TITLE: Difference between "space" and "algebraic structure"
QUESTION [14 upvotes]: What is the difference between a "space" and an "algebraic structure"? For example, metric spaces and vector spaces are both spaces and algebraic structures. Is a group a space? Is a manifold a space or an algebraic structure, both or neither?

REPLY [3 votes]: There's no precise definition; anyway, the way I look at it:

There exist many different types of spaces (e.g. sets, posets, graphs, digraphs, metric spaces, uniform spaces, topological spaces, manifolds, etc.) There's no precise definition, but anyway, spaces usually form a distributive category. Furthermore, if $U : \mathbf{C} \rightarrow \mathbf{Set}$ denotes the relevant forgetful functor, then this usually preserves coproducts, and it usually has a left-adjoint $F$ such that the inclusion $UFS \leftarrow S$ is an isomorphism for each set $S$.
(For this reason, I wouldn't consider vector spaces to really be "spaces".)
We may consider spaces that are equipped with further algebraic structure, like a set equipped with the structure of a group, or a topological space equipped with the structure of a group, or a manifold equipped with the structure of a group. Etc. These can often be defined as finite-product preserving functors out of an appropriately chosen Lawvere theory. Categories of algebraic structures usually don't form a distributive category, despite that all relevant products and coproducts often exist, and the relevant forgetful functor to the relevant category of spaces usually doesn't preserve coproducts. But I think the biggest giveaway that we're not dealing with spaces is that if $U$ and $F$ are the relevant free and forgetful functors, there tend to exist spaces $S$ such that the inclusion $UFS \leftarrow S$ is not an isomorphism.