202507191607

Tags : Category Theory

A functor admits a left adjoint iff all its comma categories have an initial object

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Theorem

A functor $U:A\to S$ admits a left adjoint iff for each $s\in S$, the comma category $s\downarrow U$ has an initial object.

The comma category $s\downarrow U$ is isomorphic to the Element Category for the functor $S(s,U-):A\to \text{Set}$. If a left adjoint $F$ exists, then the component of the unit as $s$ defines an initial object ${\eta }_s:s\to UFs$ in this category.

Conversely if $s\downarrow U$ admits an initial object, which we suggestively denote by ${\eta }_s:s\to UFs$. This defines the value of a functor $\text{ob }S\to \text{ob }A$, which we can extend to a functor using Unique construction of Adjunction to the functor that we call $F$.

Since ${\eta }_s$ is the initial object in $s\downarrow U$, this implies the existence and uniqueness of such a map. This also gives the unit natural transformation.

This allows to to define the natural transformation $\varphi :A(F-,-)\Rightarrow S(-,U-)$ with components

$${\varphi }_{s,a}:A(Fs,A)\Rightarrow S(s,Ua)$$

as in Equivalent Definitions of Adjuctions: given a map $g:Fs\to a$ in $A$, define

$${\varphi }_{s,a}(g):=s\stackrel{{\eta }_s}{\to }UFs\stackrel{Ug}{\to }Ua$$

Injectivity and surjectivity of ${\varphi }_{s,a}$ follow from uniqueness and existence for the morphism ${\eta }_s$ to any particular object $s\to Ua$ in $s\downarrow U$. This proves the adjunction.

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References

• Element Category
• Adjunctions
• Comma Category
• Equivalent Definitions of Adjuctions
• Unique construction of Adjunction