Prior to and during my first few years of graduate work, I had the conceit that perhaps one could do enriched category theory in a way completely free of the “size issues” that beset ordinary and enriched category theory. The dream was of a paradise in which one could freely take functor categories without fear, where adjoint functor theorems exist without worrying about solution set technicalities, and where enriched category theory could operate autonomously from set theory, in very pure and algebraic fashion.
Ah, youth! In the beginning, I had in mind a world $E$ (called an “epistemology”^{1}) that would be like $V$-$\mathrm{Cat}$ but was symmetric monoidal closed, where the guiding assumption was that $E$ carried a free cocompletion monad taking an object $C$ to $V$-valued presheaves on $C$ for some distinguished object $V$. So $E$ was assumed to have an involution $(-{)}^{\mathrm{op}}$ operating on it, and there was a monad $p:E\to E$ of Kock-Zöberlein type, taking $C$ to ${V}^{{C}^{\mathrm{op}}}$, and satisfying some axioms to the effect that $p$-algebras would behave like $V$-total categories. The ensuing theory involved lots of adjoint strings and plenty of stacked exponentials, and I have to admit that the original axioms were somewhat clumsy to begin with (quite aside from their being attached to a certain “ideology” and also certain foundational pretensions). However, it was my first serious attempt at doing mathematical research, and it was all my own, and I was in love with the subject and thought it beautiful^{2}.
Quite a long time later, sometime during 1999 after I had been working with cartesian bicategories for awhile, it dawned on me that the basic axiomatics of epistemologies could be made much prettier by starting not with something that behaved like a paradise form of $V$-$\mathrm{Cat}$, but like a paradise form of $V$-$\mathrm{Mod}$, with the subbicategory of “maps” or left adjoints serving as a proxy for $V$-$\mathrm{Cat}$^{3}. (And it was later still that I understood that the crucial concept of “potency”, explained below, could be developed just as well from the $V$-$\mathrm{Cat}$ side, essentially by working with yoneda structures in the sense of Street and Walters in which every 1-cell is admissible.)
In any event, this page is will present some basic epistemology theory from the $V$-$\mathrm{Mod}$ point of view, which I continue to find quite pretty.
The key unresolved issue is in the nature of models, which even after all this time I don’t have much understanding of. A good analogy is to the early days of $\lambda $-calculus, where the theory had been developed very well on the syntactic side, but not on the semantic side before Scott and his models. (Here the situation is even more difficult and intricate, and I always have this slightly edgy feeling of skirting close to a razor’s edge of algorithmic inconsistency whenever I think about epistemologies, a feeling that is exciting and uncomfortable at the same time.) My hope is that even if natural models are hard to come by, maybe one can establish algorithmic consistency, by appealing to theorems of Church-Rosser and strong normalization type. With any luck, I’ll write down some ideas I’ve had on this.
(opposite to Bénabou’s convention).
Recall the following definition:
Given bicategories $B$, $C$, a biadjunction $F\u22a3G:B\to C$ consists of homomorphisms (strong functors) $F:C\to B$, $G:B\to C$ together with a strong (i.e., pseudo-) natural adjoint equivalence of the form
between $\mathrm{Cat}$-valued homs.
In elementary terms, the data of the strongly natural adjoint equivalence is given by strong transformations $\eta :{1}_{C}\to GF$, $\epsilon :FG\to {1}_{B}$ and invertible modifications $s,t$,
that satisfy the triangulator coherence conditions (swallowtail coherence conditions in the language of Baez-Langford):
$\phantom{\rule{thinmathspace}{0ex}}$
Let $F\u22a3G$ be a biadjunction, with unit $\eta $ and counit $\epsilon $ and triangulators $s$, $t$ as above. The following conditions are equivalent:
The triangulator $t:{1}_{F}\to (\epsilon F)(F\eta )$ is the unit of an adjunction $F\eta \u22a3\epsilon F$;
The triangulator $s:(G\epsilon )(\eta G)\to {1}_{G}$ is the counit of an adjunction $G\epsilon \u22a3\eta G$.
We prove that 1. implies 2.; the proof that 2. implies 1. is dual. Let $v:F\eta \circ \epsilon F\to {1}_{FGF}$ be the counit of $F\eta \u22a3\epsilon F$. We have a 2-cell
defined by a pasting
(where the unlabeled 2-cells are obvious whiskerings of ${s}^{-1}$), and we compose this 2-cell with a strong naturality constraint
to arrive at a 2-cell $u:{1}_{GFG}\to (\eta G)\circ (G\epsilon )$. It is straightforward to prove that $u$ and $s$ form the unit and counit of an adjunction $G\epsilon \u22a3\eta G$, given that $t$ and $v$ are the unit and counit of $F\eta \u22a3\epsilon F$.
A biadjunction is KZ (Kock-Zöberlein) if either of the two conditions of lemma 1 hold.
For $B$ a bicategory, $\mathrm{Map}(B)$ denotes the locally full subbicategory whose 1-cells are precisely the 1-cells that are left adjoints in $B$ (which we will call maps). If $B$ is the bicategory of relations in a regular category, then $\mathrm{Map}(B)$ reproduces the original category. In general, we will think of the $B$ of interest to us as like bicategories of generalized relations (relations, spans, profunctors, etc.), and $\mathrm{Map}(B)$ will then be like a category whose morphisms are functions or functors.
Here is our fundamental notion.
A bicategory $B$ is potent if the inclusion $i:\mathrm{Map}(B)\to B$ is the left biadjoint of a KZ biadjunction $i\u22a3p:B\to \mathrm{Map}(B)$.
The right adjoint of a map $f:A\to B$ is denoted ${f}^{*}:B\to A$.
Given an arrow $r:A\to B$ in a potent bicategory, let ${\chi}_{r}:A\to p(B)$ denote the characteristic map of $r$, defined by the formula ${\chi}_{r}=p(r)yA$. We have
where $e:ip\to {1}_{B}$ is the counit of the biadjunction $i\u22a3p$. We note that the unit is not only a strong transformation on $\mathrm{Map}(B)$, but can be viewed also as a lax (map-valued) transformation $y:1\to ip$ on $B$, with structure 2-cells of the form
mated to the isomorphism $r\cong eB\circ {\chi}_{r}$.
We also have that the right adjoint of
is $p({\chi}_{r}^{*})\circ ypA$, since $pe\u22a3yp$ by the KZ biadjunction. In other words,
where the right adjoint is manifestly a map.
Right Kan lifts exist in a potent bicategory.
Let $r:A\to C$ and $s:B\to C$ be arrows in a potent bicategory. The right Kan lift of $r$ through $s$ is constructed as the composite
Indeed, for any $t:A\to B$, we have natural bijections
where ${\chi}_{st}\cong p(s){\chi}_{t}$ is clear from how characteristic maps were defined, and we get to the fourth line by applying the adjunction $p(s)\u22a3{\chi}_{{\chi}_{s}^{*}}$. The passage to the final line is effected by the application $eb:\mathrm{Map}(B)(a,pb)\to B(a,b)$.
Now let $B$ be a symmetric monoidal bicategory, with tensor $\otimes $ and unit $1$. We say that $B$ is compact closed if for every object $B$ there is an object ${B}^{*}$ together with a unit and counit
and triangulators
which exhibit ${B}^{*}\otimes -$ as right biadjoint to $B\otimes -$. Since $B$ is symmetric monoidal, we can exhibit $B\otimes -$ also as right adjoint to ${B}^{*}\otimes -$.
An epistemology is a potent compact closed bicategory $B$.
As we calculate with epistemologies, we will suppose given a specified biadjunction structure $i\u22a3p$ attached to the inclusion $i:\mathrm{Map}(B)\to B$.
The object $p1$ in an epistemology plays a distinguished role in the theory; we denote it $V$. It should be thought of as an object of generalized truth values (akin to $\Omega =p(1)$ in a topos) or as a base of enrichment, so that $B$ behaves something like $V$-$\mathrm{Mod}$ and $\mathrm{Map}(B)$ behaves something like $V$-$\mathrm{Cat}$.
The notion of epistemology encapsulates an idealized world of enriched category theory in which we can in particular iterate the $V$-valued presheaf construction as $V$-enriched free cocompletion.
In an epistemology, there is an equivalence $p(B\otimes C)\simeq (pC{)}^{{B}^{*}}$ in $\mathrm{Map}(B)$; in particular, $p(B)\simeq {V}^{{B}^{*}}$.
There are natural equivalences between local hom-categories whose objects appear below:
which shows that $p(B\otimes C)$ satisfies the universal property expected of the bicategorical exponential $p(C{)}^{{B}^{*}}$. The equivalence $p(B)\simeq {V}^{{B}^{*}}$ arises by taking $C=1$.
As a consequence, the unit $y:1\to pi$ of the KZ biadjunction is map-valued transformation
which gives rise to a map ${\mathrm{hom}}_{A}:{A}^{*}\otimes A\to V$. We will see that we can simulate enriched category theory in an epistemology, with $V$ playing the role of hom base of enrichment.
We develop some further consequences of compact closure. Let ${B}^{\mathrm{op}}$ be $B$ with 1-cells reversed, and let ${B}^{\mathrm{co}}$ be $B$ with 2-cells reversed. Compact closure allows one to construct an equivalence
This equivalence takes right adjoints in $B$ to left adjoints (maps) in $B$, and vice-versa. On the other hand, by taking mates we define a 2-functor
taking a 2-cell $\alpha :f\to g$ between left adjoints in $B$ to the corresponding mate ${\alpha}^{*}:{g}^{*}\to {f}^{*}$ between right adjoints in $B$. Now combine these operations: starting with an adjunction
in $B$, we obtain an adjunction
and by the process of taking mates, a 2-cell $\alpha :f\to f\prime $ between maps in $B$ corresponds to a 2-cell $({\alpha}^{*}{)}^{\u2020}:(f\prime {)}^{*\u2020}\to {f}^{*\u2020}$ between maps.
The functor $(-{)}^{\mathrm{op}}:\mathrm{Map}(B{)}^{\mathrm{co}}\to \mathrm{Map}(B)$ takes
The functor $(-{)}^{\mathrm{op}}$ is symmetric monoidal and involutive in the evident way.
There are two basic examples. For the first, let $V$ be a commutative quantale, and construct the bicategory $B$ of small $V$-enriched categories and $V$-enriched bimodules between them. $B$ inherits a tensor product from the quantale multiplication on $V$, and it is compact closed.
The second example is any compact closed bicategory $B$ whose underlying bicategory is compact (meaning that every 1-cell has a right adjoint). In this case, the inclusion $\mathrm{Map}(B)\to B$ is an identity.
The concept of epistemology is “algebraic” in that one can construct a free epistemology on a given bicategory, and show epistemologies are monadic over bicategories in an appropriate sense. (This certainly needs to be justified.)
Put $E=\mathrm{Map}(B)$, and let $L$-$\mathrm{Alg}$ (for left adjoint) be the category of algebras of the pseudomonad $pi:E\to E$. Let $R$-$\mathrm{Alg}$ be the category of algebras of the pseudomonad $(-{)}^{\mathrm{op}}\circ (pi)\circ (-{)}^{\mathrm{op}}:E\to E$. The unit of $R$ will be a morphism in $E$ denoted $\upsilon C:C\to R(C)=({V}^{C}{)}^{*}$.
For any 1-cell $f:A\to B$ in $E$, let $g={f}^{*}$ be its right adjoint in $B$. Then $p(g)\cong {V}^{{f}^{\mathrm{op}}}:{V}^{{B}^{\mathrm{op}}}\to {V}^{{A}^{\mathrm{op}}}$.
For any object $C$ of $B$, we have equivalences as follows:
which proves the claim.
For any $f:A\to B$ in $E$ and $g={f}^{*}$, the morphism ${V}^{{f}^{\mathrm{op}}}:{V}^{{B}^{\mathrm{op}}}\to {V}^{{A}^{\mathrm{op}}}$ has both a left and right adjoint in $E$:
In particular, for $f=yA:A\to {V}^{{A}^{\mathrm{op}}}$, the multiplication $LL(A)\to L(A)$ is given by
(since $y\u22a3e$ and the multiplication on $L$ is given by $pe$).
The previous result is that we can take both right and left Kan extensions along morphisms in $E$. Related is the fact that both right Kan lifts and right Kan extensions exist in an epistemology $B$, by proposition 1 and the fact that $(-{)}^{\u2020}:{B}^{\mathrm{op}}\to B$ converts right extension problems to right lifting problems. If $s\backslash r$ denotes the right Kan lift of $r$ through $s$, then the right Kan extension of $r$ along $t$ is given by the formula $r/t\u2254({t}^{\u2020}\backslash {r}^{\u2020}{)}^{\u2020}$.
Let $A$, $B$ be $L$-algebras. Then $L$-algebra maps $A\to B$ coincide with left adjoints $A\to B$ in $E$.
The monad $R$ distributes over the monad $L$, and the monad $LR$ (as induced from the distributive law) is equivalent to the double dualization monad ${V}^{{V}^{(-)}}$.
$V$ is an $R$-algebra.
We define the algebra structure $\theta :R(V)={V}^{*{V}^{*}}\to V$ to be the map $[\theta ]$ named by the composite
In that case, the unit equation
is equivalent to
If $A$ is an $L$-algebra, then so is any exponential ${A}^{C}$ that exists in $E$, so that $L$-$\mathrm{Alg}$ is an exponential ideal in $E$.
The left adjoint to the yoneda embedding on ${A}^{C}$ is (claim) the composite
The object $V$ is a symmetric monoidal object in $E=\mathrm{Map}(B)$.
For this, we observe that $E$ inherits a symmetric monoidal bicategory structure from $B$ via the inclusion $i:E\to B$: the tensor product
restricts to a 2-functor
and it is automatic that the 1-cell constraints $\alpha $, $\sigma $, etc., for the symmetric monoidal structure on $B$ are maps (because they are equivalences), and all the 2-cell constraints are then automatically in $E$. In this way, $i:E\to B$ becomes a symmetric monoidal 2-functor. Its right adjoint $p:B\to E$ thereby becomes a lax symmetric monoidal 2-functor; in particular there is a lax constraint of the form
This follows from the observation that the unit $1$ of a symmetric monoidal bicategory is a symmetric monoidal object, together with the lax constraint above. In more detail, there is a symmetric monoidal category $U$ whose objects are 1-cells $r:{1}^{\otimes n}\to 1$ in $B$, so that letting $F[1]$ be the free symmetric monoidal category on one generator, there is a symmetric monoidal functor
It takes a morphism $\alpha :u\to w$ in $F[1]$ between two words in $n$ variables to a morphism ${\alpha}_{*}:{u}_{*}\to {w}_{*}$ in the local hom-category $B({1}^{\otimes n},1)$, which is a 2-cell in $B$. Whiskering the 2-cell ${\alpha}_{*}$ by the 1-cell $(e1{)}^{\otimes n}:{V}^{\otimes n}\to {1}^{\otimes n}$, we get a corresponding morphism in
and this defines a symmetric monoidal structure on $V$.
The object $V$ is a symmetric monoidal closed object in $E$.
The first question is what is even meant by a symmetric monoidal closed object.
Called an “epistemology” for reasons that were obscure to me then and even more so now, but I’ve never called it anything else and I’ve never tried to come up with anything better. Roughly speaking, I had in mind that any “epistemology”, i.e., any “theory of (scientific) knowledge” worthy of the name, had to avoid an infinite regress, and had to be be based on some system of comparison and measurement of the entities under consideration. The measurements would be valued in some base of measurements $V$ (the archetypal example being $V=\mathbb{R}$ or $V=[0,\mathrm{\infty}]$, or $V=\mathrm{Set}$ if we think of hom-sets as measuring the degree to which two entities are related), and $V$ would be used to measure itself (to avoid an infinite regress). Such a system should be closed and autonomous (so as to avoid regressing to another background “theory of knowledge” like set theory). Thus I had in mind a world like $V$-$\mathrm{Cat}$, but free of any extraneous or background set theory to which constructions make reference. ↩
It took quite some time before it at last became clear to me that this was not a suitable subject on which to write a doctoral dissertation or to start a mathematical career with; at length, my dissertation topic morphed into the coherence problem for symmetric monoidal closed categories. Luckily for me, certain intuitions developed during my “epistemology phase” turned out to be useful during the later dissertation work. (And let me pay tribute to my adviser, Myles Tierney, who was very kind and patient all the while, and let me figure it out for myself!) ↩
Actually, the proper and certainly more up-to-date way of relating the $\mathrm{Mod}$ side and the $\mathrm{Cat}$ side is probably through the use of equipments or framed bicategories. This may be undertaken in a later revision of these notes. ↩