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?, 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.
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