Srihari Thyagarajan commited on
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0aa6802
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unverified ·
2 Parent(s): 6befcfa bfe46e3

Merge pull request #60 from metaboulie/fp/functors

Browse files
functional_programming/05_functors.py ADDED
@@ -0,0 +1,1293 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ # /// script
2
+ # requires-python = ">=3.9"
3
+ # dependencies = [
4
+ # "marimo",
5
+ # ]
6
+ # ///
7
+
8
+ import marimo
9
+
10
+ __generated_with = "0.11.17"
11
+ app = marimo.App(app_title="Category Theory and Functors", css_file="")
12
+
13
+
14
+ @app.cell(hide_code=True)
15
+ def _(mo):
16
+ mo.md(
17
+ """
18
+ # Category Theory and Functors
19
+
20
+ In this notebook, you will learn:
21
+
22
+ * Why `length` is a *functor* from the category of `list concatenation` to the category of `integer addition`
23
+ * How to *lift* an ordinary function into a specific *computational context*
24
+ * How to write an *adapter* between two categories
25
+
26
+ In short, a mathematical functor is a **mapping** between two categories in category theory. In practice, a functor represents a type that can be mapped over.
27
+
28
+ /// admonition | Intuitions
29
+
30
+ - A simple intuition is that a `Functor` represents a **container** of values, along with the ability to apply a function uniformly to every element in the container.
31
+ - Another intuition is that a `Functor` represents some sort of **computational context**.
32
+ - Mathematically, `Functors` generalize the idea of a container or a computational context.
33
+ ///
34
+
35
+ We will start with intuition, introduce the basics of category theory, and then examine functors from a categorical perspective.
36
+
37
+ /// details | Notebook metadata
38
+ type: info
39
+
40
+ version: 0.1.0 | last modified: 2025-03-13 | author: [métaboulie](https://github.com/metaboulie)<br/>
41
+ reviewer: [Haleshot](https://github.com/Haleshot)
42
+
43
+ ///
44
+ """
45
+ )
46
+ return
47
+
48
+
49
+ @app.cell(hide_code=True)
50
+ def _(mo):
51
+ mo.md(
52
+ """
53
+ # Functor as a Computational Context
54
+
55
+ A [**Functor**](https://wiki.haskell.org/Functor) is an abstraction that represents a computational context with the ability to apply a function to every value inside it without altering the structure of the context itself. This enables transformations while preserving the shape of the data.
56
+
57
+ To understand this, let's look at a simple example.
58
+
59
+ ## [The One-Way Wrapper Design Pattern](http://blog.sigfpe.com/2007/04/trivial-monad.html)
60
+
61
+ Often, we need to wrap data in some kind of context. However, when performing operations on wrapped data, we typically have to:
62
+
63
+ 1. Unwrap the data.
64
+ 2. Modify the unwrapped data.
65
+ 3. Rewrap the modified data.
66
+
67
+ This process is tedious and inefficient. Instead, we want to wrap data **once** and apply functions directly to the wrapped data without unwrapping it.
68
+
69
+ /// admonition | Rules for a One-Way Wrapper
70
+
71
+ 1. We can wrap values, but we cannot unwrap them.
72
+ 2. We should still be able to apply transformations to the wrapped data.
73
+ 3. Any operation that depends on wrapped data should itself return a wrapped result.
74
+ ///
75
+
76
+ Let's define such a `Wrapper` class:
77
+
78
+ ```python
79
+ from dataclasses import dataclass
80
+ from typing import Callable, Generic, TypeVar
81
+
82
+ a = TypeVar("a")
83
+ b = TypeVar("b")
84
+
85
+ @dataclass
86
+ class Wrapper(Generic[a]):
87
+ value: a
88
+ ```
89
+
90
+ Now, we can create an instance of wrapped data:
91
+
92
+ ```python
93
+ wrapped = Wrapper(1)
94
+ ```
95
+
96
+ ### Mapping Functions Over Wrapped Data
97
+
98
+ To modify wrapped data while keeping it wrapped, we define an `fmap` method:
99
+
100
+ ```python
101
+ @dataclass
102
+ class Wrapper(Generic[a]):
103
+ value: a
104
+
105
+ def fmap(self, func: Callable[[a], b]) -> "Wrapper[b]":
106
+ return Wrapper(func(self.value))
107
+ ```
108
+
109
+ Now, we can apply transformations without unwrapping:
110
+
111
+ ```python
112
+ >>> wrapped.fmap(lambda x: x + 1)
113
+ Wrapper(value=2)
114
+
115
+ >>> wrapped.fmap(lambda x: [x])
116
+ Wrapper(value=[1])
117
+ ```
118
+
119
+ > Try using the `Wrapper` in the cell below.
120
+ """
121
+ )
122
+ return
123
+
124
+
125
+ @app.cell
126
+ def _(Callable, Functor, Generic, a, b, dataclass):
127
+ @dataclass
128
+ class Wrapper(Functor, Generic[a]):
129
+ value: a
130
+
131
+ def fmap(self, func: Callable[[a], b]) -> "Wrapper[b]":
132
+ return Wrapper(func(self.value))
133
+
134
+ def __repr__(self):
135
+ return repr(self.value)
136
+
137
+
138
+ wrapper = Wrapper(1)
139
+ return Wrapper, wrapper
140
+
141
+
142
+ @app.cell(hide_code=True)
143
+ def _(mo):
144
+ mo.md(
145
+ """
146
+ We can analyze the type signature of `fmap` for `Wrapper`:
147
+
148
+ * `self` is of type `Wrapper[a]`
149
+ * `func` is of type `Callable[[a], b]`
150
+ * The return value is of type `Wrapper[b]`
151
+
152
+ Thus, in Python's type system, we can express the type signature of `fmap` as:
153
+
154
+ ```python
155
+ def fmap(self: Wrapper[a], func: Callable[[a], b]) -> Wrapper[b]:
156
+ ```
157
+
158
+ Essentially, `fmap`:
159
+
160
+ 1. Takes a `Wrapper[a]` instance and a function `Callable[[a], b]` as input.
161
+ 2. Applies the function to the value inside the wrapper.
162
+ 3. Returns a new `Wrapper[b]` instance with the transformed value, leaving the original wrapper and its internal data unmodified.
163
+
164
+ Now, let's examine `list` as a similar kind of wrapper.
165
+ """
166
+ )
167
+ return
168
+
169
+
170
+ @app.cell(hide_code=True)
171
+ def _(mo):
172
+ mo.md(
173
+ """
174
+ ## The List Wrapper
175
+
176
+ We can define a `ListWrapper` class to represent a wrapped list that supports `fmap`:
177
+ """
178
+ )
179
+ return
180
+
181
+
182
+ @app.cell
183
+ def _(Callable, Functor, Generic, a, b, dataclass):
184
+ @dataclass
185
+ class ListWrapper(Functor, Generic[a]):
186
+ value: list[a]
187
+
188
+ def fmap(self, func: Callable[[a], b]) -> "ListWrapper[b]":
189
+ return ListWrapper([func(x) for x in self.value])
190
+
191
+ def __repr__(self):
192
+ return repr(self.value)
193
+
194
+
195
+ list_wrapper = ListWrapper([1, 2, 3, 4])
196
+ return ListWrapper, list_wrapper
197
+
198
+
199
+ @app.cell
200
+ def _(ListWrapper, mo):
201
+ with mo.redirect_stdout():
202
+ print(ListWrapper(value=[2, 3, 4, 5]))
203
+ print(ListWrapper(value=[[1], [2], [3], [4]]))
204
+ return
205
+
206
+
207
+ @app.cell(hide_code=True)
208
+ def _(mo):
209
+ mo.md(
210
+ """
211
+ ### Extracting the Type of `fmap`
212
+
213
+ The type signature of `fmap` for `ListWrapper` is:
214
+
215
+ ```python
216
+ def fmap(self: ListWrapper[a], func: Callable[[a], b]) -> ListWrapper[b]
217
+ ```
218
+
219
+ Similarly, for `Wrapper`:
220
+
221
+ ```python
222
+ def fmap(self: Wrapper[a], func: Callable[[a], b]) -> Wrapper[b]
223
+ ```
224
+
225
+ Both follow the same pattern, which we can generalize as:
226
+
227
+ ```python
228
+ def fmap(self: Functor[a], func: Callable[[a], b]) -> Functor[b]
229
+ ```
230
+
231
+ where `Functor` can be `Wrapper`, `ListWrapper`, or any other wrapper type that follows the same structure.
232
+
233
+ ### Functors in Haskell (optional)
234
+
235
+ In Haskell, the type of `fmap` is:
236
+
237
+ ```haskell
238
+ fmap :: Functor f => (a -> b) -> f a -> f b
239
+ ```
240
+
241
+ or equivalently:
242
+
243
+ ```haskell
244
+ fmap :: Functor f => (a -> b) -> (f a -> f b)
245
+ ```
246
+
247
+ This means that `fmap` **lifts** an ordinary function into the **functor world**, allowing it to operate within a computational context.
248
+
249
+ Now, let's define an abstract class for `Functor`.
250
+ """
251
+ )
252
+ return
253
+
254
+
255
+ @app.cell(hide_code=True)
256
+ def _(mo):
257
+ mo.md(
258
+ """
259
+ ## Defining Functor
260
+
261
+ Recall that, a **Functor** is an abstraction that allows us to apply a function to values inside a computational context while preserving its structure.
262
+
263
+ To define `Functor` in Python, we use an abstract base class:
264
+
265
+ ```python
266
+ from dataclasses import dataclass
267
+ from typing import Callable, Generic, TypeVar
268
+ from abc import ABC, abstractmethod
269
+
270
+ a = TypeVar("a")
271
+ b = TypeVar("b")
272
+
273
+ @dataclass
274
+ class Functor(ABC, Generic[a]):
275
+ @abstractmethod
276
+ def fmap(self, func: Callable[[a], b]) -> "Functor[b]":
277
+ raise NotImplementedError
278
+ ```
279
+
280
+ We can now extend custom wrappers, containers, or computation contexts with this `Functor` base class, implement the `fmap` method, and apply any function.
281
+
282
+ Next, let's implement a more complex data structure: [RoseTree](https://en.wikipedia.org/wiki/Rose_tree).
283
+ """
284
+ )
285
+ return
286
+
287
+
288
+ @app.cell(hide_code=True)
289
+ def _(mo):
290
+ mo.md(
291
+ """
292
+ ## Case Study: RoseTree
293
+
294
+ A **RoseTree** is a tree where:
295
+
296
+ - Each node holds a **value**.
297
+ - Each node has a **list of child nodes** (which are also RoseTrees).
298
+
299
+ This structure is useful for representing hierarchical data, such as:
300
+
301
+ - Abstract Syntax Trees (ASTs)
302
+ - File system directories
303
+ - Recursive computations
304
+
305
+ We can implement `RoseTree` by extending the `Functor` class:
306
+
307
+ ```python
308
+ from dataclasses import dataclass
309
+ from typing import Callable, Generic, TypeVar
310
+
311
+ a = TypeVar("a")
312
+ b = TypeVar("b")
313
+
314
+ @dataclass
315
+ class RoseTree(Functor, Generic[a]):
316
+ value: a
317
+ children: list["RoseTree[a]"]
318
+
319
+ def fmap(self, func: Callable[[a], b]) -> "RoseTree[b]":
320
+ return RoseTree(
321
+ func(self.value), [child.fmap(func) for child in self.children]
322
+ )
323
+
324
+ def __repr__(self) -> str:
325
+ return f"RoseNode({self.value}, {self.children})"
326
+ ```
327
+
328
+ - The function is applied **recursively** to each node's value.
329
+ - The tree structure **remains unchanged**.
330
+ - Only the values inside the tree are modified.
331
+
332
+ > Try using `RoseTree` in the cell below.
333
+ """
334
+ )
335
+ return
336
+
337
+
338
+ @app.cell(hide_code=True)
339
+ def _(Callable, Functor, Generic, a, b, dataclass, mo):
340
+ @dataclass
341
+ class RoseTree(Functor, Generic[a]):
342
+ """
343
+ ### Doc: RoseTree
344
+
345
+ A Functor implementation of `RoseTree`, allowing transformation of values while preserving the tree structure.
346
+
347
+ **Attributes**
348
+
349
+ - `value (a)`: The value stored in the node.
350
+ - `children (list[RoseTree[a]])`: A list of child nodes forming the tree structure.
351
+
352
+ **Methods:**
353
+
354
+ - `fmap(func: Callable[[a], b]) -> RoseTree[b]`
355
+ ```Python
356
+ def fmap(RoseTree[a], (a -> b)) -> RoseTree[b]
357
+ ```
358
+ Applies a function to each value in the tree, producing a new `RoseTree[b]` with transformed values.
359
+
360
+ **Implementation logic:**
361
+
362
+ - The function `func` is applied to the root node's `value`.
363
+ - Each child in `children` recursively calls `fmap`, ensuring all values in the tree are mapped.
364
+ - The overall tree structure remains unchanged.
365
+
366
+ - `__repr__() -> str`: Returns a string representation of the node and its children.
367
+ """
368
+
369
+ value: a
370
+ children: list["RoseTree[a]"]
371
+
372
+ def fmap(self, func: Callable[[a], b]) -> "RoseTree[b]":
373
+ return RoseTree(
374
+ func(self.value), [child.fmap(func) for child in self.children]
375
+ )
376
+
377
+ def __repr__(self) -> str:
378
+ return f"RoseNode({self.value}, {self.children})"
379
+
380
+
381
+ mo.md(RoseTree.__doc__)
382
+ return (RoseTree,)
383
+
384
+
385
+ @app.cell(hide_code=True)
386
+ def _(RoseTree, mo):
387
+ ftree = RoseTree(1, [RoseTree(2, []), RoseTree(3, [RoseTree(4, [])])])
388
+
389
+ with mo.redirect_stdout():
390
+ print(ftree)
391
+ print(ftree.fmap(lambda x: [x]))
392
+ print(ftree.fmap(lambda x: RoseTree(x, [])))
393
+ return (ftree,)
394
+
395
+
396
+ @app.cell(hide_code=True)
397
+ def _(mo):
398
+ mo.md(
399
+ """
400
+ ## Generic Functions that can be Used with Any Functor
401
+
402
+ One of the powerful features of functors is that we can write **generic functions** that can work with any functor.
403
+
404
+ Remember that in Haskell, the type of `fmap` can be written as:
405
+
406
+ ```haskell
407
+ fmap :: Functor f => (a -> b) -> (f a -> f b)
408
+ ```
409
+
410
+ Translating to Python, we get:
411
+
412
+ ```python
413
+ def fmap(func: Callable[[a], b]) -> Callable[[Functor[a]], Functor[b]]
414
+ ```
415
+
416
+ This means that `fmap`:
417
+
418
+ - Takes an **ordinary function** `Callable[[a], b]` as input.
419
+ - Outputs a function that:
420
+ - Takes a **functor** of type `Functor[a]` as input.
421
+ - Outputs a **functor** of type `Functor[b]`.
422
+
423
+ We can implement a similar idea in Python:
424
+
425
+ ```python
426
+ # fmap(func: Callable[[a], b]) -> Callable[[Functor[a]], Functor[b]]
427
+ fmap = lambda func: lambda f: f.fmap(lambda x: func(x))
428
+
429
+ # inc([Functor[a]) -> Functor[b]
430
+ inc = fmap(lambda x: x + 1)
431
+ ```
432
+
433
+ - **`fmap`**: Lifts an ordinary function (`lambda x: func(x)`) to the functor world, allowing the function to operate on the wrapped value inside the functor.
434
+ - **`inc`**: A specific instance of `fmap` that operates on any functor. It takes a functor, applies the function `lambda x: x + 1` to every value inside it, and returns a new functor with the updated values.
435
+
436
+ Thus, **`fmap`** transforms an ordinary function into a **function that operates on functors**, and **`inc`** is a specific case where it increments the value inside the functor.
437
+
438
+ ### Applying the `inc` Function to Various Functors
439
+
440
+ You can now apply `inc` to any functor like `Wrapper`, `ListWrapper`, or `RoseTree`:
441
+
442
+ ```python
443
+ # Applying `inc` to a Wrapper
444
+ wrapper = Wrapper(5)
445
+ inc(wrapper) # Wrapper(value=6)
446
+
447
+ # Applying `inc` to a ListWrapper
448
+ list_wrapper = ListWrapper([1, 2, 3])
449
+ inc(list_wrapper) # ListWrapper(value=[2, 3, 4])
450
+
451
+ # Applying `inc` to a RoseTree
452
+ tree = RoseTree(1, [RoseTree(2, []), RoseTree(3, [])])
453
+ inc(tree) # RoseTree(value=2, children=[RoseTree(value=3, children=[]), RoseTree(value=4, children=[])])
454
+ ```
455
+
456
+ > Try using `fmap` in the cell below.
457
+ """
458
+ )
459
+ return
460
+
461
+
462
+ @app.cell(hide_code=True)
463
+ def _(ftree, list_wrapper, mo, wrapper):
464
+ fmap = lambda func: lambda f: f.fmap(func)
465
+ inc = fmap(lambda x: x + 1)
466
+ with mo.redirect_stdout():
467
+ print(inc(wrapper))
468
+ print(inc(list_wrapper))
469
+ print(inc(ftree))
470
+ return fmap, inc
471
+
472
+
473
+ @app.cell(hide_code=True)
474
+ def _(mo):
475
+ mo.md(
476
+ """
477
+ ## Functor laws
478
+
479
+ In addition to providing a function `fmap` of the specified type, functors are also required to satisfy two equational laws:
480
+
481
+ ```haskell
482
+ fmap id = id -- fmap preserves identity
483
+ fmap (g . h) = fmap g . fmap h -- fmap distributes over composition
484
+ ```
485
+
486
+ 1. `fmap` should preserve the **identity function**, in the sense that applying `fmap` to this function returns the same function as the result.
487
+ 2. `fmap` should also preserve **function composition**. Applying two composed functions `g` and `h` to a functor via `fmap` should give the same result as first applying `fmap` to `g` and then applying `fmap` to `h`.
488
+
489
+ /// admonition |
490
+ - Any `Functor` instance satisfying the first law `(fmap id = id)` will automatically satisfy the [second law](https://github.com/quchen/articles/blob/master/second_functor_law.mo) as well.
491
+ ///
492
+
493
+ ### Functor Law Verification
494
+
495
+ We can add a helper function `check_functor_law` in the `Functor` class to verify that an instance satisfies the functor laws.
496
+
497
+ ```Python
498
+ id = lambda x: x
499
+
500
+ @dataclass
501
+ class Functor(ABC, Generic[a]):
502
+ @abstractmethod
503
+ def fmap(self, func: Callable[[a], b]) -> "Functor[b]":
504
+ return NotImplementedError
505
+
506
+ def check_functor_law(self):
507
+ return repr(self.fmap(id)) == repr(self)
508
+
509
+ @abstractmethod
510
+ def __repr__(self):
511
+ return NotImplementedError
512
+ ```
513
+
514
+ We can verify the functor we've defined.
515
+ """
516
+ )
517
+ return
518
+
519
+
520
+ @app.cell
521
+ def _():
522
+ id = lambda x: x
523
+ compose = lambda f, g: lambda x: f(g(x))
524
+ return compose, id
525
+
526
+
527
+ @app.cell
528
+ def _(ftree, list_wrapper, mo, wrapper):
529
+ with mo.redirect_stdout():
530
+ print(wrapper.check_functor_law())
531
+ print(list_wrapper.check_functor_law())
532
+ print(ftree.check_functor_law())
533
+ return
534
+
535
+
536
+ @app.cell(hide_code=True)
537
+ def _(mo):
538
+ mo.md("""And here is an `EvilFunctor`. We can verify it's not a valid `Functor`.""")
539
+ return
540
+
541
+
542
+ @app.cell
543
+ def _(Callable, Functor, Generic, a, b, dataclass):
544
+ @dataclass
545
+ class EvilFunctor(Functor, Generic[a]):
546
+ value: list[a]
547
+
548
+ def fmap(self, func: Callable[[a], b]) -> "EvilFunctor[b]":
549
+ return (
550
+ EvilFunctor([self.value[0]] * 2 + list(map(func, self.value[1:])))
551
+ if self.value
552
+ else []
553
+ )
554
+
555
+ def __repr__(self):
556
+ return repr(self.value)
557
+ return (EvilFunctor,)
558
+
559
+
560
+ @app.cell
561
+ def _(EvilFunctor):
562
+ EvilFunctor([1, 2, 3, 4]).check_functor_law()
563
+ return
564
+
565
+
566
+ @app.cell(hide_code=True)
567
+ def _(mo):
568
+ mo.md(
569
+ """
570
+ ## Final definition of Functor
571
+
572
+ We can now draft the final definition of `Functor` with some utility functions.
573
+
574
+ ```Python
575
+ @dataclass
576
+ class Functor(ABC, Generic[a]):
577
+ @abstractmethod
578
+ def fmap(self, func: Callable[[a], b]) -> "Functor[b]":
579
+ return NotImplementedError
580
+
581
+ def check_functor_law(self) -> bool:
582
+ return repr(self.fmap(id)) == repr(self)
583
+
584
+ def const_fmap(self, b) -> "Functor[b]":
585
+ return self.fmap(lambda _: b)
586
+
587
+ def void(self) -> "Functor[None]":
588
+ return self.const_fmap(None)
589
+
590
+ @abstractmethod
591
+ def __repr__(self):
592
+ return NotImplementedError
593
+ ```
594
+ """
595
+ )
596
+ return
597
+
598
+
599
+ @app.cell(hide_code=True)
600
+ def _(ABC, Callable, Generic, a, abstractmethod, b, dataclass, id, mo):
601
+ @dataclass
602
+ class Functor(ABC, Generic[a]):
603
+ """
604
+ ### Doc: Functor
605
+
606
+ A generic interface for types that support mapping over their values.
607
+
608
+ **Methods:**
609
+
610
+ - `fmap(func: Callable[[a], b]) -> Functor[b]`
611
+ Abstract method to apply a function `func` to transform the values inside the Functor.
612
+
613
+ - `check_functor_law() -> bool`
614
+ Verifies the identity law of functors: `fmap(id) == id`.
615
+ This ensures that applying `fmap` with the identity function does not alter the structure.
616
+
617
+ - `const_fmap(b) -> Functor[b]`
618
+ Replaces all values inside the Functor with a constant `b`, preserving the original structure.
619
+
620
+ - `void() -> Functor[None]`
621
+ Equivalent to `const_fmap(None)`, transforming all values into `None`.
622
+
623
+ - `__repr__()`
624
+ Abstract method to define a string representation of the Functor.
625
+
626
+ **Functor Laws:**
627
+ A valid Functor implementation must satisfy:
628
+
629
+ 1. **Identity Law:** `F.fmap(id) == F`
630
+ 2. **Composition Law:** `F.fmap(f).fmap(g) == F.fmap(lambda x: g(f(x)))`
631
+ """
632
+
633
+ @abstractmethod
634
+ def fmap(self, func: Callable[[a], b]) -> "Functor[b]":
635
+ return NotImplementedError
636
+
637
+ def check_functor_law(self) -> bool:
638
+ return repr(self.fmap(id)) == repr(self)
639
+
640
+ def const_fmap(self, b) -> "Functor[b]":
641
+ return self.fmap(lambda _: b)
642
+
643
+ def void(self) -> "Functor[None]":
644
+ return self.const_fmap(None)
645
+
646
+ @abstractmethod
647
+ def __repr__(self):
648
+ return NotImplementedError
649
+
650
+
651
+ mo.md(Functor.__doc__)
652
+ return (Functor,)
653
+
654
+
655
+ @app.cell(hide_code=True)
656
+ def _(mo):
657
+ mo.md("""> Try with utility functions in the cell below""")
658
+ return
659
+
660
+
661
+ @app.cell(hide_code=True)
662
+ def _(ftree, list_wrapper, mo):
663
+ with mo.redirect_stdout():
664
+ print(ftree.const_fmap("λ"))
665
+ print(ftree.void())
666
+ print(list_wrapper.const_fmap("λ"))
667
+ print(list_wrapper.void())
668
+ return
669
+
670
+
671
+ @app.cell(hide_code=True)
672
+ def _(mo):
673
+ mo.md(
674
+ """
675
+ ## Functors for Non-Iterable Types
676
+
677
+ In the previous examples, we implemented functors for **iterables**, like `ListWrapper` and `RoseTree`, which are inherently **iterable types**. This is a natural fit for functors, as iterables can be mapped over.
678
+
679
+ However, **functors are not limited to iterables**. There are cases where we want to apply the concept of functors to types that are not inherently iterable, such as types that represent optional values, computations, or other data structures.
680
+
681
+ ### The Maybe Functor
682
+
683
+ One example is the **`Maybe`** type from Haskell, which is used to represent computations that can either result in a value (`Just a`) or no value (`Nothing`).
684
+
685
+ We can define the `Maybe` functor as below:
686
+ """
687
+ )
688
+ return
689
+
690
+
691
+ @app.cell
692
+ def _(Callable, Functor, Generic, a, b, dataclass):
693
+ @dataclass
694
+ class Just(Generic[a]):
695
+ value: a
696
+
697
+ def __init__(self, value: a):
698
+ # If the value is already a `Just`, we extract the value, else we wrap it
699
+ self.value = value.value if isinstance(value, Just) else value
700
+
701
+ def __repr__(self):
702
+ return f"Just {self.value}"
703
+
704
+
705
+ @dataclass
706
+ class Maybe(Functor, Generic[a]):
707
+ value: None | Just[a]
708
+
709
+ def fmap(self, func: Callable[[a], b]) -> "Maybe[b]":
710
+ # Apply the function to the value inside `Just`, or return `Nothing` if value is None
711
+ return (
712
+ Maybe(Just(func(self.value.value))) if self.value else Maybe(None)
713
+ )
714
+
715
+ def __repr__(self):
716
+ return repr(self.value) if self.value else "Nothing"
717
+ return Just, Maybe
718
+
719
+
720
+ @app.cell(hide_code=True)
721
+ def _(mo):
722
+ mo.md(
723
+ """
724
+ - **`Just`** is a wrapper that holds a value. We use it to represent the presence of a value.
725
+ - **`Maybe`** is a functor that can either hold a `Just` value or be `Nothing` (equivalent to `None` in Python). The `fmap` method applies a function to the value inside the `Just` wrapper, if it exists. If the value is `None` (representing `Nothing`), `fmap` simply returns `Nothing`.
726
+
727
+ By using `Maybe` as a functor, we gain the ability to apply transformations (`fmap`) to potentially absent values, without having to explicitly handle the `None` case every time.
728
+
729
+ > Try using `Maybe` in the cell below.
730
+ """
731
+ )
732
+ return
733
+
734
+
735
+ @app.cell
736
+ def _(Just, Maybe, ftree):
737
+ mftree = Maybe(Just(ftree))
738
+ mint = Maybe(Just(1))
739
+ mnone = Maybe(None)
740
+ return mftree, mint, mnone
741
+
742
+
743
+ @app.cell(hide_code=True)
744
+ def _(inc, mftree, mint, mnone, mo):
745
+ with mo.redirect_stdout():
746
+ print(mftree.check_functor_law())
747
+ print(mint.check_functor_law())
748
+ print(mnone.check_functor_law())
749
+ print(mftree.fmap(inc))
750
+ print(mint.fmap(lambda x: x + 1))
751
+ print(mnone.fmap(lambda x: x + 1))
752
+ return
753
+
754
+
755
+ @app.cell(hide_code=True)
756
+ def _(mo):
757
+ mo.md(
758
+ """
759
+ ## Limitations of Functor
760
+
761
+ Functors abstract the idea of mapping a function over each element of a structure. Suppose now that we wish to generalise this idea to allow functions with any number of arguments to be mapped, rather than being restricted to functions with a single argument. More precisely, suppose that we wish to define a hierarchy of `fmap` functions with the following types:
762
+
763
+ ```haskell
764
+ fmap0 :: a -> f a
765
+
766
+ fmap1 :: (a -> b) -> f a -> f b
767
+
768
+ fmap2 :: (a -> b -> c) -> f a -> f b -> f c
769
+
770
+ fmap3 :: (a -> b -> c -> d) -> f a -> f b -> f c -> f d
771
+ ```
772
+
773
+ And we have to declare a special version of the functor class for each case.
774
+
775
+ We will learn how to resolve this problem in the next notebook on `Applicatives`.
776
+ """
777
+ )
778
+ return
779
+
780
+
781
+ @app.cell(hide_code=True)
782
+ def _(mo):
783
+ mo.md(
784
+ """
785
+ # Introduction to Categories
786
+
787
+ A [category](https://en.wikibooks.org/wiki/Haskell/Category_theory#Introduction_to_categories) is, in essence, a simple collection. It has three components:
788
+
789
+ - A collection of **objects**.
790
+ - A collection of **morphisms**, each of which ties two objects (a _source object_ and a _target object_) together. If $f$ is a morphism with source object $C$ and target object $B$, we write $f : C → B$.
791
+ - A notion of **composition** of these morphisms. If $g : A → B$ and $f : B → C$ are two morphisms, they can be composed, resulting in a morphism $f ∘ g : A → C$.
792
+
793
+ ## Category laws
794
+
795
+ There are three laws that categories need to follow.
796
+
797
+ 1. The composition of morphisms needs to be **associative**. Symbolically, $f ∘ (g ∘ h) = (f ∘ g) ∘ h$
798
+
799
+ - Morphisms are applied right to left, so with $f ∘ g$ first $g$ is applied, then $f$.
800
+
801
+ 2. The category needs to be **closed** under the composition operation. So if $f : B → C$ and $g : A → B$, then there must be some morphism $h : A → C$ in the category such that $h = f ∘ g$.
802
+
803
+ 3. Given a category $C$ there needs to be for every object $A$ an **identity** morphism, $id_A : A → A$ that is an identity of composition with other morphisms. Put precisely, for every morphism $g : A → B$: $g ∘ id_A = id_B ∘ g = g$
804
+
805
+ /// attention | The definition of a category does not define:
806
+
807
+ - what `∘` is,
808
+ - what `id` is, or
809
+ - what `f`, `g`, and `h` might be.
810
+
811
+ Instead, category theory leaves it up to us to discover what they might be.
812
+ ///
813
+ """
814
+ )
815
+ return
816
+
817
+
818
+ @app.cell(hide_code=True)
819
+ def _(mo):
820
+ mo.md(
821
+ """
822
+ ## The Python category
823
+
824
+ The main category we'll be concerning ourselves with in this part is the Python category, or we can give it a shorter name: `Py`. `Py` treats Python types as objects and Python functions as morphisms. A function `def f(a: A) -> B` for types A and B is a morphism in Python.
825
+
826
+ Remember that we defined the `id` and `compose` function above as:
827
+
828
+ ```Python
829
+ def id(x: Generic[a]) -> Generic[a]:
830
+ return x
831
+
832
+ def compose(f: Callable[[b], c], g: Callable[[a], b]) -> Callable[[a], c]:
833
+ return lambda x: f(g(x))
834
+ ```
835
+
836
+ We can check second law easily.
837
+
838
+ For the first law, we have:
839
+
840
+ ```python
841
+ # compose(f, g) = lambda x: f(g(x))
842
+ f ∘ (g ∘ h)
843
+ = compose(f, compose(g, h))
844
+ = lambda x: f(compose(g, h)(x))
845
+ = lambda x: f(lambda y: g(h(y))(x))
846
+ = lambda x: f(g(h(x)))
847
+
848
+ (f ∘ g) ∘ h
849
+ = compose(compose(f, g), h)
850
+ = lambda x: compose(f, g)(h(x))
851
+ = lambda x: lambda y: f(g(y))(h(x))
852
+ = lambda x: f(g(h(x)))
853
+ ```
854
+
855
+ For the third law, we have:
856
+
857
+ ```python
858
+ g ∘ id_A
859
+ = compose(g: Callable[[a], b], id: Callable[[a], a]) -> Callable[[a], b]
860
+ = lambda x: g(id(x))
861
+ = lambda x: g(x) # id(x) = x
862
+ = g
863
+ ```
864
+ the similar proof can be applied to $id_B ∘ g =g$.
865
+
866
+ Thus `Py` is a valid category.
867
+ """
868
+ )
869
+ return
870
+
871
+
872
+ @app.cell(hide_code=True)
873
+ def _(mo):
874
+ mo.md(
875
+ """
876
+ # Functors, again
877
+
878
+ A functor is essentially a transformation between categories, so given categories $C$ and $D$, a functor $F : C → D$:
879
+
880
+ - Maps any object $A$ in $C$ to $F ( A )$, in $D$.
881
+ - Maps morphisms $f : A → B$ in $C$ to $F ( f ) : F ( A ) → F ( B )$ in $D$.
882
+
883
+ > Endofunctors are functors from a category to itself.
884
+ """
885
+ )
886
+ return
887
+
888
+
889
+ @app.cell(hide_code=True)
890
+ def _(mo):
891
+ mo.md(
892
+ """
893
+ ## Functors on the category of Python
894
+
895
+ Remember that a functor has two parts: it maps objects in one category to objects in another and morphisms in the first category to morphisms in the second.
896
+
897
+ Functors in Python are from `Py` to `func`, where `func` is the subcategory of `Py` defined on just that functor's types. E.g. the RoseTree functor goes from `Py` to `RoseTree`, where `RoseTree` is the category containing only RoseTree types, that is, `RoseTree[T]` for any type `T`. The morphisms in `RoseTree` are functions defined on RoseTree types, that is, functions `RoseTree[T] -> RoseTree[U]` for types `T`, `U`.
898
+
899
+ Recall the definition of `Functor`:
900
+
901
+ ```Python
902
+ @dataclass
903
+ class Functor(ABC, Generic[a])
904
+ ```
905
+
906
+ And RoseTree:
907
+
908
+ ```Python
909
+ @dataclass
910
+ class RoseTree(Functor, Generic[a])
911
+ ```
912
+
913
+ **Here's the key part:** the _type constructor_ `RoseTree` takes any type `T` to a new type, `RoseTree[T]`. Also, `fmap` restricted to `RoseTree` types takes a function `a -> b` to a function `RoseTree[a] -> RoseTree[b]`.
914
+
915
+ But that's it. We've defined two parts, something that takes objects in `Py` to objects in another category (that of `RoseTree` types and functions defined on `RoseTree` types), and something that takes morphisms in `Py` to morphisms in this category. So `RoseTree` is a functor.
916
+
917
+ To sum up:
918
+
919
+ - We work in the category **Py** and its subcategories.
920
+ - **Objects** are types (e.g., `int`, `str`, `list`).
921
+ - **Morphisms** are functions (`Callable[[A], B]`).
922
+ - **Things that take a type and return another type** are type constructors (`RoseTree[T]`).
923
+ - **Things that take a function and return another function** are higher-order functions (`Callable[[Callable[[A], B]], Callable[[C], D]]`).
924
+ - **Abstract base classes (ABC)** and duck typing provide a way to express polymorphism, capturing the idea that in category theory, structures are often defined over multiple objects at once.
925
+ """
926
+ )
927
+ return
928
+
929
+
930
+ @app.cell(hide_code=True)
931
+ def _(mo):
932
+ mo.md(
933
+ """
934
+ ## Functor laws, again
935
+
936
+ Once again there are a few axioms that functors have to obey.
937
+
938
+ 1. Given an identity morphism $id_A$ on an object $A$, $F ( id_A )$ must be the identity morphism on $F ( A )$, i.e.: ${\displaystyle F(\operatorname {id} _{A})=\operatorname {id} _{F(A)}}$
939
+ 2. Functors must distribute over morphism composition, i.e. ${\displaystyle F(f\circ g)=F(f)\circ F(g)}$
940
+ """
941
+ )
942
+ return
943
+
944
+
945
+ @app.cell(hide_code=True)
946
+ def _(mo):
947
+ mo.md(
948
+ """
949
+ Remember that we defined the `fmap` (not the `fmap` in `Functor` class) and `id` as
950
+ ```python
951
+ # fmap :: Callable[[a], b] -> Callable[[Functor[a]], Functor[b]]
952
+ fmap = lambda func: lambda f: f.fmap(func)
953
+ id = lambda x: x
954
+ compose = lambda f, g: lambda x: f(g(x))
955
+ ```
956
+
957
+ Let's prove that `fmap` is a functor.
958
+
959
+ First, let's define a `Category` for a specific `Functor`. We choose to define the `Category` for the `Wrapper` as `WrapperCategory` here for simplicity, but remember that `Wrapper` can be any `Functor`(i.e. `ListWrapper`, `RoseTree`, `Maybe` and more):
960
+
961
+ **Notice that** in this case, we can actually view `fmap` as:
962
+ ```python
963
+ # fmap :: Callable[[a], b] -> Callable[[Wrapper[a]], Wrapper[b]]
964
+ fmap = lambda func: lambda wrapper: wrapper.fmap(func)
965
+ ```
966
+
967
+ We define `WrapperCategory` as:
968
+
969
+ ```python
970
+ @dataclass
971
+ class WrapperCategory():
972
+ @staticmethod
973
+ def id() -> Callable[[Wrapper[a]], Wrapper[a]]:
974
+ return lambda wrapper: Wrapper(wrapper.value)
975
+
976
+ @staticmethod
977
+ def compose(
978
+ f: Callable[[Wrapper[b]], Wrapper[c]],
979
+ g: Callable[[Wrapper[a]], Wrapper[b]],
980
+ ) -> Callable[[Wrapper[a]], Wrapper[c]]:
981
+ return lambda wrapper: f(g(Wrapper(wrapper.value)))
982
+ ```
983
+
984
+ And `Wrapper` is:
985
+
986
+ ```Python
987
+ @dataclass
988
+ class Wrapper(Generic[a]):
989
+ value: a
990
+
991
+ def fmap(self, func: Callable[[a], b]) -> "Wrapper[b]":
992
+ return Wrapper(func(self.value))
993
+ ```
994
+ """
995
+ )
996
+ return
997
+
998
+
999
+ @app.cell(hide_code=True)
1000
+ def _(mo):
1001
+ mo.md(
1002
+ """
1003
+ notice that
1004
+
1005
+ ```python
1006
+ fmap(f)(wrapper) = wrapper.fmap(f)
1007
+ ```
1008
+
1009
+ We can get:
1010
+
1011
+ ```python
1012
+ fmap(id)
1013
+ = lambda wrapper: wrapper.fmap(id)
1014
+ = lambda wrapper: Wrapper(id(wrapper.value))
1015
+ = lambda wrapper: Wrapper(wrapper.value)
1016
+ = WrapperCategory.id()
1017
+ ```
1018
+ And:
1019
+ ```python
1020
+ fmap(compose(f, g))
1021
+ = lambda wrapper: wrapper.fmap(compose(f, g))
1022
+ = lambda wrapper: Wrapper(compose(f, g)(wrapper.value))
1023
+ = lambda wrapper: Wrapper(f(g(wrapper.value)))
1024
+
1025
+ WrapperCategory.compose(fmap(f), fmap(g))
1026
+ = lambda wrapper: fmap(f)(fmap(g)(wrapper))
1027
+ = lambda wrapper: fmap(f)(wrapper.fmap(g))
1028
+ = lambda wrapper: fmap(f)(Wrapper(g(wrapper.value)))
1029
+ = lambda wrapper: Wrapper(g(wrapper.value)).fmap(f)
1030
+ = lambda wrapper: Wrapper(f(Wrapper(g(wrapper.value)).value))
1031
+ = lambda wrapper: Wrapper(f(g(wrapper.value)))
1032
+ = fmap(compose(f, g))
1033
+ ```
1034
+
1035
+ So our `Wrapper` is a valid `Functor`.
1036
+
1037
+ > Try validating functor laws for `Wrapper` below.
1038
+ """
1039
+ )
1040
+ return
1041
+
1042
+
1043
+ @app.cell(hide_code=True)
1044
+ def _(Callable, Wrapper, a, b, c, dataclass):
1045
+ @dataclass
1046
+ class WrapperCategory:
1047
+ @staticmethod
1048
+ def id() -> Callable[[Wrapper[a]], Wrapper[a]]:
1049
+ return lambda wrapper: Wrapper(wrapper.value)
1050
+
1051
+ @staticmethod
1052
+ def compose(
1053
+ f: Callable[[Wrapper[b]], Wrapper[c]],
1054
+ g: Callable[[Wrapper[a]], Wrapper[b]],
1055
+ ) -> Callable[[Wrapper[a]], Wrapper[c]]:
1056
+ return lambda wrapper: f(g(Wrapper(wrapper.value)))
1057
+ return (WrapperCategory,)
1058
+
1059
+
1060
+ @app.cell(hide_code=True)
1061
+ def _(WrapperCategory, compose, fmap, id, mo, wrapper):
1062
+ with mo.redirect_stdout():
1063
+ print(fmap(id)(wrapper) == id(wrapper))
1064
+ print(
1065
+ fmap(compose(lambda x: x + 1, lambda x: x * 2))(wrapper)
1066
+ == WrapperCategory.compose(
1067
+ fmap(lambda x: x + 1), fmap(lambda x: x * 2)
1068
+ )(wrapper)
1069
+ )
1070
+ return
1071
+
1072
+
1073
+ @app.cell(hide_code=True)
1074
+ def _(mo):
1075
+ mo.md(
1076
+ """
1077
+ ## Length as a Functor
1078
+
1079
+ Remember that a functor is a transformation between two categories. It is not only limited to a functor from `Py` to `func`, but also includes transformations between other mathematical structures.
1080
+
1081
+ Let’s prove that **`length`** can be viewed as a functor. Specifically, we will demonstrate that `length` is a functor from the **category of list concatenation** to the **category of integer addition**.
1082
+
1083
+ ### Category of List Concatenation
1084
+
1085
+ First, let’s define the category of list concatenation:
1086
+ """
1087
+ )
1088
+ return
1089
+
1090
+
1091
+ @app.cell
1092
+ def _(Generic, a, dataclass):
1093
+ @dataclass
1094
+ class ListConcatenation(Generic[a]):
1095
+ value: list[a]
1096
+
1097
+ @staticmethod
1098
+ def id() -> "ListConcatenation[a]":
1099
+ return ListConcatenation([])
1100
+
1101
+ @staticmethod
1102
+ def compose(
1103
+ this: "ListConcatenation[a]", other: "ListConcatenation[a]"
1104
+ ) -> "ListConcatenation[a]":
1105
+ return ListConcatenation(this.value + other.value)
1106
+ return (ListConcatenation,)
1107
+
1108
+
1109
+ @app.cell(hide_code=True)
1110
+ def _(mo):
1111
+ mo.md(
1112
+ """
1113
+ - **Identity**: The identity element is an empty list (`ListConcatenation([])`).
1114
+ - **Composition**: The composition of two lists is their concatenation (`this.value + other.value`).
1115
+ """
1116
+ )
1117
+ return
1118
+
1119
+
1120
+ @app.cell(hide_code=True)
1121
+ def _(mo):
1122
+ mo.md(
1123
+ """
1124
+ ### Category of Integer Addition
1125
+
1126
+ Now, let's define the category of integer addition:
1127
+ """
1128
+ )
1129
+ return
1130
+
1131
+
1132
+ @app.cell
1133
+ def _(dataclass):
1134
+ @dataclass
1135
+ class IntAddition:
1136
+ value: int
1137
+
1138
+ @staticmethod
1139
+ def id() -> "IntAddition":
1140
+ return IntAddition(0)
1141
+
1142
+ @staticmethod
1143
+ def compose(this: "IntAddition", other: "IntAddition") -> "IntAddition":
1144
+ return IntAddition(this.value + other.value)
1145
+ return (IntAddition,)
1146
+
1147
+
1148
+ @app.cell(hide_code=True)
1149
+ def _(mo):
1150
+ mo.md(
1151
+ """
1152
+ - **Identity**: The identity element is `IntAddition(0)` (the additive identity).
1153
+ - **Composition**: The composition of two integers is their sum (`this.value + other.value`).
1154
+ """
1155
+ )
1156
+ return
1157
+
1158
+
1159
+ @app.cell(hide_code=True)
1160
+ def _(mo):
1161
+ mo.md(
1162
+ """
1163
+ ### Defining the Length Functor
1164
+
1165
+ We now define the `length` function as a functor, mapping from the category of list concatenation to the category of integer addition:
1166
+ """
1167
+ )
1168
+ return
1169
+
1170
+
1171
+ @app.cell
1172
+ def _(IntAddition):
1173
+ length = lambda l: IntAddition(len(l.value))
1174
+ return (length,)
1175
+
1176
+
1177
+ @app.cell(hide_code=True)
1178
+ def _(mo):
1179
+ mo.md("""This function takes an instance of `ListConcatenation`, computes its length, and returns an `IntAddition` instance with the computed length.""")
1180
+ return
1181
+
1182
+
1183
+ @app.cell(hide_code=True)
1184
+ def _(mo):
1185
+ mo.md(
1186
+ """
1187
+ ### Verifying Functor Laws
1188
+
1189
+ Now, let’s verify that `length` satisfies the two functor laws.
1190
+
1191
+ #### 1. **Identity Law**:
1192
+ The identity law states that applying the functor to the identity element of one category should give the identity element of the other category.
1193
+ """
1194
+ )
1195
+ return
1196
+
1197
+
1198
+ @app.cell
1199
+ def _(IntAddition, ListConcatenation, length):
1200
+ length(ListConcatenation.id()) == IntAddition.id()
1201
+ return
1202
+
1203
+
1204
+ @app.cell(hide_code=True)
1205
+ def _(mo):
1206
+ mo.md("""This ensures that the length of an empty list (identity in the `ListConcatenation` category) is `0` (identity in the `IntAddition` category).""")
1207
+ return
1208
+
1209
+
1210
+ @app.cell(hide_code=True)
1211
+ def _(mo):
1212
+ mo.md(
1213
+ """
1214
+ #### 2. **Composition Law**:
1215
+ The composition law states that the functor should preserve composition. Applying the functor to a composed element should be the same as composing the functor applied to the individual elements.
1216
+ """
1217
+ )
1218
+ return
1219
+
1220
+
1221
+ @app.cell
1222
+ def _(ListConcatenation):
1223
+ lista = ListConcatenation([1, 2])
1224
+ listb = ListConcatenation([3, 4])
1225
+ return lista, listb
1226
+
1227
+
1228
+ @app.cell
1229
+ def _(IntAddition, ListConcatenation, length, lista, listb):
1230
+ length(ListConcatenation.compose(lista, listb)) == IntAddition.compose(
1231
+ length(lista), length(listb)
1232
+ )
1233
+ return
1234
+
1235
+
1236
+ @app.cell(hide_code=True)
1237
+ def _(mo):
1238
+ mo.md("""This ensures that the length of the concatenation of two lists is the same as the sum of the lengths of the individual lists.""")
1239
+ return
1240
+
1241
+
1242
+ @app.cell(hide_code=True)
1243
+ def _(mo):
1244
+ mo.md(
1245
+ """
1246
+ # Further reading
1247
+
1248
+ - [The Trivial Monad](http://blog.sigfpe.com/2007/04/trivial-monad.html)
1249
+ - [Haskellwiki. Category Theory](https://en.wikibooks.org/wiki/Haskell/Category_theory)
1250
+ - [Haskellforall. The Category Design Pattern](https://www.haskellforall.com/2012/08/the-category-design-pattern.html)
1251
+ - [Haskellforall. The Functor Design Pattern](https://www.haskellforall.com/2012/09/the-functor-design-pattern.html)
1252
+
1253
+ /// attention | ATTENTION
1254
+ The functor design pattern doesn't work at all if you aren't using categories in the first place. This is why you should structure your tools using the compositional category design pattern so that you can take advantage of functors to easily mix your tools together.
1255
+ ///
1256
+
1257
+ - [Haskellwiki. Functor](https://wiki.haskell.org/index.php?title=Functor)
1258
+ - [Haskellwiki. Typeclassopedia#Functor](https://wiki.haskell.org/index.php?title=Typeclassopedia#Functor)
1259
+ - [Haskellwiki. Typeclassopedia#Category](https://wiki.haskell.org/index.php?title=Typeclassopedia#Category)
1260
+ """
1261
+ )
1262
+ return
1263
+
1264
+
1265
+ @app.cell(hide_code=True)
1266
+ def _():
1267
+ import marimo as mo
1268
+ return (mo,)
1269
+
1270
+
1271
+ @app.cell(hide_code=True)
1272
+ def _():
1273
+ from abc import abstractmethod, ABC
1274
+ return ABC, abstractmethod
1275
+
1276
+
1277
+ @app.cell(hide_code=True)
1278
+ def _():
1279
+ from dataclasses import dataclass
1280
+ from typing import Callable, Generic, TypeVar
1281
+ return Callable, Generic, TypeVar, dataclass
1282
+
1283
+
1284
+ @app.cell(hide_code=True)
1285
+ def _(TypeVar):
1286
+ a = TypeVar("a")
1287
+ b = TypeVar("b")
1288
+ c = TypeVar("c")
1289
+ return a, b, c
1290
+
1291
+
1292
+ if __name__ == "__main__":
1293
+ app.run()
functional_programming/CHANGELOG.md ADDED
@@ -0,0 +1,11 @@
 
 
 
 
 
 
 
 
 
 
 
 
1
+ # Changelog of the functional-programming course
2
+
3
+ ## 2025-03-11
4
+
5
+ * Demo version of notebook `05_functors.py`
6
+
7
+ ## 2025-03-13
8
+
9
+ * `0.1.0` version of notebook `05_functors`
10
+
11
+ Thank [Akshay](https://github.com/akshayka) and [Haleshot](https://github.com/Haleshot) for reviewing
functional_programming/README.md ADDED
@@ -0,0 +1,61 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ # Learn Functional Programming
2
+
3
+ _🚧 This collection is a
4
+ [work in progress](https://github.com/marimo-team/learn/issues/51)._
5
+
6
+ This series of marimo notebooks introduces the powerful paradigm of functional
7
+ programming through Python. Taking inspiration from Haskell and Category Theory,
8
+ we'll build a strong foundation in FP concepts that can transform how you
9
+ approach software development.
10
+
11
+ ## What You'll Learn
12
+
13
+ **Using only Python's standard library**, we'll construct functional programming
14
+ concepts from first principles.
15
+
16
+ Topics include:
17
+
18
+ - Recursion and higher-order functions
19
+ - Category theory fundamentals
20
+ - Functors, applicatives, and monads
21
+ - Composable abstractions for robust code
22
+
23
+ ## Timeline & Collaboration
24
+
25
+ I'm currently studying functional programming and Haskell, estimating about 2
26
+ months or even longer to complete this series. The structure may evolve as the
27
+ project develops.
28
+
29
+ If you're interested in collaborating or have questions, please reach out to me
30
+ on Discord (@eugene.hs).
31
+
32
+ **Running notebooks.** To run a notebook locally, use
33
+
34
+ ```bash
35
+ uvx marimo edit <URL>
36
+ ```
37
+
38
+ For example, run the `Functor` tutorial with
39
+
40
+ ```bash
41
+ uvx marimo edit https://github.com/marimo-team/learn/blob/main/Functional_programming/05_functors.py
42
+ ```
43
+
44
+ You can also open notebooks in our online playground by appending `marimo.app/`
45
+ to a notebook's URL:
46
+ [marimo.app/github.com/marimo-team/learn/blob/main/functional_programming/05_functors.py](https://marimo.app/https://github.com/marimo-team/learn/blob/main/functional_programming/05_functors.py).
47
+
48
+ # Description of notebooks
49
+
50
+ Check [here](https://github.com/marimo-team/learn/issues/51) for current series
51
+ structure.
52
+
53
+ | Notebook | Description | References |
54
+ | ----------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
55
+ | [05. Category and Functors](https://github.com/marimo-team/learn/blob/main/Functional_programming/05_functors.py) | Learn why `len` is a _Functor_ from `list concatenation` to `integer addition`, how to _lift_ an ordinary function into a _computation context_, and how to write an _adapter_ between two categories. | - [The Trivial Monad](http://blog.sigfpe.com/2007/04/trivial-monad.html) <br> - [Haskellwiki. Category Theory](https://en.wikibooks.org/wiki/Haskell/Category_theory) <br> - [Haskellforall. The Category Design Pattern](https://www.haskellforall.com/2012/08/the-category-design-pattern.html) <br> - [Haskellforall. The Functor Design Pattern](https://www.haskellforall.com/2012/09/the-functor-design-pattern.html) <br> - [Haskellwiki. Functor](https://wiki.haskell.org/index.php?title=Functor) <br> - [Haskellwiki. Typeclassopedia#Functor](https://wiki.haskell.org/index.php?title=Typeclassopedia#Functor) <br> - [Haskellwiki. Typeclassopedia#Category](https://wiki.haskell.org/index.php?title=Typeclassopedia#Category) |
56
+
57
+ **Authors.**
58
+
59
+ Thanks to all our notebook authors!
60
+
61
+ - [métaboulie](https://github.com/metaboulie)