# When is the property of being a dense functor cancellable?

Let $$\mathcal C \to \mathcal D \to \mathcal E$$ be a pair of functors. Assume that $$\mathcal C$$ is dense in $$\mathcal E$$. Is then $$\mathcal D$$ also dense in $$\mathcal E$$? One definition of denseness is that the nerve, that is the restricted Yoneda embedding $$\mathcal E \to \mathrm{PSh}(\mathcal C)$$ is fully faithful. We want that the same holds with $$\mathcal D$$ instead of $$\mathcal C$$. Everything fits into a commutative diagram

$$\require{AMScd} \begin{CD} \mathcal E @>>> \mathrm{PSh}(\mathcal D)\\ @. {_{}\style{display: inline-block; transform: rotate(30deg)}{{\xrightarrow[\rule{4em}{0em}]{}}}} @VVV\\ @. \mathrm{PSh}(\mathcal C), \end{CD}$$

where the vertical functor is the restriction. Obviously the diagonal functor is faithful because the top is faithful but from there I would need something like the restriction being faithful for example, which is not always the case. So is there a counterexample to the question or some (reference to a) proof?

• To be clear, you take the composite $C\to E$ to be a functor realising $C$ as dense in $E$, and want to know if this implies $D\to E$ realises $D$ as dense in $E$? Commented Jun 28, 2023 at 1:36
• @FShrike Correct. If you mean that the functors are inclusions, then this would be sufficient for me. Or you can just take the definition of a dense functor which doesn‘t necessarily require subcategories as in ncatlab.org/nlab/show/dense+functor.
– HDB
Commented Jun 28, 2023 at 9:06

A useful reference for cancellability of dense functors is §5.2 of Kelly's Basic Concepts of Enriched Category Theory. For instance, in your example, if $$\mathcal C \to \mathcal D$$ is essentially surjective, or if $$\mathcal D \to \mathcal E$$ is fully faithful, then density of $$\mathcal C \to \mathcal D \to \mathcal E$$ implies density of $$\mathcal D \to \mathcal E$$ (Proposition 5.11 and Theorem 5.13 ibid.).