This is a small diagram I put together using the wonderful GraphViz. The nodes in this graph denote categories of object, while an arrow $A \to B$ denotes some kind of "inclusion" of categories.

(More precisely, it denotes the existence of a forgetful functor from $B$ to $A$, so perhaps I should reverse the arrows!)

G Sets Sets Magmas Magmas Sets->Magmas Posets Posets Sets->Posets Topological Spaces Topological Spaces Sets->Topological Spaces Measurable Spaces Measurable Spaces Sets->Measurable Spaces Semigroups Semigroups Magmas->Semigroups Quasigroups Quasigroups Magmas->Quasigroups Lattices Lattices Posets->Lattices Metric Spaces Metric Spaces Topological Spaces->Metric Spaces Manifolds Manifolds Topological Spaces->Manifolds Normed Spaces Normed Spaces Topological Spaces->Normed Spaces Measure Spaces Measure Spaces Topological Spaces->Measure Spaces Topological Groups Topological Groups Topological Spaces->Topological Groups Topological Vector Spaces Topological Vector Spaces Topological Spaces->Topological Vector Spaces Monoids Monoids Semigroups->Monoids Loops Loops Quasigroups->Loops Groups Groups Loops->Groups Abelian Groups Abelian Groups Groups->Abelian Groups Groups->Topological Groups Lie Groups Lie Groups Groups->Lie Groups Monoids->Groups Rings Rings Monoids->Rings Abelian Groups->Rings Modules Modules Abelian Groups->Modules Vector Spaces Vector Spaces Abelian Groups->Vector Spaces Integral Domains Integral Domains Rings->Integral Domains Rings->Modules Algebras Algebras Rings->Algebras UFDs UFDs Integral Domains->UFDs PIDs PIDs UFDs->PIDs Euclidean Domains Euclidean Domains PIDs->Euclidean Domains Fields Fields Euclidean Domains->Fields Fields->Vector Spaces Fields->Algebras Modules->Vector Spaces Vector Spaces->Algebras Vector Spaces->Normed Spaces Vector Spaces->Topological Vector Spaces Tensor Algebras Tensor Algebras Algebras->Tensor Algebras Lie Algebras Lie Algebras Algebras->Lie Algebras Clifford Algebras Clifford Algebras Tensor Algebras->Clifford Algebras Exterior Algebras Exterior Algebras Tensor Algebras->Exterior Algebras Geometric Algebras Geometric Algebras Clifford Algebras->Geometric Algebras Metric Spaces->Manifolds Riemannian Manifolds Riemannian Manifolds Metric Spaces->Riemannian Manifolds Manifolds->Riemannian Manifolds Manifolds->Lie Groups Banach Spaces Banach Spaces Normed Spaces->Banach Spaces Inner Product Spaces Inner Product Spaces Normed Spaces->Inner Product Spaces Hilbert Spaces Hilbert Spaces Inner Product Spaces->Hilbert Spaces Measurable Spaces->Measure Spaces Measure Spaces->Normed Spaces Probability Spaces Probability Spaces Measure Spaces->Probability Spaces Lie Groups->Lie Algebras