nLab absolutely dense functor

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Contents

Definition

An absolutely dense functor F:ABF \colon A \to B is a functor which is equivalently characterised by the following conditions:

  1. FF is dense and Lan FFLan_F F is an absolute Kan extension.

  2. F *=[F op,Set]:[B op,Set][A op,Set]F^* = [F^{op}, Set] \colon [B^{op}, Set] \to [A^{op}, Set] is fully faithful.

  3. The counit of the adjunction F *F *:[B op,Set][A op,Set]F_* \dashv F^* : [B^{op}, Set] \to [A^{op}, Set] is invertible.

  4. FF is corepresentably fully faithful (also called a lax epimorphism?) in Cat.

  5. For every functor G:B op×BCG : B^{op} \times B \to C, there is a canonical isomorphism (MathOverflow answer)

    bBG(b,b) aAG(Fa,Fa) \int_{b \in B} G(b, b) \cong \int_{a \in A} G(F a, F a)

    (This is a notion of initiality for ends.)

More generally, an absolutely dense morphism in a proarrow equipment KMK \to M is a 1-cell f:abf : a \to b in KK for which the counit f *f *1 bf_* \circ f^* \cong 1_b.

Properties

Every simultaneously reflective and coreflective subcategory of a presheaf category is itself a presheaf category and is induced by precomposition along an absolutely dense functor: this is the main result of the paper of El Bashir–Velebil below.

References

(Absolutely dense functors are called Cauchy dense in the following paper of Brian Day.)

  • Brian Day. Density presentations of functors. Bulletin of the Australian Mathematical Society 16.3 (1977): 427-448.

  • Jiri Adamek, Robert El Bashir?, Manuela Sobral, Jiri Velebil?. On functors which are lax epimorphisms. TAC. (2001)

  • Robert El Bashir and Jiri Velebil?. Simultaneously reflective and coreflective subcategories of presheaves. Theory and Applications of Categories 10.16 (2002): 410-423.

  • Fernando Lucatelli Nunes and Sousa Lurdes?. On lax epimorphisms and the associated factorization. Journal of Pure and Applied Algebra 226.12 (2022)

Last revised on March 10, 2023 at 06:48:06. See the history of this page for a list of all contributions to it.