Public:Known Game Comonads

\( \newcommand\Ck{\mathbb C_k} \newcommand\Lk{\mathcal L_k} \newcommand\Hom{\mathrm{Hom}} \newcommand\qtq[1]{\quad\text{#1}\quad} \)

In this page we gather the list all comonads we've explored so far.

Convention:

• $$A \leftrightarrows_k B$$ iff there exist homomorphisms $$\Ck A \to B$$ and $$\Ck B \to A$$
• $$A \leftrightarrow_k B$$ iff there exist non-empty sets $$F\subseteq \Hom(\Ck A, B)$$ and $$G\subseteq \Hom(\Ck B, A)$$ which are locally invertible
• $$A \cong_{\mathrm{Kl}(\Ck)} B$$ iff $$A$$ and $$B$$ are isomorphic in the coKleisly category for $$\Ck$$, i.e. there exist homomorphisms $$f\colon\Ck A \to B$$ and $$g\colon\Ck B \to A$$ such that both $$f^* g^*$$ and $$g^* f^*$$ are identities

Typically, each of those equivalences corresponds to a certain logical equivalence. That is, we have that

1. $$A \leftrightarrows_k B \qtq{iff} A \equiv^{\exists \Lk} B$$
2. $$A \leftrightarrow_k B \qtq{iff} A \equiv^{\Lk} B$$
3. $$A \cong_{\mathrm{Kl}(\Ck)} B \qtq{iff} A \equiv^{\Lk^\#} B$$

where $$\exists \Lk$$ is the existential-positive fragment of a certain logic $$\Lk$$ and $$\Lk^\#$$ is the extension of $$\Lk$$ by adding counting quantifiers $$\exists_{{\geq} n}$$ and $$\exists_{{\leq} n}$$.

Comonad	$$\Ck$$	Logic	$$\leftrightarrows_k$$	$$\leftrightarrow_k$$	$$\cong_{\mathrm{Kl}(\Ck)}$$	Coalgebras
Pebbling	$$\mathbb P_k$$	$$k$$-variable fragment of FO	✓	✓ with $$I$$-morphisms	✓ with $$I$$-morphisms	tree width $${\leq} k$$
Ehrenfeucht-Fraissé	$$\mathbb E_k$$	quantifier rank $${\leq} k$$ fragment of FO	✓	✓ with $$I$$-morphisms	✓ with $$I$$-morphisms	tree depth $${\leq} k$$
Modal (*)	$$\mathbb M_k$$	modal depth $${\leq} k$$ fragment of ML	✓	✓	✓	synchronization tree depth $${\leq} k$$
Pebble Relation	$$\mathbb {PR}_k$$		✓ wrt a fragment $$M^k$$ of $$\exists^+ \mathcal L^k$$			path width $${\leq} k$$
"Hella's game"	$$\mathbb {PR}_{n,k}$$ [1]	extension of $$\mathcal L^k$$ with generalised quantifiers	for $$\exists^+ \mathcal L^k(\mathbf Q_n^H)$$	for $$\mathcal L^k(\mathbf Q_n^H)$$	for $$\mathcal L^k_{\infty,\omega}(\mathbf Q_n^H)$$	$$n$$-ary generalised tree width $${\leq} k$$
Guarded Fragment Modal (*)	GF [2]	modal depth $${\leq} k$$ fragment of ML

The comonads marked with ``(*)`` denote the comonads over the category of pointed labeled graphs. The $$\leftrightarrows_k$$, $$\leftrightarrow_k$$, and $$\cong_{\mathrm{Kl}(\Ck)}$$ columns mark whether the equivalence in (1), (2), resp. (3) above holds.