CTFP11 /

# Tobias

**Chapter 1**

- In the definition of Functor: Notice how the condition of mapping units to units and
*preserving*composition correspondence to how structure preserving homomorphism work for other algebraic structures. - The comment that
*It's the arrows that really matter!*seems justified. It seems to me like any object**A**is fully characterized by it's identity arrow (and how it behaves together with with arrows with**A**as a source). - I noticed that product of categories seems to relate very well to products of algebraic structures like for example the direct product of groups.
- Using the
*forgetful functor*to get to the underlying functions set and functions of categories seems like a nice way of somehow getting access to internal structure with out breaking the layer of abstraction provided by category theory. On the whole however it seems like category theory demands that you don't focus on concrete structures but rather abstract one. Takes a bit getting used to I guess. - Ramona gave a nice intuition of the UMP of the free monoid : Give a set of variables
**A**,**M(A)**is the set of expressions over**A**. Since A can be embedded in**M(A)**, for any evalution f that takes the variables of**A**we can have an evaluation g that takes the variables seen as expressions instead but behaves in the same way. - I personally think that it's nice and illusive to look at the a free category as being generated by paths in a graph. The UMP that captures all this seems a bit technical but I guess it's more useful in this form when you actually want to prove stuff.