}{\to} N \stackrel{f}{\to} G, or the result of applying the map

G NG m NG N×NG^N \stackrel{G^{m_N}}{\to} G^{N \times N}

to ff. The right represents the function N×Nf×fG×Gm GGN \times N \stackrel{f \times f}{\to} G \times G \stackrel{m_G}{\to} G, or the result of applying the map

G Nsq(G×G) N×N(m G) N×NG N×NG^N \stackrel{sq}{\to} (G \times G)^{N \times N} \stackrel{(m_G)^{N \times N}}{\to} G^{N \times N}

to ff. Hence the set GrHom(N,G)GrHom(N, G) of homomorphisms NGN \to G may be constructed as the equalizer of the two legs of the following triangle

GrHom(N,G) i G N sq (G×G) N×N G m m N×N G N×N\array{ GrHom(N, G) & \stackrel{i}{\to} & G^N & \stackrel{sq}{\to} & (G \times G)^{N \times N} \\ & & & _{\mathllap{G^m}} \; \searrow & \downarrow \; _{\mathrlap{m^{N \times N}}} \\ & & & & G^{N \times N} }

The same construction applies more generally in a cartesian closed category. In particular, the “squaring map” sqsq is defined to be

G π 1,G π 2:G NG N×N×G N×N(G×G) N×N.\langle G^{\pi_1}, G^{\pi_2} \rangle : G^N \to G^{N \times N} \times G^{N \times N} \cong (G \times G)^{N \times N}.

Thus, in a finitely complete cartesian closed category, we may construct the object GrHom(N,G)GrHom(N, G) of group object homomorphisms as the equalizer displayed above. Taking G=NG = N, the exponential N NN^N naturally forms a monoid, and the subobject GrHom(N,N)GrHom(N, N) becomes a submonoid.

Similarly, one may internalize automorphism objects. An automorphism on XX can be construed as a pair of morphisms f,g:XXf, g: X \to X obeying the equations fg=1 X=gff \circ g = 1_X = g \circ f, and thus we may construct a group object Aut(X)Aut(X) as an equalizer of two legs of a triangle

Aut(X) X X×X X 1,σ (X X×X X)×(X X×X X) ! comp×comp 1 e,e X X×X X\array{ Aut(X) & \hookrightarrow & X^X \times X^X & \stackrel{\langle 1, \sigma\rangle}{\to} & (X^X \times X^X) \times (X^X \times X^X) \\ & & _{\mathllap{!}} \; \downarrow & & \downarrow \; _{\mathrlap{comp \times comp}} \\ & & 1 & \stackrel{\langle e, e\rangle}{\to} & X^X \times X^X }

where comp:X X×X XX Xcomp: X^X \times X^X \to X^X is internal composition and e:1X Xe: 1 \to X^X names the identity 1 X:XX1_X: X \to X.

Let j:Aut(X)X Xj: Aut(X) \to X^X be the composite

Aut(X)X X×X Xπ 1X X;Aut(X) \hookrightarrow X^X \times X^X \stackrel{\pi_1}{\to} X^X;

this map jj is a monomorphism and the subobject j:Aut(X)X Xj: Aut(X) \to X^X is closed under composition, i.e., the composite

Aut(X)×Aut(X)j×jX X×X XcompX XAut(X) \times Aut(X) \stackrel{j \times j}{\to} X^X \times X^X \stackrel{comp}{\to} X^X

factors through j:Aut(X)X Xj: Aut(X) \to X^X. Thus Aut(X)Aut(X) is a submonoid of X XX^X and in fact forms a group (object).

Further, for a group NN the intersection or pullback of subobjects forms a subgroup GrAut(N)GrAut(N) of Aut(N)Aut(N):

GrAut(N) GrHom(N,N) i Aut(N) j N N\array{ GrAut(N) & \to & GrHom(N, N) \\ \downarrow & & \downarrow \; _{\mathrlap{i}} \\ Aut(N) & \underset{j}{\to} & N^N }

Semidirect products

Suppose G,NG, N are groups in C\mathbf{C}, and ϕ:GGrAut(N)\phi: G \to GrAut(N) is a homomorphism, we can form the semidirect product N ϕGN \ltimes_\phi G as a purely categorical construction. The composite

GGrAut(N)N NG \to GrAut(N) \hookrightarrow N^N

corresponds (under the ×\times-hom\hom adjunction) to a map

α:G×NN\alpha: G \times N \to N

and we form a composite

N×G×N×G1 N×δ G×1 N×GN×G×G×N×G1 N×G×σ×1 GN×G×N×G×G1 N×α×1 G×GN×N×G×Gm N×m GN×GN \times G \times N \times G \stackrel{1_N \times \delta_G \times 1_{N \times G}}{\to} N \times G \times G \times N \times G \stackrel{1_{N \times G} \times \sigma \times 1_G}{\to} N \times G \times N \times G \times G \stackrel{1_N \times \alpha \times 1_{G \times G}}{\to} N \times N \times G \times G \stackrel{m_N \times m_G}{\to} N \times G

which gives the multiplication for the semidirect product.

Revised on March 12, 2014 at 05:54:27 by Todd Trimble