Notes on group objects

Let $\mathbf{C}$ be a complete cartesian closed category. A running example will be the category of cocommutative coalgebras over a field $k$ (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 $k$ are precisely cocommutative Hopf algebras.

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

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

$A \otimes B \stackrel{1_A \otimes \epsilon_B}{\to} A \otimes I \cong A, \qquad A \otimes B \stackrel{\epsilon_A \otimes 1_B}{\to} I \otimes B \cong B$

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

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

The category of cocommutative $k$-coalgebras is the $Ind$-completion of the category of finite-dimensional cocommutative $k$-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

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

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

$CoCom_{fd}^{op} \simeq CAlg_{fd}.$

where $CAlg_{fd}$ is the category of finite-dimensional commutative $k$-algebras.

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

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

Exponentials $C^D$ in $Cocom(Vect_k)$ are often called *measuring coalgebras*.

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

$c_{Y_1, \ldots, Y_n} = \langle \pi_1^X, \ldots, \pi_n^X \rangle : (Y_1 \times \ldots \times Y_n)^X \to Y_1^X \times \ldots \times Y_n^X$

is invertible.

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

$G^X \times G^X \stackrel{(c_{G, G})^{-1}}{\to} (G \times G)^X \stackrel{m}{\to} G^X$

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

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

is also product-preserving.)

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

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

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

$G^N \stackrel{G^{m_N}}{\to} G^{N \times N}$

to $f$. The right represents the function $N \times N \stackrel{f \times f}{\to} G \times G \stackrel{m_G}{\to} G$, or the result of applying the map

$G^N \stackrel{sq}{\to} (G \times G)^{N \times N} \stackrel{(m_G)^{N \times N}}{\to} G^{N \times N}$

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

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

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

$\langle G^{\pi_1}, G^{\pi_2} \rangle : G^N \to G^{N \times N} \times G^{N \times N} \cong (G \times G)^{N \times N}.$

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

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

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

where $comp: X^X \times X^X \to X^X$ is internal composition and $e: 1 \to X^X$ names the identity $1_X: X \to X$.

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

$Aut(X) \hookrightarrow X^X \times X^X \stackrel{\pi_1}{\to} X^X;$

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

$Aut(X) \times Aut(X) \stackrel{j \times j}{\to} X^X \times X^X \stackrel{comp}{\to} X^X$

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

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

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

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

$G \to GrAut(N) \hookrightarrow N^N$

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

$\alpha: G \times N \to N$

and we form a composite

$N \times G \times N \times G \stackrel{1_N \times \delta_G \times 1_{N \times G}}{\to} N \times G \times G \times N \times G \stackrel{1_{N \times G} \times \sigma \times 1_G}{\to} N \times G \times N \times G \times G \stackrel{1_N \times \alpha \times 1_{G \times G}}{\to} N \times N \times G \times G \stackrel{m_N \times m_G}{\to} N \times G$

which gives the multiplication for the semidirect product.

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