“Free Object Functors”
Chapter 12 (Functors) gives as an example of a functor the “free group functor” from the category of sets () to the category of groups (). We will call this functor , and its construction, as described, is the following:
For any set , the object is the free group on .
Let be a -morphism, that is, any mapping, between sets and . Let and be the respective free groups, with and the associated mappings.
Then, by virtue of the universal property of free groups, there exists a unique -morphism such that, as mappings, .
So we decide that will be this . Of course, for this to be a functor, we must check that the composition of morphisms are preserved, as are identities.
Later in the chapter, the general concept of a free object in category on an object in via a functor from to is defined, as an extension of the notion of a free group; a free group thus becomes the free object in category on an object in category following the forgetful functor from to . We are called to consider the “free object functor” from to (hence in the reverse direction respective to ), based as above on the mapping of objects in to their respective free objects in and of -morphisms to the corresponding -morphisms following the above construction.
This set-up at first appears clear, but left me feeling uneasy. After some thought I managed to pinpoint two issues:
- Free objects, when they exist, are unique up to an isomorphism, which means that they aren’t «unique, period». For a functor to map a set to its free group, one must have a rule that says what exactly that group is; you can’t just say what it is up to an isomorphism.
- «Free objects» are not actually objects in the target category; they are ordered pairs , where is an object of the target category, that is, in our example, groups, and is a morphism between objects of the source category. If I build a free group functor retaining only the target object, in our case , the part is lost.
The first issue can be solved easily for groups, and, I suspect, in most or all other cases. The existence of a free object is usually (always?) demonstrated by building some particular instance; one can decide that the functor will map the source object to that specific free object, that we will call the canonical free object.
For the second issue, one must note that the associated morphisms is not complete. We need these to build from . Thus the ‘s remain at least partly included in the definition of the free group functor.
I don’t have the impression that if we are given the functor , without the knowledge of the canonical free groups that it is built upon, we can reconstruct the morphisms that are part of the definition of thise canonical free objects. The picture is rather this:
- We define a particular rule for building a free group on any set .
- Based on this rule, we define the mappings from objects in to objects in , and from morphisms in to morphisms in .
- The entire above procedure (including the rule for building a specific free group) constitutes our free group functor, that is our rule for mapping to and morphisms in to morphisms in .
In practice, it doesn’t matter much what particular rule we take for building our free objects. It does matter, however, that it be conceived of as well specified, and that we understand that not only the part but also the in the free object serve to define the free object functor.