# Group objects in category of pointed topological spaces

A pointed space is a (non-empty) topological space with a choice of one of its points. Together with pointed spaces and continuous maps preserving base points is called the category of pointed spaces and denoted by $$\mathbf{Top}_*$$. This category has a zero object, which is the point space itself. Also, it has binary products, $$(X, x_0)\times (Y, y_0)=(X\times Y, (x_0, y_0))$$ equipped with product topology.

Therefore this category is qualified to have internal group objects, and I am trying to understand them explicitly. The forgetful functor $$U: \mathbf{Top}_*\to\mathbf{Top}$$ that forgets the base points admit a free left adjoint $$F: \mathbf{Top}\to\mathbf{Top}_*$$ given by forming the disjoint union space with a point space. Hence $$U$$ preserves limits, (in particular, terminal object and binary products) and hence group objects. If I understood correctly this argument shows that group objects in $$\mathbf{Top}_*$$ are nothing but topological groups (with a choice of a base point that may be different from the identity element). Am I missing something here?

Your reasoning is not really correct. It is true that $$U$$ preserves limits, and hence group objects. This does not mean that a group object in pointed spaces is "nothing but" a group object in spaces together with a basepoint. The issue is that being in pointed spaces changes the morphisms, not just the objects: to have a group object in pointed spaces, you need all the structure morphisms of the group object to preserve your chosen basepoint.
Now it turns out that in this case, there is always only one way to pick a basepoint to make this true. Namely, if $$G$$ is a topological group, for the unit morphism $$1\to G$$ to be pointed, the basepoint chosen for $$G$$ must be the identity element. The multiplication and inverse maps then happen to always preserve this basepoint. Moreover, any homomorphism of topological groups also preserves their basepoints, since it must preserve the identity element. So group objects in pointed spaces are the same thing as group objects in spaces in a very strong sense: every group object in spaces uniquely admits the structure of a group object in pointed spaces (by picking the identity as the basepoint) and then the morphisms of group objects are also the same. Another way to say this is that the forgetful functor from the category of group objects in pointed spaces to the category of group objects in spaces is an isomorphism.
Note that in general, forgetful functors do not need to preserve group objects in this strong sense. For instance, consider the forgetful functor from the category of groups to the category of sets. The analogous question would then be: given a group object $$G$$ in the category of sets (i.e., a group), can you give $$G$$ a second group structure such that the unit, multiplication, and inverse maps are homomorphisms with respect to this group structure? It turns out that this second group structure must automatically be the same as the first group structure, but only has the required properties if the first group structure was abelian. In other words, a group object in the category of groups turns out to be an abelian group.