Dagger symmetric monoidal category

Okay kiddo, are you ready to learn about something called a dagger symmetric monoidal category?

First, let's start with some simple definitions. A category is a way of organizing and understanding mathematical concepts and structures. It's like a big box where things (objects) and the connections between them (morphisms) live.

Now, let's add some more details. A dagger category is a special kind of category where each morphism has a "reverse" or "conjugate". We call this reverse the dagger, and we write it as a superscript * next to the morphism.

So, what about the "symmetric monoidal" part? Well, in a symmetric monoidal category, we have a special way of combining objects together (like putting together puzzle pieces!). This combination is called the monoidal product, and it has some special properties.

First, it's symmetric, which means that the order we put things together doesn't matter. So if we have two objects A and B, then A⊗B (that's the symbol we use for the monoidal product) is the same as B⊗A.

Second, it has something called a unit object, which is like a special "do nothing" object. We'll call this object I. When we combine an object with the unit object, it's like nothing happened! So A⊗I is the same as A.

Finally, we can also add a dagger to the monoidal product. This means that if we have two morphisms f and g, then their combination with the monoidal product and dagger looks like (f⊗g)*. This is called the dagger monoidal product.

When we put all these pieces together, we get a dagger symmetric monoidal category! It's a category where we have objects, morphisms, a dagger, a monoidal product, and a unit object, and all of these things have special properties that help us understand mathematical concepts and structures.