Kan Construction, Profunctors, and Kan Extensions – ct.category-theory,profunctors,kan-extension

ct.category-theorykan-extensionprofunctors

It's been a long time since I tried to understand the deep meaning of the "Kan construction", or "nerve-realization" adjunction
$$
\text{Lan}_y F \dashv N_F = \hom(F,1)
$$
that exists among the left Kan extension of $F\colon \mathcal{A}\to \mathbf{D}$ ($\cal A$ small, $\bf D$ cocomplete) along the Yoneda embedding $y\colon {\cal A} \to \hat{\cal A}$. It can be expressed as
$$
\text{Lan}_y F \dashv \text{Lan}_F y,
$$
and this property seems pretty peculiar; especially if I think about the definition of a "Yoneda structure"[here, several comments in the discussion are mine].

  1. Is there a reason why this is true?
  2. What are other examples of a span of functors ${\bf C} \xleftarrow{G} {\cal A} \xrightarrow{F} {\bf B}$ such that $\text{Lan}_GF\dashv \text{Lan}_FG$?
  3. My sensation is that this question acquires a (more?) meaning plunging the 2-category $\bf Cat$ in $\bf Prof$ in the usual way. The functor $N_F = \hom(F,1)$ is the image of $F$ via the canonical 2-functor $\varphi^{(-)} : {\bf Cat}^\text{co} \to \bf Prof$, and $N_F$ has a right adjoint $\hom(1,F)$; the Kan construction amounts to say that this extends to a triple of adjoints
    $$
    \text{Lan}_yF = \varphi_F^! \dashv \varphi^F\dashv \varphi_F.
    $$
    What's the meaning of this extension, and its universal property, in $\bf Prof$?

Best Answer

I will try to answer the second question.

Prop 1. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}_if$ is pointwise. Then, the following are equivalent.

  1. $\text{lan}_if \dashv \text{lan}_fi$.
  2. $f$ is the $i$-relative left adjoint of $\text{lan}_fi$, i.e. ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, - ).$
  3. $f = \text{lift}_{\text{lan}_fi}i$ and the lift is absolute.

If $i$ is only fully faithful $1 \Rightarrow 2$, if $i$ is only dense $2 \Rightarrow 1$.

Proof.

$1 \Rightarrow 2$) $${\bf B}(f, -) \stackrel{i \text{ is ff.}}{\cong} {\bf B}((\text{lan}_if) i, -) \stackrel{1}{\cong} {\bf C}(i, \text{lan}_fi).$$

$2 \Rightarrow 1$) $${\bf B}(\text{lan}_if, -) \stackrel{\text{point.}}{\cong} \text{lan}_i{\bf B}(f, -) \stackrel{2}{\cong} \text{lan}_i{\bf C}(i, \text{lan}_fi) \stackrel{\text{point.}}{\cong} {\bf C}(\text{lan}_ii, \text{lan}_fi) \stackrel{i \text{ is dense}}{\cong} {\bf C}(-, \text{lan}_fi).$$

$3$ is just a rewriting of $2$.

Now we study a very special setting.

Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. Moreover $\text{lan}_if$ is pointwise, ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are cocomplete.

In this setting ${\bf C}$ is a reflective subcategory $ V: {\bf C} \leftrightarrows \text{Set}^{{\bf A}^\circ} : L $ of $\text{Set}^{{\bf A}^\circ}$ via the nerve $V = \text{lan}_i(y_{{\bf A}})$ (V is the right adjoint).

Prop 2. Let ${\bf C} \xleftarrow{i} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $i$ is dense and fully faithful. ${\bf A}$ is small, ${\bf C}$ and ${\bf B}$ are cocomplete. Then, the following are equivalent.

  1. $\text{lan}_if \dashv \text{lan}_fi$.
  2. V preserves $\text{lan}_fi$.
  3. ${\bf C}(i,\text{lan}_fi) \cong \text{lan}_f{\bf C}(i,i)$.

Proof. Recall that $\text{lan}_if$ is pointwise,because ${\bf B}$ is cocomplete.

$1 \Rightarrow 2)$.

Using Prop 1. we know that ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, - )$. Since the presheaf construction is a Yoneda structure, we have that $\text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong {\bf B}(f, -)$.

Thus, $$ {\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, -) \cong \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fVi)$$

Observe also that $${\bf C}(i, \text{lan}_fi) \cong {\bf C}(Ly_A, \text{lan}_fi) \stackrel{L \dashv V}{\cong} \text{Set}^{{\bf A}^\circ}(y_A, V\text{lan}_fi),$$ putting the last two equation together, one gets: $$\text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fVi) \cong \text{Set}^{{\bf A}^\circ}(y_A, V\text{lan}_fi). $$ By Yoneda Lemma the two functors on the right have to coincide.

$2 \Rightarrow 1).$

Using Prop 1. it is enough to prove that ${\bf C}(i, \text{lan}_fi) \cong {\bf B}(f, - )$. Since the presheaf construction is a Yoneda structure, we have that $\text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong {\bf B}(f, -)$. Now, $${\bf C}(i, \text{lan}_fi) \cong {\bf C}(Ly_A, \text{lan}_fi) \stackrel{L \dashv V}{\cong} \text{Set}^{{\bf A}^\circ}(y_A, V\text{lan}_fi) \stackrel{2}{\cong} \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fVi) \cong \text{Set}^{{\bf A}^\circ}(y_A, \text{lan}_fy_A) \cong {\bf B}(f, -). $$

$3$ is just a rewriting of $2$.

Cor. 1 (Quite surprising). Let ${\bf Set}^{A^\circ} \xleftarrow{y} {\bf A} \xrightarrow{f} {\bf B}$ be a span where $y$ is the Yoneda embedding of a small category and ${\bf B}$ is cocomplete. Then $\text{lan}_yf \dashv \text{lan}_fy$.

Proof 1. In Prop 1 the condition 2 is verified and is essentially a rewriting of the Yoneda Lemma. Moreover $\text{lan}_yf$ is pointwise because ${\bf B}$ is cocomplete.

Proof 2. In Prop 2 the condition 2 is trivially verified because $V$ is the identity.

I shall conclude by saying that the adjunction cannot verify too often.

Rem. 4 Let $y: {\bf A} \to {\bf C}$ be a dense and fully faithful functor. Then the following are equivalent.

  1. For every map $f: {\bf A} \to {\bf B}$, where ${\bf B}$ is cocomplete, the Kan extensions $\text{lan}_yf$ and $\text{lan}_fy$ exist and are adjoint.
  2. $y$ is the Yoneda embedding.

Proof. One implications is Cor 1. The other implication can be read as follows, if 1 is verified, then for every functor $f: {\bf A} \to {\bf B}$, there is a unique cocontinous extension ($\text{lan}_yf$), this is the characterization of the free cocompletion under colimits, that is the (small) presheaf construction.

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