dabd · May 7, 2020 15:16
diff --git a/gistfile1.txt b/gistfile1.txt
 5.3.1 Co-Kleisli morphisms
 Definition 5.3.5. Let (C, ε, ν) be a comonad on C. A co-Kleisli morphism of C from X to Y is a
 morphism CX → Y of C.
 Here is a typical situation in science. Suppose that, during an experiment, we have a process
 f taking an input x ∈ X and giving an output y ∈ Y. This could be for example a survey in
 which we ask people of different age (X) what their political views are (Y). Suppose that we
 repeat the experiment, feeding again the same input x to f . It could happen that this time we
 get a different outcome than before, y
 ′ , y. For example, suppose a person of age x expresses
 political view y. Another person of the same age x may express a different political view y
 ′
 (this
 may even happen by asking the same person twice, in different moments). The usual conclusion
 that we draw, in science, is: X alone (for example, age alone) is not enough to determine Y. There
 was some “hidden”, “extra” data that the process f has access to in order to determine Y. It
 could depend on hidden information, it could depend on past outcomes, and so on. Therefore
 a better mathematical model of the situation is not quite f : X → Y, but rather f : CX → Y,
 where CX contains more information than just X, as specified by the comonad C.
 Note that this is somewhat dual to Kleisli morphisms: there, functions were allowed to have
 more possible outcomes. Here, they are allowed to depend on possibly more information.
 Example 5.3.6 (several fields). A co-Kleisli morphism for the reader comonad of Example 5.3.1
 is a map k : X × A → Y. We can view it as a map that, when fed x ∈ X, also needs to read
 a a ∈ A to give its output. This can model the experimental situation of “hidden variables”
 described above.
 3The condition of local contractibility can be weakened, see [Hat02, Section 1.3].
 162
 5 Monads and comonads
 An example in computer science is a function that needs an extra input in order to carry out
 the computation, either via user interface (say, keyboard), or by having access to some external
 environment. This motivates the name “reader comonad”.
	5.3.1 Co-Kleisli morphisms
	Definition 5.3.5. Let (C, ε, ν) be a comonad on C. A co-Kleisli morphism of C from X to Y is a
	morphism CX → Y of C.
	Here is a typical situation in science. Suppose that, during an experiment, we have a process
	f taking an input x ∈ X and giving an output y ∈ Y. This could be for example a survey in
	which we ask people of different age (X) what their political views are (Y). Suppose that we
	repeat the experiment, feeding again the same input x to f . It could happen that this time we
	get a different outcome than before, y
	′ , y. For example, suppose a person of age x expresses
	political view y. Another person of the same age x may express a different political view y
	′
	(this
	may even happen by asking the same person twice, in different moments). The usual conclusion
	that we draw, in science, is: X alone (for example, age alone) is not enough to determine Y. There
	was some “hidden”, “extra” data that the process f has access to in order to determine Y. It
	could depend on hidden information, it could depend on past outcomes, and so on. Therefore
	a better mathematical model of the situation is not quite f : X → Y, but rather f : CX → Y,
	where CX contains more information than just X, as specified by the comonad C.
	Note that this is somewhat dual to Kleisli morphisms: there, functions were allowed to have
	more possible outcomes. Here, they are allowed to depend on possibly more information.
	Example 5.3.6 (several fields). A co-Kleisli morphism for the reader comonad of Example 5.3.1
	is a map k : X × A → Y. We can view it as a map that, when fed x ∈ X, also needs to read
	a a ∈ A to give its output. This can model the experimental situation of “hidden variables”
	described above.
	3The condition of local contractibility can be weakened, see [Hat02, Section 1.3].
	162
	5 Monads and comonads
	An example in computer science is a function that needs an extra input in order to carry out
	the computation, either via user interface (say, keyboard), or by having access to some external
	environment. This motivates the name “reader comonad”.