Wednesday, August 13, 2014

Lex total categories and Grothendieck toposes II

Next post in this series.

I said in the first post in this series that totally complete categories have a strong adjoint functor theorem.

Here is the adjoint functor theorem: if $A$ and $B$ are locally small categories, $A$ is totally cocomplete and $F: A\to B$ is a functor which preserves colimits of discrete fibrations with small fibres then $F$ has a right adjoint.

Before indicating a proof of this result let us look at the adjoint functor theorem for posets.
If $F:A\to B $ is an order preserving function, $A$ is a complete poset and $F$ preserves sups then $F$ has a right adjoint $U$ with $U(b)=sup_{F(a)\leq b}(a)$.
proof Certainly $U$ so defined has the property that if $F(a)\leq b$ then $a\leq U(b)$. But also since $F$ preserves $sup$ then $FU(b)=F(sup_{F(a)\leq b}(a))=sup_{F(a)\leq b}F(a)\leq b$ and hence if $a\leq U(b)$ then $F(a)\leq FU(b)\leq b$. Hence $F(a)\leq b$ if and only if $a\leq U(b)$. Notice that we used preservation of sups only for the sups in the definition of $U$.

There is a straightforward extension to categories of this result, namely a functor $F:A\to B$ has a right adjoint if and only if for each $b\in B$ the colimit $colimit_{Fa\to b}a$ exists and is preserved by $F$. I have been imprecise about the diagram of the colimit - we will see a definition below.  Let's use this result to prove the adjoint functor theorem.

What we need to show is if $A$, $B$ are locally small and $A$ is totally complete then $colimit_{Fa\to b}a$ is the colimit of a discrete fibration with small fibres.

Notice that the fact that $A$ is totally complete suggest a possible adjoint to $F$, namely
where by $PA$ I mean $Sets^{A^{op}}$.
But $PF(yoneda_B(b))=B(F(-),b))$ so the suggested formula for $U(b)$ is the colimit of the discrete fibration $Elements(B(F(-),b))\to A$ corresponding to the functor $B(F(-),b):A^{op}\to Sets$.

What is this discrete fibration? Over $a\in A$ the fibre is $B(Fa,b)$, that is consists of arrows $Fa\to b$. Over $\alpha:a_1\to a_2$ there are arrows $(f:Fa_1\to b)\to (g:Fa_2\to b)$ exactly when $g F(\alpha) =f$. But this is exactly the diagram I intended for $colim_{Fa\to b}(a)=U(b)$. Certainly this colimit exists and we have the theorem.

I am busy today, so I will give examples of the use of this theorem in producing constructions in lex total categories next time.



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