nLab absolutely dense functor

Definition

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

1. $F$ is dense and $Lan_F F$ is an absolute Kan extension.

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

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

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

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

   $$\int_{b \in B} G(b, b) \cong \int_{a \in A} G(F a, F a)$$

   (This is a notion of initiality for ends.)

6. For every morphism $f : b \to b'$ in $B$, the category of $F$-factorisations of $f$ (which has objects triples $(a \in A, f_1 : b \to f a, f_2 : f a \to b')$ such that $f_2 f_1 = f$) is connected. (MathOverflow answer)

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

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, Jiří Velebil?. On functors which are lax epimorphisms. TAC. (2001)

• Robert El Bashir and Jiří 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)

• Adrián Doña Mateo, Cauchy density, arXiv:2507.07869 (2025).