# Subcategories of a group

Find all subcategories of a group. Which of them are full?

A group is a category with one object in which every arrow is an isomorphism. To specify a subcategory, I need to specify a bunch of objects (in this cae 0 or 1) and a bunch of morphisms.

1) Let's specify 0 objects. Then there are no arrows to specify. This gives the empty subcategory. Here I'm worried about the fact that the empty subcategory is not a group; is it okay for a subcategory of a group to not be a group?

2) Let's specify 1 object. The identity arrow must be in the subcategory. If there are two arrows, then the subcategory must contain their composition. But how to describe this "class" of subcategories more explicitly? And do we need to use that all arrows are isomorphisms?

Regarding full subcategories:

1') The empty subcategory is full vacuously, as far as I understand.

2') If a subcategory contains one object, then the only full such subcategory is the whole initial group.

Let me reword the answer of Clive Newstead (just to strangthen my understanding).

First, there is the empty subcategory. Further, a sub-category of a group containing one object must have the property that for every two morphisms in the subcategory, their composition is in the subcategory. Such a subcategory is by definition a monoid. And not only is it a monoid, but it is also a submonoid of the original group because (1) the collection of arrows of a subcategory is a subclass of the collection of arrows of the original category [which tells us that the monoid being considered is a subset of the original group]; (2) there is a binary operateion (the monoid operation) on that set defined as the restriction of the group operation to the mentioned set, and the set is closed under this monoid operation by the definition of a subcategory; and (3) the identity arrow on the only object of the group is in the subcategory [i.e., the set (=monoid) referred to above contains an identity element of the group].

So the subcategories of a group are the empty category and the sub-monoids of the group. The full subcategories are the empty category and the whole group.

## 1 Answer

A category with one object is precisely a monoid. Since the category structure is inherited from the unit element and group operation of $$G$$, the subcategories of $$G$$ with one object are precisely the submonoids of $$G$$.

The fact that the morphisms of $$G$$ are all invertible is irrelevant to this question: it is also the case that the subcategories of a monoid $$M$$ are (the empty category and) its submonoids of $$M$$.

• Thanks! I've edited my question and explained how I understand your answer. If you have time to take a look and give some feedback, that would be great. – user634426 Jun 24 '19 at 23:02