… that every comonad in the 2-category of doctrines determines an … comonads and adjunctions
in a general 2-category. In Section 4 we define the 2-categories of doctrines and doctrines …

… modalities in the framework of primary linear doctrines to … variant of elementary doctrines
that we call Lipschitz doctrines. We … doctrines arise as coalgebras for a 2comonad on graded …

… for a 2-comonad on the 2-category of R-graded doctrines. This … modalities in the framework
of primary linear doctrines to … of elementary doctrines that we call Lipschitz doctrines. We …

… In general, given a category with a linear exponential comonad, we … exponential comonad
and any desired limits and colimits. … modalities F,U, are those that preserve the comonad up to …

… to provide a minimal categorical doctrine for differential linear … coalgebra modalities and
monoidal coalgebra modalities (we … Proof By the naturality of e and the comonad identities, we …

… comonads are an extension of linear exponential comonads wih grading, and provide a
categorical semantics of resource-sensitive exponential modality in … exponential comonads as a …

… comonads [16]), while we also hope that this work leads to a better understanding of modalities
… The corresponding axiomatics in terms of comonads is more difficult, as the definition of …

… comonadic exponential modality from Girard’s linear logic. For dependency analysis, type
systems employ the monadic modality … the graded modality they employ is comonadic and not …

… Comonad for exchange. We conclude this section by showing that the comonad induced by
some LAM is monoidal and extends L with exchange. The former is proved in [12]. The latter …

… was introduced as a minimal categorical doctrine in which differentiability could be studied.
… maps is an exact modality. In the next section, as we study this comonad in more detail, the …