The costructure–cosemantics adjunction for comodels for computational effects

Abstract It is well established that equational algebraic theories and the monads they generate can be used to encode computational effects. An important insight of Power Shkaravska comodels an theory $\mathbb{T}$ – i.e., models in opposite category $\mathcal{S}\mathrm{et}^{\mathrm{op}}$ provide a suitable environment for evaluating effects encoded by . As already noted Shkaravska, taking yields functor from accessible comonads on $\mathcal{S}\mathrm{et}$ In this paper, we show part adjunction “costructure–cosemantics adjunction” title undertake thorough investigation its properties. We that, one hand, cosemantics takes image what term presheaf induced small categories; other, costructure categories. particular, comonad monad will explicitly-described called behaviour encodes static dynamic properties comodels. Similarly, encoding coalgebras. tie these results together showing costructure–cosemantics idempotent , with fixpoints either side given precisely comonads. Along way, illustrate value our numerous examples drawn computation mathematics.