4

Adjuncts are morphisms related by the natural isomorphism: $$Hom(FA, B) \cong Hom(A, GB)$$ defining the adjunction. The adjuncts of (co)units are identities: $$ (ε_A : FGA → A) \longleftrightarrow (1_{GA} : GA → GA) \\ (η_A : A → GFA) \longleftrightarrow (1_{FA} : FA → FA) $$ This correspondence is basically what connects the $Hom$ definition of ...


3

Your question is not specific to the evaluation map. Consider any morphism $f : A \otimes B \to C$ in a in a symmetric monoidal (not "monodical") category. There are two adjuncts because $A \otimes B \cong B \otimes A$: We can take $A$ to the other side of the arrow to get $f_1 : B \to [A, C]$. We can take $B$ to the other side of the arrow to get $f_2 : A ...


3

A data structure Queue is very close to the concept of a List, however a Queue is a First-In-First-Out (FIFO) data structure. The available operations to interact with the List structure then, are limited to enqueue() and dequeue() (it is not a case in fact that a queue can be implemented using a List structure). In the context of Category Theory you might ...


3

I guess no one else is going to answer this, so I'll take a crack. First, there's nothing inherently 'functional' about categories, necessarily. For instance, you can make any monoid into a category, where you have one object, and then the arrows from that one object to itself are the elements of the monoid. The identity element is the identity arrow, and ...


3

$C$-monoids and their variants is what you are looking for. You can find further references and an account of what is what in Martin Hyland's Towards a Notion of Lambda Monoid.


2

Here is a new paper that covers a similar topic. The idea is that by doing algebra in enriched categories (2-categories are like categories enriched in categories), you can talk about more fine grained semantic structure on the algebra. (I haven't read through the whole paper myself, but I know enough to see some of the ideas behind it.) The way it relates ...


1

Consider (.) :: (a -> b) -> (d -> a) -> d -> b is the fmap for functor $F_d\colon \mathfrak{C} \rightarrow d/\mathfrak{C}$ from category of types $\mathfrak{C}$ to its coslice category $d/\mathfrak{C}$ on object $d$. Similarly, (.) :: ((d -> a) -> d -> b) -> (c -> d -> a) -> c -> d -> b is considered the fmap ...


Only top voted, non community-wiki answers of a minimum length are eligible