category of filtered vector spaces

  • notation: FiltVectK\FiltVect_K
  • objects: A filtered vector space over a field KK consists of a vector space VV over KK together with an Z\IZ-indexed sequence of subspaces Fn(V)⊆VF^n(V) \subseteq V satisfying Fn+1(V)⊆Fn(V)F^{n+1}(V) \subseteq F^n(V) for all n∈Zn \in \IZ. ⋯⊆F1(V)⊆F0(V)⊆F−1(V)⊆⋯⊆V\cdots \subseteq F^1(V) \subseteq F^0(V) \subseteq F^{-1}(V) \subseteq \cdots \subseteq V
  • morphisms: A morphism (V,F)→(W,G)(V,F) \to (W,G) is a linear map f:V→Wf : V \to W satisfying f∗(Fn(V))⊆Gn(W)f_*(F^n(V)) \subseteq G^n(W) for all n∈Zn \in \IZ. We call such morphisms filtered linear maps.
  • Related categories: VectK\Vect_K
  • nLab Link

The definition of a filtered object is available in any abelian category; in this entry, the ambient category is VectK\Vect_K. We do not require the filtration to be exhaustive (i.e. V=⋃nFn(V)V = \bigcup_n F^n(V)) or separated (i.e. ⋂nFn(V)=0\bigcap_n F^n(V) = 0). In the proofs below, we will often write FF for the filtration regardless of the vector space on which it is defined.

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: the trivial vector space equipped with the unique filtration
  • initial object: the trivial vector space equipped with the unique filtration
  • products: direct products of the underlying vector spaces equipped with the filtration Fn(∏iVi)≔∏iFn(Vi)F^n(\prod_i V_i) \coloneqq \prod_i F^n(V_i)
  • coproducts: direct sums of the underlying vector spaces equipped with the filtration Fn(⨁iVi)≔⨁iFn(Vi)F^n(\bigoplus_i V_i) \coloneqq \bigoplus_i F^n(V_i)

Special morphisms

  • isomorphisms: A filtered linear map f:(V,F)→(W,G)f : (V,F) \to (W,G) is an isomorphism if f:V→Wf : V \to W is bijective and f∗(Fn(V))=Gn(W)f_*(F^n(V)) = G^n(W) for all n∈Zn \in \IZ.
  • monomorphisms: injective filtered linear maps
  • epimorphisms: surjective filtered linear maps
  • regular monomorphisms: A filtered linear map f:(V,F)→(W,G)f : (V,F) \to (W,G) is a regular monomorphism if and only if f:V→Wf : V \to W is injective and Fn(V)=f∗(Gn(W))F^n(V) = f^*(G^n(W)) for all n∈Zn \in \IZ.
  • regular epimorphisms: A filtered linear map f:(V,F)→(W,G)f : (V,F) \to (W,G) is a regular epimorphism if and only if f:V→Wf : V \to W is surjective and Gn(W)=f∗(Fn(V))G^n(W) = f_*(F^n(V)) for all n∈Zn \in \IZ.

Undistinguishable categories

These categories in the database currently have exactly the same properties as the category of filtered vector spaces. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.

Comments

  • The forgetful functor U:FiltVect→VectU : \FiltVect \to \Vect has a left adjoint that equips a vector space with the trivial filtration. It also has a right adjoint that equips a vector space with the maximal filtration. In particular, it follows that UU preserves limits and colimits.
  • The filtration functor Fn:FiltVect→VectF^n : \FiltVect \to \Vect has a left adjoint that equips a vector space VV with the filtration Fn(V)=VF^n(V)=V and Fn+1(V)=0F^{n+1}(V)=0. In particular, it follows that FnF^n preserves limits. However, it does not have a right adjoint. In fact, FnF^n does not preserve coequalizers. On the other hand, FnF^n does preserve filtered colimits; this can be verified using the concrete construction of filtered colimits.