Todd Trimble
Stuff on coalgebras

Local presentability and cartesian closure

We establish the following result:

Theorem

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

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

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)Cocom(V) is locally presentable and the fact that U:Cocom(V)VU: Cocom(V) \to V preserves colimits, it has a right adjoint by the special adjoint functor theorem. In fact, UU is comonadic. The third task is by a direct construction of the exponential in CocomCocom, using the symmetric monoidal closed structure of VV, the right adjoint CofCof, and equalizers in Cocom(V)Cocom(V).

Accessibility of Cocom(V)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)=(vv)×IF(v) = (v \otimes v) \times I

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

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 kk be a regular infinite cardinal. If F:CCF: \mathbf{C} \to \mathbf{C} is kk-accessible and (C,γ:CFC)(C, \gamma: C \to F C) is a coalgebra whose underlying object is kk-presentable in CC (i.e., hom(C,):CSet\hom(C, -): C \to Set preserves kk-filtered colimits which exist in CC), then (C,γ)(C, \gamma) is kk-presentable in Coalg(F)Coalg(F). Conversely, the underlying object of a kk-presentable FF-coalgebra is kk-presentable in VV.

Proof

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

The underlying functor Coalg(F)CCoalg(F) \to \mathbf{C} preserves and reflects colimits, so BB is the colimit of the B jB_j in C\mathbf{C}. Since CC is kk-presentable in C\mathbf{C}, any map f:CBf: C \to B factors through one of the cone components g i:B iBg_i: B_i \to B in C\mathbf{C}, say f=g ihf = g_i h. Suppose now f:CBf: C \to B is a coalgebra map. Then

F(g i)β ih=βg ih=βf=F(f)γ=F(g i)F(h)γF(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 ig_i and ff are coalgebra maps. Since FF and hom(C,)\hom(C, -) preserve kk-filtered colimits, the comparison map

colim jhom(C,FB j)(hom(C,Fg j))hom(C,FB)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)β ih=F(g i)F(h)γF(g_i)\beta_i h = F(g_i)F(h)\gamma in hom(C,FB)\hom(C, F B) implies that β ih\beta_i h and F(h)γF(h)\gamma belong to the same equivalence class in colim jhom(C,FB j)\colim_j \hom(C, F B_j). In other words, there exists an arrow u:B iB ju: B_i \to B_j in the diagram (a coalgebra map) such that

F(u)β ih=F(u)F(h)γF(u)\beta_i h = F(u)F(h)\gamma

whence βuh=F(uh)γ\beta u h = F(u h)\gamma, making uh:CB ju h: C \to B_j a coalgebra map. We moreover have

f=g ih=g juhf = g_i h = g_j u h

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

To be continued…

Theorem

If C\mathbf{C} is locally finitely presentable and F:CCF: \mathbf{C} \to \mathbf{C} is finitary, then Coalg(F)Coalg(F) is locally ω 1\omega_1-presentable (and often locally finitely presentable). If kk is an uncountable regular cardinal, CC is locally kk-presentable, and FF is kk-accessible, then Coalg(F)Coalg(F) is locally kk-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 AccAcc of accessible categories and accessible functors is closed under lax limits in CatCat. If F,G:CDF, G: \mathbf{C} \to \mathbf{D} are 1-cells of CatCat and η,θ:FG\eta, \theta: F \to G are 2-cells, the equifier Eq(η,θ)Eq(\eta, \theta) is the full subcategory of C\mathbf{C} whose objects cc satisfy ηc=θc\eta c = \theta c.

Lemma (Adamek and Rosicky)

If C\mathbf{C} and D\mathbf{D} are locally presentable, FF and GG are kk-accessible functors CD\mathbf{C} \to \mathbf{D}, and η,θ:FG\eta, \theta: F \to G are transformations, then Eq(η,θ)Eq(\eta, \theta) are locally presentable.

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

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

Proposition

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

The proof is trivial.

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

Lemma

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

Proof

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

AfBB/A0.A \stackrel{f}{\to} B \to B/A \to 0.

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

A ABB ABB/A AB0.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 AB=0B/A \otimes_A B = 0, and therefore B/A AB/A=0B/A \otimes_A B/A = 0. On the other hand, considering BB to be finitely generated as an AA-module, with a minimal set of generators b 1,,b nb_1, \ldots, b_n, then we have surjections

B/AB/Ab 1,,b n1AB/A \to B/A\langle b_1, \ldots, b_{n-1}\rangle \leftarrow A

where the second arrow takes 1A1 \in A to b nb_n. By minimality, the second arrow is nonzero, with kernel an ideal II of AA, so we have a surjection

B/AA/IB/A \to A/I

and hence a surjection 0=B/A AB/AA/I AA/I0 = B/A \otimes_A B/A \to A/I \otimes_A A/I. Hence A/I AA/I=0A/I \otimes_A A/I = 0. But because AA/IA \to A/I is an epi in the category of commutative rings, the pushout projection A/I AAA/I AA/I=0A/I \otimes_A A \to A/I \otimes_A A/I = 0 is an iso, and this contradicts properness of II.

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

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