In this paper, we look at two categorical accounts of computational effects (strong monad as a model of the monadic metalanguage, adjunction as a model of call-bypush-value with stacks), and we adapt them to incorporate global exceptions. In each case, we extend the calculus with a construct, due to Benton and Kennedy, that fuses exception handling with sequencing. This immediately gives us an ...

Given an adjunction F ⊣ G connecting reasonable categories with weak equivalences, we define a new derived bar and cobar construction associated to the adjunction. This yields homotopical models of the completion and cocompletion associated to the monad and comonad of the adjunction. We discuss applications of these resolutions to spectral sequences for derived completions and Goodwillie calcul...

We show that an adjoint functor between quasi-categories may be extended to a simplicially enriched functor whose domain is an explicitly presented “homotopy coherent adjunction”. This enriched functor encapsulates both the coherent monad and the coherent comonad generated by the adjunction. Furthermore, because its domain is cofibrant, this data can be used to construct explicit quasi-categori...

In this paper, let $L$ be a completeresiduated lattice, and let {bf Set} denote the category of setsand mappings, $LF$-{bf Pos} denote the category of $LF$-posets and$LF$-monotone mappings, and $LF$-{bf CSLat}$(sqcup)$, $LF$-{bfCSLat}$(sqcap)$ denote the category of $LF$-completelattices and $LF$-join-preserving mappings and the category of$LF$-complete lattices and $LF$-meet-preserving mapping...

In this paper we show duality results between categories of equations and categories of coequations. These dualities are obtained as restrictions of dualities between categories of algebras and coalgebras, which arise by lifting contravariant adjunctions on the base categories. By extending this approach to (co)algebras for (co)monads, we retrieve the duality between equations and coequations f...

We pursue the definition of a KZ-doctrine in terms of a fully faithful adjoint string Dd ⊣ m ⊣ dD. We give the definition in any Gray-category. The concept of algebra is given as an adjunction with invertible counit. We show that these doctrines are instances of more general pseudomonads. The algebras for a pseudomonad are defined in more familiar terms and shown to be the same as the ones defi...

We define a coherent adjunction in strict $3$-category and we use string diagrams to show that any can be extended an essentially unique way.

1. Categories 2 1.1. First Definition and Examples 2 1.2. An Alternative Definition: The Arrows-Only Perspective 7 1.3. Some Constructions 8 1.4. The Category of Relations 9 1.5. Special Objects and Arrows 10 1.6. Exercises 14 2. Functors and Natural Transformations 16 2.1. Functors 16 2.2. Full and Faithful Functors 20 2.3. Contravariant Functors 21 2.4. Products of Categories 23 3. Natural Tr...