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There are lots. +\geometry{a4paper} % ... or a4paper or a5paper or ... +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{epstopdf} +\usepackage{mdframed} +\usepackage{hyperref} +\usepackage{diagrams} + +\newmdtheoremenv{defn}{Definition} +\newmdtheoremenv{thm}{Theorem} + +\title{Motivation for the General Adjoint Functor Theorem} +\author{Patrick Stevens} + +\begin{document} + +\maketitle +\tiny \begin{center} \url{http://www.patrickstevens.co.uk/misc/AdjointFunctorTheorems/AdjointFunctorTheorems.pdf} \end{center} + +\normalsize +\emph{You should draw diagrams yourself throughout this document. It will be unreadable as mere symbols.} + +\section{Primeval Adjoint Functor Theorem} + +Recall the theorem (``RAPL'') that Right Adjoints Preserve Limits. + +The PAFT is an attempt at a converse to this theorem. +It states that if $G: \mathcal{D} \to \mathcal{C}$ preserves limits, and $\mathcal{D}$ is small and complete, then $G$ is a right adjoint. + +The problem with the PAFT is that it's actually a very weak theorem: small complete categories are preorders. + +So we want to weaken that requirement of smallness and completeness. +How do we usually weaken smallness? +The next best thing is local smallness, but we're going to lose something if we just replace ``small'' by ``locally small''. + +\section{General Adjoint Functor Theorem} +The question is therefore, ``what requirement do we need to add to augment local smallness?'' + +Recall the most ``concrete'' definition of a left adjoint to $G: \mathcal{D} \to \mathcal{C}$: it is a specification of $F: \mathcal{C} \to \mathcal{D}$ together with $\eta_A: A \to GFA$ for each $A$, such that any $g: A \to GB$ has a unique $h: FA \to B$ with $(Gh) \circ \eta_A = g$. + +\newarrow {Dashto} {}{dash}{}{dash}> + +\begin{diagram} +FA & \rDashto^h & B \\ +GFA & \rTo^{Gh} & GB \\ +\uTo^{\eta_A} & \ruTo~f & {} \\ +A & {} & {} +\end{diagram} + +This definition is how I remember what an adjoint is, and it very closely parallels the UMP of the free group on a given set. + +Now, by TAMO\footnote{\url{http://star.psy.ohio-state.edu/coglab/Miracle.html}}, I define the following Tiny Set Condition: + +\ + +\begin{defn}[Tiny Set Condition] A functor $G: \mathcal{D} \to \mathcal{C}$ has the TSC iff for every object $A \in \mathcal{C}$, there is $B \in \mathcal{D}$ and $\eta_A : A \to GB$ such that every $g: A \to GX$ factors as $A \xrightarrow[\eta_A]{} GB \xrightarrow[Gh]{} GX$ for some $h: B \to X$. + +\begin{diagram} +B & \rTo^h & X \\ +GB & \rTo^{Gh} & GX \\ +\uTo^{\eta_A} & \ruTo~g & {} \\ +A & {} & {} +\end{diagram} + +\end{defn} + +\ + +Notice how closely this mirrors the definition of an adjoint: it is a very slight weakening of my favourite definition, in that we don't require $h$ to be unique. +In particular, using that definition it is a completely content-free statement that if $G$ has a left adjoint, then it satisfies the TSC: simply take $\eta_A$ to be the unit of the adjunction. + +We are assuming access to RAPL, so this gives the following theorem: + +\ + +\begin{thm} A functor $G: \mathcal{D} \to \mathcal{C}$, which has a left adjoint, must satisfy the TSC and must preserve small limits. +\end{thm} + +\ + +\subsection{Relation to the GAFT} + +\emph{This is an overview of where we are heading.} + +The GAFT talks about what happens if we replace the TSC with a weaker condition: + +\ + +\begin{defn}[Solution Set Condition] A functor $G: \mathcal{D} \to \mathcal{C}$ has the SSC iff for every object $A \in \mathcal{C}$, there is a set $\{ B_i \in \mathcal{D} : i \in I \}$ and $\{ \eta_A^i : A \to G B_i : i \in I \}$ such that all $f: A \to GB$ factor as some $A \xrightarrow[\eta_A^i]{} G B_i \xrightarrow[Gh]{} GB$. +\end{defn} + +\ + +That is, we relax the uniqueness of $B$ in the statement of the TSC. + +Then the GAFT states that: + +\ + +\begin{thm}[General adjoint functor theorem] A functor $G: \mathcal{D} \to \mathcal{C}$ (where $\mathcal{D}$ is complete and locally small) has a left adjoint iff it satisfies the SSC and preserves small limits. +\end{thm} + +\ + +\section{Converse to Theorem 1} + +We wish to show the following: + +\ + +\begin{thm}[Tiny Adjoint Functor Theorem] If $G: \mathcal{D} \to \mathcal{C}$ has the TSC and preserves small limits, and $\mathcal{D}$ is complete and locally small, then $G$ has a left adjoint. +\end{thm} + +\ + +This will give us a slightly more general version of the PAFT, because we've relaxed smallness of $\mathcal{D}$ and still given an equivalent condition for being an adjoint. + +To be clear, what we have done is taken the PAFT, replaced ``small'' with ``locally small'', and imposed the TSC. +The TSC is ostensibly only a slight weakening of the definition of an adjoint, so it's not too much to ask that they would be equivalent. + +I will write $FA$ for the object $B$ guaranteed by the TSC; this anticipates the definition of $F$ as a functor. + +\subsection{Proof} +Recall the theorem that a specification of a left adjoint is equivalent to a specification of an initial object of $(A \downarrow G)$ for each object $A \in \mathcal{C}$. +There's one obvious choice for such an initial object: $(FA, \eta_A)$. +(Note for the future that this might not actually be initial, but it's the obvious choice.) +Is this actually initial? + +\begin{diagram} +A & {} & {} \\ +\dTo^{\eta_A} & \rdTo^f \\ +GFA & \rDashto & GX \\ +\end{diagram} + +Well, it certainly has an arrow from it into any other $(X \in \mathcal{D}, f: A \to GX)$, because that's just the statement of the TSC: any arrow $f: A \to GX$ factors through the map $\eta_A : A \to GFA$. + +Is that arrow unique? +Well, OK, maybe it isn't. The TSC didn't tell us much, after all. +But we are in a complete and locally small category, so what we can do is equalise out all the ways in which the arrow fails to be unique. + +\subsubsection{Try and make the arrow unique} +Say $\{ G(h_i): i \in I \}$ is the set of distinct arrows $GFA \to GX$ with $G(h_i) \eta_A = f$. + +\begin{diagram} +A & {} & {} \\ +\dTo^{\eta_A} & \rdTo^f & {} \\ +GFA & \pile{\rTo^{G(h_1)} \\ \rTo_{G(h_2)}} & GX \\ +\end{diagram} + +Let $E$ be the ``industrial-strength equaliser'' of the $h_i$ in $\mathcal{D}$: an arrow $e: E \to FA$ such that $h_i e = h_j e$ for all $i, j$. + +Since $G$ preserves limits, $Ge: GE \to GFA$ must be an industrial-strength equaliser of the $G(h_i)$. + +\begin{diagram} +{} & {} & A & {} & {} \\ +{} & {} & \dTo^{\eta_A} & \rdTo^f & {} \\ +GE & \rTo_{Ge} & GFA & \pile{\rTo^{G(h_1)} \\ \rTo_{G(h_2)}} & GX \\ +\end{diagram} + +Since $\eta_A$ equalises all the $G(h_i)$, it must lift uniquely over $Ge$: say $\eta_A = G(e) \circ \overline{\eta_A}$. + +\begin{diagram} +{} & {} & A & {} & {} \\ +{} & \ldTo^{\overline{\eta_A}} & \dTo^{\eta_A} & \rdTo^f & {} \\ +GE & \rTo_{Ge} & GFA & \pile{\rTo^{G(h_1)} \\ \rTo_{G(h_2)}} & GX \\ +\end{diagram} + +Now, in our dream world, the $G(h_i)$ would all be equal: that is, $Ge$ would be an iso. +Can we find an inverse? +The TSC tells us that there is an arrow $G(\gamma) : GFA \to GE$ such that $G(\gamma) \eta_A = \overline{\eta_A}$. +That is, $G(e) G(\gamma) \eta_A = \eta_A$. + +\begin{diagram} +{} & {} & A & {} & {} \\ +{} & \ldTo^{\overline{\eta_A}} & \dTo^{\eta_A} & \rdTo^f & {} \\ +GE & \pile{\rTo^{Ge} \\ \lTo_{G(\gamma)}} & GFA & \pile{\rTo^{G(h_1)} \\ \rTo_{G(h_2)}} & GX \\ +\end{diagram} + +This is as far as we can go, because of the weakness of the TSC, telling us nothing about the arrows whose existence it guarantees. + +\subsubsection{Try and produce an object which works} +We see that $G(e) G(\gamma)$ is a map $GFA \to GFA$, and we wanted it to be the identity. + +What we could do is equalise out all the maps $GFA \to GFA$, and that would tell us that $\eta_A$ lifted over the equaliser. +This would produce an object which we really hope would actually have the uniqueness property. + +Let $G(i): GI \to GFA$ be the industrial strength equaliser of all maps $r_i: FA \to FA$ such that $G(r_i) \eta_A = \eta_A$. + +\begin{diagram} +{} & {} & A & {} & {} \\ +{} & {} & \dTo_{\eta_A} & \rdTo^{\eta_A} & {} \\ +GI & \rTo_{G(i)} & GFA & \pile{\rTo^{G(r_1)} \\ \rTo_{G(r_2)}} & GFA\\ +\end{diagram} + +Now $G(e \gamma) \eta_A = \eta_A$, so $\eta_A$ lifts over $Gi$ to $\eta_A': A \to GI$. + +\begin{diagram} +{} & {} & A & {} & {} \\ +{} & \ldTo^{\eta_A'} & \dTo_{\eta_A} & \rdTo^{\eta_A} & {} \\ +GI & \rTo_{G(i)} & GFA & \pile{\rTo^{G(r_1)} \\ \rTo_{G(r_2)}} & GFA\\ +\end{diagram} + +Finally, I claim that this is initial. Indeed, it certainly has maps into every $(GX, f)$, because we can just compose $A$'s TSC-map with $G(i)$. + +\begin{diagram} +{} & {} & A & {} & {} & {} & {} \\ +{} & \ldTo^{\eta_A'} & \dTo_{\eta_A} & \rdTo^{\eta_A}\rdTo(4,2)^f & {} & {} & {} \\ +GI & \rTo_{G(i)} & GFA & \pile{\rTo^{G(r_1)} \\ \rTo_{G(r_2)}} & GFA & \rTo_{G(h)} & GX\\ +\end{diagram} + +The map is unique: if we had $G(x), G(y): GI \to GX$ such that $G(x)\eta_A' = G(y) \eta_A' = f$, say, then make their equaliser $G(e)$ as above. +$\eta_A'$ lifts over $G(e): GE \to GI$, say to $\eta_A'': A \to GE$. + +\begin{diagram} +{} & {} & A & {} & {} \\ +{} & \ldTo^{\eta_A''} & \dTo_{\eta_A'} &\rdTo^{f} & {} \\ +GE & \rTo_{G(e)} & GI & \pile{\rTo^{G(x)} \\ \rTo_{G(y)}} & GX +\end{diagram} + +By the TSC, there is a map $G(\gamma) : GFA \to GE$ with $G(\gamma) \eta_A = \eta_A''$, so $G(i) G(e) G(\gamma) \eta_A = \eta_A$. + +\begin{diagram} +{} & {} & {} & {} & {} & {} & A & {} & {} \\ +{} & {} & {} & {} & {} & \ldTo(6, 2)^{\eta_A'} \ldTo(4,2)_{\eta_A} \ldTo_{\eta_A''} & \dTo_{\eta_A'} &\rdTo^{f} & {} \\ +GI & \rTo_{G(i)} & GFA & \rTo_{G(\gamma)} & GE & \rTo_{G(e)} & GI & \pile{\rTo^{G(x)} \\ \rTo_{G(y)}} & GX +\end{diagram} + +Now $G(i)$ is an industrial equaliser of things of this form, so $G(i) G(e) G(\gamma) G(i) = G(i)$; $G(i)$ is monic and so $G(e) G(\gamma) G(i) = 1_{GI}$. + +Therefore $G(e)$ is split epic; it's also monic because it's an equaliser, so it's iso. +That is, $G(x), G(y)$ are equal after all, because an equaliser is iso iff the arrows it's equalising are the same. + +\subsection{Summary} +We've seen that $G: \mathcal{D} \to \mathcal{C}$ (where $\mathcal{D}$ is complete and locally small) has a left adjoint iff it preserves small limits and has the TSC. + +Notice how often in the proof we had lines like ``this thing lifts over this thing'': effectively telling us that the limits we had constructed were genuinely members of $(A \downarrow G)$, so that we could use the TSC. +There is actually a theorem about this, and it follows basically the same pattern as those lines do. +It tells us that limits in the arrow category are basically the same as limits in $\mathcal{D}$. +However, I wanted to stay as concrete as possible here. + +\section{More generality} + +This condition has told us that ``if you look like you've got a unit of an adjunction, then you really do have a unit''. +That's still quite restrictive, and it turns out to be possible to relax the TSC to the SSC. + +Recall that the difference between the TSC and the SSC is just that the SSC asserts the existence of several objects which work, rather than just one. + +\ + +\begin{thm}[GAFT from TAFT] If $G: \mathcal{D} \to \mathcal{C}$ has the SSC and preserves products, and $\mathcal{D}$ has products and is locally small, then $G$ has the TSC. +\end{thm} + +\ + +The proof is by the only way we have of combining objects: that is, by taking products. + +If $G$ has the SSC, we construct for each $A$ an object $P = \prod_j B_j$. +Since $G$ preserves products, $GP$ is a product of the $G B_i$. +Define $\eta_A: A \to GP$ componentwise by $\pi_{G B_i} \eta_A = \eta_A^i$. + +Now any $f:A \to GB$ factors as $A \xrightarrow[\eta_A^i]{} G B_i \xrightarrow[Gh]{} GB$, because we had a solution set. +Therefore it factors as $A \xrightarrow[\pi_{G B_i} \eta_A]{} G B_i \xrightarrow[Gh]{} GB$; this is already a factorisation through $P$ in the first arrow! +(Of course, the remaining components of $A \to GP$ are defined by anything we like; the solution-set gives us arrows $A \to G B_j$ for $j \not = i$.) + +So we are done: given the SSC, we have produced an object $P$ and arrow $\eta_A: A \to GP$ which together witness that $G$ satisfies the TSC. + +Combining Theorem 4 with Theorem 3 gives exactly the General Adjoint Functor Theorem. + +\end{document} diff --git a/YonedaWithoutTears2016/YonedaLemma.pdf b/YonedaWithoutTears2016/YonedaLemma.pdf new file mode 100644 index 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