A general theory of Kan extensions

Question

In these days I am studying some properties of Kan extensions. In order to do so I am looking at some general observations on Kan extensions that I would like to share with the community. A feedback would be very important to me.

Give a look at the following diagram to set the notation.

$\require{AMScd}$ \begin{CD} A @>f>> C\\ @VgVV \\ B \end{CD}

I will suppose that $g$ is fully faithful, that $C$ is cocomplete and that the kan extension is pointwise.

My interest is to understand how properties of $f$ interact with properties of Lan$_g f.$

To do so, I would start by giving a factorization of the Kan extension in the following way:

$$B \stackrel{\text{Elts}^g}{\to} \text{Fib}(A) \stackrel{f_*}{\to} \text{Fib}(C) \stackrel{\text{colim}}{\to} C.$$

Let's give a description of these functors.

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Elts$^g$: $B \to \text{Fib}(A)$

By e $\text{Fib}(A)$ we mean the category of fibrations over $A$. This functor takes an object $b \in B$ into the category Elts($B(g\_, b)$).

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The functor $f^*$ is just composition. So we map the fibration $E \to A$ to $E \to A \stackrel{f}{\to} C.$

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Colim is just the functor that takes the colimit of the diagram induced by the fibration.

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Under these notations it looks to me (please confirm, I am following the construction presented by Borceux (3.7.4, HoCA vol II)) that one can writhe down the equation: $$\text{Lan}_gf = \text{colim} \circ f_* \circ \text{Elts}^g. $$

Here come my questions:

What do we know about the functors Elts$^g$ and colim? More precisely,

Q1 Are they faithful or conservative?

Q2 What (co)limits do they preserve?

As a final remark, I would point out that any reference about Kan extensions which goes deeply in their properties and their behave is absolutely welcome.

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Motivations:

My motivating questions are the following ones:

In the setting of the question, under what assumptions can I hope that $f$ conservative implies that Lan$_g f$ is conservative?

What about faithfulness?

Again, any reference for an answer to these questions is absolutely welcome.

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These questions are not so desperate ad they may appear. If $B = \text{Set}^{A^{\text{op}}}$ and $g$ is the yoneda embedding, then the functor Elts$^y$ should be faithful and conservative because $$\text{PseudoPres}(A) \cong \text{Fib}(A) $$ precisely under that functor.

Answer 1

$ \def\colim{\text{colim}} \def\A{\mathcal{A}} \def\coker{\text{coker }} $ There is a blatant example in which $\colim : \A^J \to \A$ is not conservative; take a pushout square in an abelian category. Then, the canonical comparison map $$ \begin{CD} X @>f>> Y @>>> \coker f\\ @VuVV @VVvV @VVV\\ X'@>>g> Y' @>>> \coker g \end{CD} $$ between cokernels is invertible whatever $u,v$ are. (This appears as Lemma 1.2, p. 86 of Hilton-Stammbach's Course in Homological Algbera, but I bet there are better references and the proof is very easily done by hand).

So, for $J=\{* \rightrightarrows \bullet\}$ the $\colim$ functor is (I'd say, highly and irredemably) not conservative.

As you maybe remember, a necessary and sufficient condition for a functor admitting an adjoint, so that it is (faithful and) conservative is that

• it is a right adjoint, and the components of the counit are extremal epis (here a neat proof), or
• it is a left adjoint, and the components of the unit are extremal monos.

Of course here $\colim\dashv \Delta_J$, the constant functor $\A \to \A^J$, and the unit components at $P\in \A^J$ are precisely the arrows of the initial cocone for $P$.

Somehow this is telling you two things, that $\colim$ is conservative iff each colimit inclusion is in fact an honest inclusion, and that sometimes (in fact, quite often, if you like pointed categories) $\colim$ won't be conservative because some maps in the initial cocone can be the zero morphism.

More than often, however, colimits are honest inclusions; I'd check what happens in a topos (where each $X\to \varnothing$ must be an iso), for example.