Here's a nice example of how Galois connections can arise in practice, using the notion of _opposite poset_ that you introduced here: Say we have a collection of things \\(A\\) and a collection of properties \\(B\\) that the things might have; for example, \\(A\\) could be a set of animals

\\[A = \\{\text{dog, cat, octopus, snake, elephant}\\}\\]

and \\(B\\) could be a set of animal properties

\\[B = \\{\text{is a mammal, has fur, has a tail, has legs}\\}\\]

Then we get a function \\(f:P(A)\to P(B)\\) sending a subset of the animals to the set of properties they all share, e.g.

\\[f(\\{\text{dog, cat, elephant}\\})=\\{\text{is a mammal, has a tail, has legs}\\}\\]

\\[f(\\{\text{dog, cat, octopus, elephant}\\} )=\\{\text{has legs}\\}\\]

and a function \\(g:P(B)\to P(A)\\) sending a subset of the properties to the set of animals that have them all, e.g.

\\[g(\\{\text{is a mammal, has a tail}\\}) = \\{\text{dog, cat, elephant}\\}\\]

\\[g(\\{\text{is a mammal, has fur, has a tail}\\}) = \\{\text{dog, cat}\\}\\]

Usually we make power sets like \\(P(A)\\) and \\(P(B)\\) into posets using containment \\(\subseteq\\), but then these functions aren't monotone: the larger the set of animals, the fewer properties they have in common, and vice versa. But we can make both functions monotone by using _reverse_ containment, \\(\supseteq\\), for one of the posets (say, \\(P(B)\\)). This means that we regard \\(f\\) as a monotone function \\(P(A)\to P(B)^\text{op}\\) and \\(g\\) as a monotone function \\(P(B)^\text{op}\to P(A)\\). And these monotone functions are always a Galois connection! Spelled out, this says that for two subsets \\(S\subseteq A\\) and \\(T\subseteq B\\), we always have

\\[f(S) \supseteq T \iff S\subseteq g(T),\\]

and this equivalence holds because both containments just mean that every animal in the set \\(S\\) has every property in the set \\(T\\).

You can even go further and think to yourself "Any meaningful set of animals that includes dogs and elephants should include cats too, because cats have every property that dogs and elephants both have (namely, tails, legs, and being mammals)." This amounts to composing the functions \\(g\\) and \\(f\\), and the laws of adjoints tell us that \\(S\subseteq g\circ f (S)\\). However, if we do the same process again, we don't get any more the second time: \\(g\circ f(g\circ f(S)) = g\circ f(S)\\). You could say that the set {dog, cat, elephant} is already _closed_ under the operation of adding in all the other animals that have all the properties these animals have: in particular, these are all mammals, and there aren't any more mammals in our set of animals. In other words, this set of animals is exactly the set \\(g(\\{\text{is a mammal}\\})\\). In fact, \\(g\\) always outputs a closed set of animals!

We can do the same with sets of properties, and describe a set of animal properties as _closed_ if there are no extra properties all animals with those properties also share. For example, a closed set of properties including "has fur" must also include "is a mammal", because the only animals with fur we're considering are also mammals; similarly, for our list of animals, the only animals that are mammals also have legs and tails. Here are the closed sets of properties:

\\[\varnothing, \\{\text{has a tail}\\}, \\{\text{has legs}\\}, \\{\text{is a mammal, has a tail, has legs}\\}, \\{\text{is a mammal, has fur, has a tail, has legs}\\}\\]

And here are the closed sets of animals:

\\[\\{\text{dog, cat, octopus, snake, elephant}\\}, \\{\text{dog, cat, snake, elephant}\\},\\]

\\[\\{\text{dog, cat, octopus, elephant}\\}, \\{\text{dog, cat, elephant}\\}, \\{\text{dog, cat}\\}\\]

It's not a coincidence that there are five of each: the way I've ordered the closed sets, they correspond to each other via \\(f\\) and \\(g\\): for example,

\\[f(\\{\text{has a tail}\\}) = \\{\text{dog, cat, snake, elephant}\\}\\]

\\[g(\\{\text{dog, cat, snake, elephant}\\}) = \\{\text{has a tail}\\}\\]

We end up with an order-reversing one-to-one correspondence between the closed sets of animals and properties! You can think of these closed sets as telling us some information about how we should taxonomize our animals, with each closed set of properties corresponding hierarchically to some meaningful collection of animals:

![animal taxonomy](https://docs.google.com/drawings/d/e/2PACX-1vRVvs7JTqINXpN7S-LgyZFm32STzcs5U3asHzhJt2ZgpgbiWwoxrYsqIjMrt7S8__--SRTO5_BaKSC8/pub?w=509&h=408)

But a lot of this works just as well for any Galois connectionâ€”I'll post what I know as a series of puzzles.

\\[A = \\{\text{dog, cat, octopus, snake, elephant}\\}\\]

and \\(B\\) could be a set of animal properties

\\[B = \\{\text{is a mammal, has fur, has a tail, has legs}\\}\\]

Then we get a function \\(f:P(A)\to P(B)\\) sending a subset of the animals to the set of properties they all share, e.g.

\\[f(\\{\text{dog, cat, elephant}\\})=\\{\text{is a mammal, has a tail, has legs}\\}\\]

\\[f(\\{\text{dog, cat, octopus, elephant}\\} )=\\{\text{has legs}\\}\\]

and a function \\(g:P(B)\to P(A)\\) sending a subset of the properties to the set of animals that have them all, e.g.

\\[g(\\{\text{is a mammal, has a tail}\\}) = \\{\text{dog, cat, elephant}\\}\\]

\\[g(\\{\text{is a mammal, has fur, has a tail}\\}) = \\{\text{dog, cat}\\}\\]

Usually we make power sets like \\(P(A)\\) and \\(P(B)\\) into posets using containment \\(\subseteq\\), but then these functions aren't monotone: the larger the set of animals, the fewer properties they have in common, and vice versa. But we can make both functions monotone by using _reverse_ containment, \\(\supseteq\\), for one of the posets (say, \\(P(B)\\)). This means that we regard \\(f\\) as a monotone function \\(P(A)\to P(B)^\text{op}\\) and \\(g\\) as a monotone function \\(P(B)^\text{op}\to P(A)\\). And these monotone functions are always a Galois connection! Spelled out, this says that for two subsets \\(S\subseteq A\\) and \\(T\subseteq B\\), we always have

\\[f(S) \supseteq T \iff S\subseteq g(T),\\]

and this equivalence holds because both containments just mean that every animal in the set \\(S\\) has every property in the set \\(T\\).

You can even go further and think to yourself "Any meaningful set of animals that includes dogs and elephants should include cats too, because cats have every property that dogs and elephants both have (namely, tails, legs, and being mammals)." This amounts to composing the functions \\(g\\) and \\(f\\), and the laws of adjoints tell us that \\(S\subseteq g\circ f (S)\\). However, if we do the same process again, we don't get any more the second time: \\(g\circ f(g\circ f(S)) = g\circ f(S)\\). You could say that the set {dog, cat, elephant} is already _closed_ under the operation of adding in all the other animals that have all the properties these animals have: in particular, these are all mammals, and there aren't any more mammals in our set of animals. In other words, this set of animals is exactly the set \\(g(\\{\text{is a mammal}\\})\\). In fact, \\(g\\) always outputs a closed set of animals!

We can do the same with sets of properties, and describe a set of animal properties as _closed_ if there are no extra properties all animals with those properties also share. For example, a closed set of properties including "has fur" must also include "is a mammal", because the only animals with fur we're considering are also mammals; similarly, for our list of animals, the only animals that are mammals also have legs and tails. Here are the closed sets of properties:

\\[\varnothing, \\{\text{has a tail}\\}, \\{\text{has legs}\\}, \\{\text{is a mammal, has a tail, has legs}\\}, \\{\text{is a mammal, has fur, has a tail, has legs}\\}\\]

And here are the closed sets of animals:

\\[\\{\text{dog, cat, octopus, snake, elephant}\\}, \\{\text{dog, cat, snake, elephant}\\},\\]

\\[\\{\text{dog, cat, octopus, elephant}\\}, \\{\text{dog, cat, elephant}\\}, \\{\text{dog, cat}\\}\\]

It's not a coincidence that there are five of each: the way I've ordered the closed sets, they correspond to each other via \\(f\\) and \\(g\\): for example,

\\[f(\\{\text{has a tail}\\}) = \\{\text{dog, cat, snake, elephant}\\}\\]

\\[g(\\{\text{dog, cat, snake, elephant}\\}) = \\{\text{has a tail}\\}\\]

We end up with an order-reversing one-to-one correspondence between the closed sets of animals and properties! You can think of these closed sets as telling us some information about how we should taxonomize our animals, with each closed set of properties corresponding hierarchically to some meaningful collection of animals:

But a lot of this works just as well for any Galois connectionâ€”I'll post what I know as a series of puzzles.