I'd say the definition you give is the best one. Things like Hilbert basis or Schauder basis aren't really basis, but more like special generating familly.

Note that your definition is exactly the local definition of "left adjoint to the forgetful functor set".

The problem with Hilbert basis, aren't the infinitary combinations - there are exemple with infinitary combination that are handled very well by this notion:

Take $C$ to be the category of suplattice (preordered set with arbitrary supremum) then the set of $\{x\}$ for $x \in X$ is a basis in your sense of the power set $\mathcal{P}(X)$. A general element of $\mathcal{P}(X)$ is a supremum of an arbitrary family of singletons.

Take $C$ to be the category of compact Hausdorff spaces, then for a set $X$, the principal ultrafilter $\delta_x$ for $x \in X$ form a "basis" of the Stone-Czech compactification (the space of all ultrafitler) of $\beta X$ of $X$. A general element of $\beta X$ is a limit (along an ultrafilter) of an infinite family of elements of $X$.

Ok, so what's wrong with say Hilbert Basis from this perspective? well the point is that they aren't really basis, in the sense that they don't generate the space "freely".

The universal property for Hilbert basis involve as you said restriction on the type of family of element we consider. The simplest way to put it is to work in the category of Hilbert spaces and isometric map and then if $(b_i)_{i \in I}$ is a Hilbert (orthonormale) basis of $\mathcal{B}$, then the data of a map $\mathcal{B} \to \mathcal{H}$ is the same as the choice of a familly of element of $t_i \in \mathcal{H}$ subject to the relation $\langle t_i,t_j\rangle = \delta_{i,j}$. (it is possible to express other universal property using different class of maps than isometric maps, but this one is the simplest by far)

This is completely analogous to the universal property you'll get for a presentation of a vector space by generator and relation: it means that $\mathcal{B}$ is freely generated by the familly $b_i$ subject to the relation $\langle b_i,b_j\rangle = \delta_{i,j}$.

A more general notion you can use that encompass both is the notion of "generating familly": a familly of elements $b_i \in |X|$ (I'm using $|X|$ to denote the underlying set of $X$) is generating if for all object $Y$ the induced map

$$ Hom(X,Y) \to |Y|^I $$

is injective.

I'd say the definition you give is the best one. Things like Hilbert basis or Schauder basis aren't really basis, but more like special generating familly.

Note that your definition is exactly the local definition of "left adjoint to the forgetful functor set".

The problem with Hilbert basis, aren't the infinitary combinations — there are examples with infinitary combination that are handled very well by this notion:

Take $C$ to be the category of suplattices (preordered sets with arbitrary suprema) then the set of $\{x\}$ for $x \in X$ is a basis in your sense of the power set $\mathcal{P}(X)$. A general element of $\mathcal{P}(X)$ is a supremum of an arbitrary family of singletons.

Take $C$ to be the category of compact Hausdorff spaces, then for a set $X$, the principal ultrafilter $\delta_x$ for $x \in X$ form a "basis" of the Stone–Čech compactification (the space of all ultrafilters) of $\beta X$ of $X$. A general element of $\beta X$ is a limit (along an ultrafilter) of an infinite family of elements of $X$.

Ok, so what's wrong with say Hilbert Basis from this perspective? Well the point is that they aren't really basis, in the sense that they don't generate the space "freely".

The universal property for Hilbert basis involve as you said restriction on the type of family of element we consider. The simplest way to put it is to work in the category of Hilbert spaces and isometric maps and then if $(b_i)_{i \in I}$ is a Hilbert (orthonormale) basis of $\mathcal{B}$, then the data of a map $\mathcal{B} \to \mathcal{H}$ is the same as the choice of a family of elements of $t_i \in \mathcal{H}$ subject to the relation $\langle t_i,t_j\rangle = \delta_{i,j}$. (it is possible to express other universal properties using different class of maps than isometric maps, but this one is the simplest by far.)

This is completely analogous to the universal property you'll get for a presentation of a vector space by generator and relation: it means that $\mathcal{B}$ is freely generated by the familly $b_i$ subject to the relation $\langle b_i,b_j\rangle = \delta_{i,j}$.

A more general notion you can use that encompasses both is the notion of "generating family": a family of elements $b_i \in \lvert X\rvert$ (I'm using $\lvert X\rvert$ to denote the underlying set of $X$) is generating if for all object $Y$ the induced map

$$ \operatorname{Hom}(X,Y) \to \lvert Y\rvert^I $$

is injective.

I'd say the definition you give is the best one. Things like Hilbert basis or Schauder basis aren't really basis, but more like special generating familly.

Note that your definition is exactly the local definition of "left adjoint to the forgetful functor set".

The problem with Hilbert basis, aren't the infinitary combinations - there are exemple with infinitary combination that are handled very well by this notion:

Take $C$ to be the category of suplattice (preordered set with arbitrary supremum) then the set of $\{x\}$ for $x \in X$ is a basis in your sense of the power set $\mathcal{P}(X)$. A general element of $\mathcal{P}(X)$ is a supremum of an arbitrary family of singletons.

Take $C$ to be the category of compact Hausdorff spaces, then for a set $X$, the principal ultrafilter $\delta_x$ for $x \in X$ form a "basis" of the Stone-Czech compactification (the space of all ultrafitler) of $\beta X$ of $X$. A general element of $\beta X$ is a limit (along an ultrafilter) of an infinite family of elements of $X$.

Ok, so what's wrong with say Hilbert Basis from this perspective? well the point is that they aren't really basis, in the sense that they don't generate the space "freely".

The universal property for Hilbert basis involve as you said restriction on the type of family of element we consider. The simplest way to put it is to work in the category of Hilbert spaces and isometric map and then if $(b_i)_{i \in I}$ is a Hilbert (orthonormale) basis of $\mathcal{B}$, then the data of a map $\mathcal{B} \to \mathcal{H}$ is the same as the choice of a familly of element of $t_i \in \mathcal{H}$ subject to the relation $\langle t_i,t_j\rangle = \delta_{i,j}$.

This is completely analogous to the universal property you'll get for a presentation of a vector space by generator and relation: it means that $\mathcal{B}$ is freely generated by the familly $b_i$ subject to the relation $\langle b_i,b_j\rangle = \delta_{i,j}$.

A more general notion you can use that encompass both is the notion of "generating familly": a familly of elements $b_i \in |X|$ (I'm using $|X|$ to denote the underlying set of $X$) is generating if for all object $Y$ the induced map

$$ Hom(X,Y) \to |Y|^I $$

is injective.

I'd say the definition you give is the best one. Things like Hilbert basis or Schauder basis aren't really basis, but more like special generating familly.

Note that your definition is exactly the local definition of "left adjoint to the forgetful functor set".

The problem with Hilbert basis, aren't the infinitary combinations - there are exemple with infinitary combination that are handled very well by this notion:

Take $C$ to be the category of suplattice (preordered set with arbitrary supremum) then the set of $\{x\}$ for $x \in X$ is a basis in your sense of the power set $\mathcal{P}(X)$. A general element of $\mathcal{P}(X)$ is a supremum of an arbitrary family of singletons.

Take $C$ to be the category of compact Hausdorff spaces, then for a set $X$, the principal ultrafilter $\delta_x$ for $x \in X$ form a "basis" of the Stone-Czech compactification (the space of all ultrafitler) of $\beta X$ of $X$. A general element of $\beta X$ is a limit (along an ultrafilter) of an infinite family of elements of $X$.

Ok, so what's wrong with say Hilbert Basis from this perspective? well the point is that they aren't really basis, in the sense that they don't generate the space "freely".

The universal property for Hilbert basis involve as you said restriction on the type of family of element we consider. The simplest way to put it is to work in the category of Hilbert spaces and isometric map and then if $(b_i)_{i \in I}$ is a Hilbert (orthonormale) basis of $\mathcal{B}$, then the data of a map $\mathcal{B} \to \mathcal{H}$ is the same as the choice of a familly of element of $t_i \in \mathcal{H}$ subject to the relation $\langle t_i,t_j\rangle = \delta_{i,j}$. (it is possible to express other universal property using different class of maps than isometric maps, but this one is the simplest by far)

This is completely analogous to the universal property you'll get for a presentation of a vector space by generator and relation: it means that $\mathcal{B}$ is freely generated by the familly $b_i$ subject to the relation $\langle b_i,b_j\rangle = \delta_{i,j}$.

A more general notion you can use that encompass both is the notion of "generating familly": a familly of elements $b_i \in |X|$ (I'm using $|X|$ to denote the underlying set of $X$) is generating if for all object $Y$ the induced map

$$ Hom(X,Y) \to |Y|^I $$

is injective.