Todd Trimble Stuff on coalgebras

Local presentability and cartesian closure

We establish the following result:

Theorem

If $V$ is locally presentable and is given a symmetric monoidal closed structure, then the category of cocommutative comonoids in $V$ is cartesian closed and locally presentable.

Let $Cocom(V)$ be the category of commutative comonoids in $V$ , and $U: Cocom(V) \to V$ the forgetful functor. There are three main tasks:

Show that $Cocom(V)$ is accessible. Then, since cocomplete accessible categories are locally presentable, they are also complete (Adamek and Rosicky, …). In particular, $Cocom(V)$ has equalizers. (Finite products are given by tensor products in $V$ .)
Show that $U$ has a right adjoint $Cof: V \to Cocom(V)$ .
Given the fact that $Cocom(V)$ has equalizers and $U \dashv Cof$ , the category $Cocom(V)$ is cartesian closed. Products in $Cocom(V)$ are given by tensor products in $V$ .

I am roughly following Porst here; see particularly the proposition on page 8 and remarks on page 9. If the first task is completed, then the second is easy: using the fact that $Cocom(V)$ is locally presentable and the fact that $U: Cocom(V) \to V$ preserves colimits, it has a right adjoint by the special adjoint functor theorem. In fact, $U$ is comonadic. The third task is by a direct construction of the exponential in $Cocom$ , using the symmetric monoidal closed structure of $V$ , the right adjoint $Cof$ , and equalizers in $Cocom(V)$ .

Accessibility of $Cocom(V)$

The first task is divided into two subtasks: first is to show that the category of coalgebras for the finitary endofunctor

F(v) = (v \otimes v) \times I

is accessible. Porst says (fact 5, page 6):

If $F$ preserves directed colimits (and it does), then $Coalg(F)$ is accessible.

A detailed proof is given by Adamek and Porst (see Theorem IV.2, page 16). The relevant and more general results are given in the following lemma and theorem.

Lemma

Let $k$ be a regular infinite cardinal. If $F: \mathbf{C} \to \mathbf{C}$ is $k$ -accessible and $(C, \gamma: C \to F C)$ is a coalgebra whose underlying object is $k$ -presentable in $C$ (i.e., $\hom(C, -): C \to Set$ preserves $k$ -filtered colimits which exist in $C$ ), then $(C, \gamma)$ is $k$ -presentable in $Coalg(F)$ . Conversely, the underlying object of a $k$ -presentable $F$ -coalgebra is $k$ -presentable in $V$ .

Proof

Suppose given a $k$ -filtered diagram $J \to Coalg(F)$ with colimit $(B, \beta)$ ; let $(B_j, \beta_j: B_j \to F(B_j))$ denote nodes of the diagram. We must show that $\hom(C, -): Coalg(F) \to Set$ preserves this colimit. That is, any coalgebra map $C \to B$ must be shown to factor through one of the $B_j \to B$ in $Coalg(F)$ , and one should check that any two factorings resolve at a later stage.

The underlying functor $Coalg(F) \to \mathbf{C}$ preserves and reflects colimits, so $B$ is the colimit of the $B_j$ in $\mathbf{C}$ . Since $C$ is $k$ -presentable in $\mathbf{C}$ , any map $f: C \to B$ factors through one of the cone components $g_i: B_i \to B$ in $\mathbf{C}$ , say $f = g_i h$ . Suppose now $f: C \to B$ is a coalgebra map. Then

F(g_i) \beta_i h = \beta g_i h = \beta f = F(f)\gamma = F(g_i) F(h)\gamma

using the fact that $g_i$ and $f$ are coalgebra maps. Since $F$ and $\hom(C, -)$ preserve $k$ -filtered colimits, the comparison map

colim_j \hom(C, F B_j) \stackrel{(\hom(C, F g_j))}{\to} \hom(C, F B)

is a bijection; therefore the equality of elements $F(g_i)\beta_i h = F(g_i)F(h)\gamma$ in $\hom(C, F B)$ implies that $\beta_i h$ and $F(h)\gamma$ belong to the same equivalence class in $\colim_j \hom(C, F B_j)$ . In other words, there exists an arrow $u: B_i \to B_j$ in the diagram (a coalgebra map) such that

F(u)\beta_i h = F(u)F(h)\gamma

whence $\beta u h = F(u h)\gamma$ , making $u h: C \to B_j$ a coalgebra map. We moreover have

f = g_i h = g_j u h

so $f$ factors in $Coalg(F)$ through one of the nodes $B_j$ . The fact that any two factorings resolve at a later stage is clear.

Example: Let $k$ be a field. Then $Vect_k$ is locally finitely presentable, since every vector space is a filtered colimit of its finite-dimensional subspaces. The functor
$Vect_k \to Vect_k: v \mapsto \otimes^n v$

is finitary; therefore so is $F(v) =$

To be continued…

Theorem

If $\mathbf{C}$ is locally finitely presentable and $F: \mathbf{C} \to \mathbf{C}$ is finitary, then $Coalg(F)$ is locally $\omega_1$ -presentable (and often locally finitely presentable). If $k$ is an uncountable regular cardinal, $C$ is locally $k$ -presentable, and $F$ is $k$ -accessible, then $Coalg(F)$ is locally $k$ -presentable.

Next, Porst quotes a result from Adamek and Rosicky (Locally Presentable and Accessible Categories, 2.76) which figures in the theorem that the 2-category $Acc$ of accessible categories and accessible functors is closed under lax limits in $Cat$ . If $F, G: \mathbf{C} \to \mathbf{D}$ are 1-cells of $Cat$ and $\eta, \theta: F \to G$ are 2-cells, the equifier $Eq(\eta, \theta)$ is the full subcategory of $\mathbf{C}$ whose objects $c$ satisfy $\eta c = \theta c$ .

Lemma (Adamek and Rosicky)

If $\mathbf{C}$ and $\mathbf{D}$ are locally presentable, $F$ and $G$ are $k$ -accessible functors $\mathbf{C} \to \mathbf{D}$ , and $\eta, \theta: F \to G$ are transformations, then $Eq(\eta, \theta)$ are locally presentable.

Interlude: monos and equalizers in $Cocom(Vect_k)$

Suppose $U: \mathbf{C} \to Set$ is faithful.

Proposition

If $U(f)$ is monic, then so is $f$ .

The proof is trivial.

Now suppose $f: C \to D$ is monic in $Cocom(Vect_k)$ . We know subcoalgebra inclusions $D' \hookrightarrow D$ are monic by Proposition 1. Hence the pullback $f^{-1}(D') \to C$ is monic as well, as is of course the pullback $f^{-1}(D') \to D'$ .

Lemma

A morphism $f: C \to D$ in the category of finite-dimensional cocommutative $k$ -coalgebras is monic iff $U(f)$ is monic.

Proof

By taking linear duals, this is equivalent to the statement that epimorphisms $f: A \to B$ in the category of finite-dimensional commutative $k$ -algebras are surjective. Suppose otherwise, so we have an exact sequence of modules

A \stackrel{f}{\to} B \to B/A \to 0.

with $B/A$ nonzero. Apply the functor $- \otimes_A B$ to arrive at an exact sequence

A \otimes_A B \to B \otimes_A B \to B/A \otimes_A B \to 0.

That the first arrow is an iso is just the characterization of epis in terms of pushouts. Hence $B/A \otimes_A B = 0$ , and therefore $B/A \otimes_A B/A = 0$ . On the other hand, considering $B$ to be finitely generated as an $A$ -module, with a minimal set of generators $b_1, \ldots, b_n$ , then we have surjections

B/A \to B/A\langle b_1, \ldots, b_{n-1}\rangle \leftarrow A

where the second arrow takes $1 \in A$ to $b_n$ . By minimality, the second arrow is nonzero, with kernel an ideal $I$ of $A$ , so we have a surjection

B/A \to A/I

and hence a surjection $0 = B/A \otimes_A B/A \to A/I \otimes_A A/I$ . Hence $A/I \otimes_A A/I = 0$ . But because $A \to A/I$ is an epi in the category of commutative rings, the pushout projection $A/I \otimes_A A \to A/I \otimes_A A/I = 0$ is an iso, and this contradicts properness of $I$ .

Now consider the directed system of inclusions of finite-dimensional subcoalgebra inclusions $D' \hookrightarrow D$ (whose colimit is $D$ ). The pulled-back system $f^{-1}(D') \hookrightarrow C$

Revised on August 28, 2015 at 20:01:01 by Todd Trimble

Todd Trimble Stuff on coalgebras

Local presentability and cartesian closure

Theorem

Accessibility of Cocom(V)Cocom(V)

Lemma

Proof

Theorem

Lemma (Adamek and Rosicky)

Interlude: monos and equalizers in Cocom(Vect k)Cocom(Vect_k)

Proposition

Lemma

Proof

Accessibility of $Cocom(V)$

Interlude: monos and equalizers in $Cocom(Vect_k)$