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Stream: theory: category theory

Topic: Are dense functors final?

James Deikun (Nov 24 2025 at 20:18):

This seems to pass a basic smell test (for $j : A \to C$ dense, the colimit of $j$ is the same as the colimit of $\mathrm{id}$), but it seems it's not always true.

Let $j$ be the inclusion of the non-coapex objects in the walking cospan. This is dense as is easily checked (3 cases, 2 of which are the same). However, it is clearly not final, as the colimit of a restricted functor is the coproduct of the 2 non-coapicial values, and the colimit of the unrestricted functor is the value at the coapex.

fosco (Nov 25 2025 at 09:28):

The domain of $j$ is discrete (in case you're defining it as $\Delta^{\{0\}}+\Delta^{\{1\}} \hookrightarrow \Lambda^2_2$), and a functor with discrete domain is final if and only if it's a right adjoint.

fosco (Nov 25 2025 at 09:42):

...and I think one can just do case analysis to disprove the fact that $j$ has a left adjoint.

fosco (Nov 25 2025 at 09:44):

(is there a quicker way than just arguing that $\hom(2,jy)=\emptyset$ can't be of the form $\hom(L2,y)$, whatever value $L\dashv j$ has on objects?)

Tom Hirschowitz (Nov 27 2025 at 08:58):

But in this case, the characterisation by connectedness of comma cats seems to readily apply, no? The comma $a/j$ (where $a$ denotes the apex) is empty, hence not connected.