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Let $C$ be locally small. Consider the Yoneda embedding $Y:C\rightarrow [C^{op},Set]$. Since limits in functor categories are computed pointwise and since the hom-functor preserves limits, the Yoneda embedding is limit-preserving. A natural question to ask then is, whether or when the Yoneda embedding has a left adjoint. Categories whose Yoneda embedding has a left adjoint are called total. Apparently the category of sets and the category of groups are both total.

What are the left adjoints of the Yoneda embedding (of these or other interesting examples) explicitely?

Margaret
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  • Allegedly locally small cocomplete categories with a dense subcategory are total. $\mathsf{Set}$ has the dense subcategory consisting of a singleton set of your choice. I'm not sure how to join the dots into an explicit adjoint, though – FShrike Sep 21 '23 at 12:38
  • A natural transformation $F(X)\to \hom_{\mathsf{Set}}(X,S)$ corresponds to a collection of maps $FX\times X\to S$ satisfying naturality conditions. So I think the left adjoint will be a colimit - the disjoint union of the $FX\times X$ modulo some relations. – Daniel Teixeira Sep 21 '23 at 12:47

1 Answers1

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Given a contravariant functor $F:\mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$, we need to find a set $Y$ with a universal natural transformation $F \to \mathrm{Hom}(-,Y)$.

Note that the contravariant functor $\mathrm{Hom}(-,Y)$ (treated as a functor $\mathbf{Set} \to \mathbf{Set}^{\mathrm{op}}$) is left adjoint to itself (treated as a functor $\mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$).

Let $Y=F(1)$ where $1$ is a singleton set. Then, for any set $X$, there is a natural map $F(X) \to \mathrm{Hom}(X,F(1))$ given by the adjunct of the map $X \cong \mathrm{Hom}(1,X) \to \mathrm{Hom}(F(X),F(1))$ under the above adjunction.

To see that this is universal, let $\eta:F \to \mathrm{Hom}(-,Z)$ be a natural transformation. Then, the induced map $F(1) \to Z$ is just $\eta_1$ (modulo the isomorphism $\mathrm{Hom}(1,Z) \cong Z$).

This shows that $\mathbf{Set}$ is total.

Geoffrey Trang
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    To be explicit, the map $F\to\mathrm{Hom}(-,F(1))$ takes $X$ and $y\in F(X)$ to the function $X\to F(1)$ which maps $x\mapsto F(\overline{x})(y)$ where $\overline{x}:1\to X$ is the point function with image $x$ – FShrike Sep 21 '23 at 15:16
  • More conceptually, cocones with vertex $Z$ under a digram $J\colon\mathcal D\to\mathcal C$ weighted by $F\colon\mathcal D^{op}\to\mathcal C$ are natural transformations $F\Rightarrow\mathcal C(J^{op}-,Z)$. The $F$-weighted colimit of $J$ is $Jd$ if $F\cong\mathcal D(-,d)$. But $F\Rightarrow\mathcal Set(J^{op}-,Z)$ are in natural bijection with $J\Rightarrow\mathcal Set(F^{op}-,Z)$. So the $F$-weighted colimit of $J$ is also $Fd$ if $J\cong\mathcal D(d,-)$. Cototality of $\mathcal Set$ follows from $\mathrm{id}_{Set}=J\cong\mathcal Set({*},-)$. – Vladimir Sotirov Sep 23 '23 at 15:08