Bousfield localization

Okay, so let's say you have a big toy box with a lot of toys in it. And you really like playing with a specific type of toy, let's say it's your Barbie dolls. But your toy box has a lot of other things in it, like cars and trucks and blocks.

Now imagine if you had a special toy box that just had your Barbie dolls and nothing else. You would be able to find your dolls so much easier and play with them more easily too, right?

Well, in math, something similar happens. Instead of toys, we have something called "spaces" which are like shapes. And instead of a toy box, we have something called a "category" which is like a container where all the spaces live.

So now let's say you have a bunch of spaces in your category, just like you have a bunch of toys in your toy box. And you really like one specific space, let's say it's a circle. But your category has a lot of other spaces in it, like squares and triangles.

Just like how you wanted a special toy box with just your Barbie dolls, in math we can make a special category that only has the spaces we're interested in. We call this special category a "localization".

In Bousfield localization, we make a special localization that keeps the spaces that have certain properties we care about. So just like you only wanted your Barbie dolls, we only keep the spaces that have the properties we care about.

And just like how your special toy box made it easier for you to find and play with your Barbie dolls, the Bousfield localization makes it easier for us to study and understand the properties we care about in the spaces we're looking at.