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 ...
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