The Clowder Project

Examples of forgetful functors that forget structure include:

1. Forgetting Group Structures. The functor $\mathsf{Grp}\to \mathsf{Sets}$ sending a group $\webleft (G,\mu _{G},\eta _{G}\webright )$ to its underlying set $G$, forgetting the multiplication and unit maps $\mu _{G}$ and $\eta _{G}$ of $G$.
2. Forgetting Topologies. The functor $\mathsf{Top}\to \mathsf{Sets}$ sending a topological space $\webleft (X,\mathcal{T}_{X}\webright )$ to its underlying set $X$, forgetting the topology $\mathcal{T}_{X}$.
3. Forgetting Fibrations. The functor $\mathsf{FibSets}\webleft (K\webright )\to \mathsf{Sets}$ sending a $K$-fibred set $\phi _{X}\colon X\to K$ to the set $X$, forgetting the map $\phi _{X}$ and the base set $K$.

Examples of forgetful functors that forget properties include:

1. Forgetting Commutativity. The inclusion functor $\iota \colon \mathsf{CMon}\hookrightarrow \mathsf{Mon}$ which forgets the property of being commutative.
2. Forgetting Inverses. The inclusion functor $\iota \colon \mathsf{Grp}\hookrightarrow \mathsf{Mon}$ which forgets the property of having inverses.