Can the functor to empty sets be defined?

Question

Can these functors from C to Sets be defined?

1. Functor $F$ from one object $A$ and its identity arrow to a empty set

• $F(A)$ = (empty set)
• $F(id_A)$ = (function from empty set to empty set)

2. Functor G from two objects $A$, $B$ and their identity arrows to a one-element set and a empty set, respectively

• $G(A)$ = $\{*\}$ (one element set)
• $G(B)$ = (empty set)
• $G(id_A)$ = $id_{\{*\}}$
• $G(id_B)$ = (function from empty set to empty set)

I think the latter appears when thinking of Yoneda embedding of the category which has two objects and their identity arrows, is it right?

Answer 1

The question is a little ill-formed as it stands, because whether it is possible to define functors $\mathcal C \to \mathbf{Set}$ that have the properties you describe depends on the category $\mathcal C$. However, there are certainly some choices of $\mathcal C$ and functors $F \colon \mathcal C \to \mathbf{Set}$ satisfying these properties.

For the first example, let $\mathcal C$ be a discrete one-object category containing an object $A$. Then the assignment sending $A \mapsto \varnothing$ and $\mathrm{id}_A \mapsto \mathrm{id}_\varnothing$ defines a functor.

For the second example, let $\mathcal C$ be a two-object discrete category having objects $A$ and $B$. Then the assignment sending $A \mapsto \{ * \}$, $\mathrm{id}_A \mapsto \mathrm{id}_{\{ * \}}$, $B \mapsto \varnothing$ and $\mathrm{id}_B \mapsto \mathrm{id}_\varnothing$ defines a functor.

In both cases, we can verify that these define functors by checking that they preserve identities and composites: preservation of identities follows by definition, whereas preservation of composites is trivial, since there are no non-identity morphisms in $\mathcal C$.

However, note that there are some choices of $\mathcal C$ for which we may not define a functor $F \colon \mathcal C \to \mathbf{Set}$ along these lines. For instance, if we take $\mathcal C$ to be a category with two objects $A$ and $B$, and a morphism $f \colon A \to B$ (together with identity morphisms), then there is no functor $F \colon \mathcal C \to \mathbf{Set}$ satisfying the properties of the second example, because that would require us to specify a function from $\{ * \}$ to $\varnothing$, of which there are none.

In summary, we can always define functions (and hence functors) that send some element to the empty set, e.g. there is a (unique) function $\{ * \} \to \{ \varnothing \}$. However, we cannot define non-trivial functions to the empty set, e.g. there is no function $\{ * \} \to \varnothing$. When we define a functor into $\mathbf{Set}$, we are doing something similar to the former, so there is no issue, as long as we can define an action of the functor on morphisms.