Lenses, Transducers, and Algebraic Effects#
Homomorphisms#
Homomorphisms are a structure preserving map between two algebraic structures of the same type (e.g. two groups, vector spaces, etc.) that preserves the operations of the structures
 suppose \(f: A \to B\) is a homomorphism between two sets \(A\), \(B\)
 suppose \(\cdot\) is an operation of the structure (e.g. a binary operation)
 \(f(x\cdot y)=f(x)\cdot f(y)\) for every pair \(x\), \(y\) of elements of \(A\)
Common examples
 linear maps: homomorphisms of vector spaces
 isomorphism: a subset of homomorphisms that also have an inverse
 homomorphic encryption: is a form of encryption that permits users to perform computations on its encrypted data without first decrypting it
Functors#

Functors are structure preserving maps between Categories

Functors are functions that lift functions to a context

Functors compose as functions
You are correct, functors are the structure preserving entities between categories. They could have been called homomorphisms.
I don't know much about the reason for the terminology used, but I think that Mac Lane and Eilenberg, when inventing category theory, borrowed terminology from philosophy where category and functor are known (at least to some)
Polymorphism is common in mathematics. For instance, the structure preserving entities between topological spaces are called continuous rather than homomorphisms.
And an invertible continuous function with continuous inverse is called a homeomorphism rather than an isomorphism. A structure preserving entity between vectors spaces is called a linear transformation.
A structure preserving mappings between metric spaces is called a short map. And an invertible one is called an isometry rather than an isomorphism. Clearly, historical reasons play a role.
 Connections to transducers
>On the fancy Haskell lens hierarchy (hackage.haskell.org/package/lens) we have the notion of a fold or a "getter which touches multiple items".
>The existence of a fold for a type likeFold s a
indicates that we can extract from the types
some number (0 to many)a
values in sequence. This is the idea of "Foldable" in Haskell.
>Given a foldable types
and a transducer we execute the transducer by passing the "build" reducer in and then "visiting" each valuea
inside ofs
with the reducer that
>the transducer returns (modulo the early stopping bit which is just sort of a Clojurespecific optimization).
>Essentially, the transducer is a notion of "visitation" which is invariant to how the final summary is constructed—essentially the same thing that's captured in the "getter which touches multiple items" of a Fold.
>So there really ought to be a way to treat any specter optic as a possibly very limited transducer. Essentially, the "read" component of a lens will correspond pretty directly.
>We can also see this by remembering that any pure transducer is semantically equivalent to a functiona > [b]
which you can read as a way of finding 0tomanyb
values "inside" ofa
.
Lenses#
 a composable getter: whole → part & setter: (whole, part) → whole
 "Lens" because you're "zooming" into a part
 Only looks complicated bc of var Laarhoven style lenses which define the getter & setter in one function and uses clever functor tricks to extract getter/setter
 Protocol/Key operations:
 get ( == "view")
 set ( == "set")
 update (== "over"): same as set except takes in a f: part → part that allows updating based on existing part value
 Why? The performant but less readable version of a lens relies on view & over, where set is defined using the const functor & over
 Why?
 Decouple state shape dependencies, especially in UI. Instead of littering your codebase with code that dives deep into the shape of a particular object, import a lens. If you later need to change the state shape, you can do so in the lens, and none of the code that depends on the lens will need to change.
 Immutable & composable niceness (getters are easy, but how do you do setters)?
Lenses allow you to abstract state shape behind getters and setters.
This follows the principle that a small change in requirements should require only a small change in the system.
 Sensible explanation
https://sinusoid.es/lager/lenses.html
https://medium.com/@reidev275/composableimmutablepropertyaccesswithlensesintypescript798da4ddc30e
https://medium.com/javascriptscene/lensesb85976cb0534
Code Explanation#
 suboptimal but readable version
export interface Lens<A, B> {
get(a: A): B;
set(b: B, a: A): A;
}
export const compose = <A, B, C>(x: Lens<A, B>, y: Lens<B, C>): Lens<A, C> => ({
get: (a: A) => y.get(x.get(a)),
set: (c: C, a: A) => {
const b: B = x.get(a);
const b2: B = y.set(c, b);
return x.set(b2, a);
}
});
 Usual Haskell implementation but expanded out
get(w): p =
return w.p
update :: (p > p) > w > w
update(P: p>p, w): w =
w` = w
w`.p = P(w.p)
return w
set :: Lens s a > a > s > s
set l a s = update l (const a) s
compose :: Lens l t > Lens w l > Lens w t
compose lt wl = Lens
{ get = (w,t) > t : lt.get( wl.get(w) )
, update = f(t>t), w: wl.update(f,w) . lt.update(f,t)
}
g.f =>
composeGet<glens,flens>(w) =
return flens.get(glens.get(w))
composeUpdate<glens,flens>(T: fp>fp, gw): gw =
auto fpupdateCurry = [T](gp): gp =
# fw == gp
return flens.update(T, gp)
glens.update(
fpupdateCurry,
gw)
Transducers#
A transducer is an object that incrementally transforms one (potentially infinite) sequence of elements into another sequence; effectively "state machines" involving (Reference)
 starting the transducer to get an initial state
 repeatedly updating that state by either consuming an element from the input sequence or by emitting an element to the output sequence
 when input is exhausted, the transducer enters a half closed state where it may emit more output elements but it will never consume more input elements
 when the transducer stops emitting elements, its finisher is called to clean up any resources held in the final transduction state.
 Optionally, a transducer may half close early, before the input sequence is fully consumed
Algebraic Effects#
ELI5: generalized catch/throw
with value semantics i.e. you can pass around a catch block
 WHAT: Delimited continuations for humans
 WHY: Effect systems decomplect where, what, and how
 HOW: By these semantics
 all semantics are strongly typed value semantics i.e. can be passed around as values
 delimited continuations as strongly typed interfaces (~= defining an "
Exception
" type)  semantics for scoped signaling (~= message passing) of an effect (~=
throw
)  semantics for specifying scoped implementations of effects (~= handlers)
 semantics for resuming from an implementation
Details#
 Types: what
func()
computes  Constrain world inside your function
 Handled by callee
 Effects: how
func()
computes  Constrain world outside
 Handled by caller
 motivation:
 gives you the power of virtualization at the expression level (e.g. can easily write a debugger that way)
 you get static strong typing for dynamic things like callbacks
 can actually be implemented with delimited continuations
 mostly aimed towards statically typed languages like Haskell as a replacement of monad transformers, which has both cognitive and performance benefits
 analogies/concept similarities
 syscalls
 C program puts stuff away in mem/registers, and tells kernel comeback and resume me when you're done
 entire program is a continuation
 the process is delimited continuation
 syscall are parameterized sideeffects
 the process is a parameters (e.g.
fork
the program continues twice)
 virtualization
 works by mocking syscall table
 means can isolate sideeffects and create a functional world for that process
 algebraic effects give you the power of virtualization at the expression level
 so you can virtualize any function like any process
 common lisp condition system: decouple the treatment of exceptional situations (or conditions) into three orthogonal roles:
 signaling the condition (akin to “throwing”),
 handling the condition (akin to “catching”),
 recovering from the condition (which has no resemblance in popular languages).
 The signaler, the handler, and the recoverer can be three disjoint bodies of code sitting in different parts of your call stack.
 interpreters
 Tagless final algebras are another much more popular alternative that has been proven very effective in practical software.
 In tagless final, one writes composable DSLs (which are just records of functions) with the nature and interpretation of effects left abstract.
 One then writes interpreters which interpret the DSL, giving meaning to the effects. This achieves the same fundamental goals as algebraic effects, but just using the ordinary language features of static FP languages.
Walkthrough#
type choice = effect
operation decide : unit > bool
end
let c = new choice
handle
let x = (if c#decide () then 10 else 20) in
let y = (if c#decide () then 0 else 5) in
x  y
with
 c#decide () k > k true
effect
definition "interface" defining supported operations
 dictates the "type"
effect
must be instanced to use needed because we may want to use same result of effect in multiple places
effect
used in a scoped evaluation context is a scoped "implementation" of
effect
interfaces  important!
c#effect()
invocation is through typed message passing, not function invocation
F#'s Effect handlers#
 TLDR: Allow user plug in a definition of
yield
and be able to callresume
withinyield
block to resume continuation  Are really just delimited continuations) but seem better/more ergonomic than transducers: http://mikeinnes.github.io/2020/06/12/transducers.html
 Very much similar to
async/await
Motivating Examples#
sum = 0
handle {
mapping(f, xs)
} with yield(x) {
sum += x
resume()
}
return sum
// Concat
for x in xs { yield(x) }
for y in ys { yield(y) }
// Interleave
for (x, y) in zip(xs, ys) {
yield(x)
yield(y)
}
// Cartesian Product
for x in xs {
for y in ys {
yield((x, y))
}
}
ys = handle {
mapping(f, xs)
nil
} with yield(x) {
cons(x, LazySeq(resume))
}
// Take while
for x in xs {
if f(x) {
yield(x)
} else {
break
}
}
// Dedupe
last = nil
for x in xs {
if x != last {
yield(x)
}
last = x
}
References#
 ELIU presentation: https://speakerdeck.com/paperswelove/brandonbloomonprogrammingwithalgebraiceffectsandhandlersbyandrejbauerandmatijapretnar
 https://github.com/brandonbloom/eclj
 More in style of CommonLisp's condition system: https://github.com/clojureman/special
 https://lilac.town/writing/effectsinclojure/
 From Scratch Tutorial in javascript:
 https://gist.github.com/yelouafi/57825fdd223e5337ba0cd2b6ed757f53
 https://gist.github.com/yelouafi/bbc559aef92f00d9682b8d0531a36503
 https://gist.github.com/yelouafi/7261da07c97c5e6322da3894f6ea60e2
 https://gist.github.com/yelouafi/5f8550b887ab7ffcf3284602330bd37d
Created: 20220126