The n-Category Café

First, a note about categorical dimensions. The last post was all about $(\infty,1)$-toposes and their (conjectural) internal homotopy type theory. This post could also be in that world, but I’m not going to use any $\infty$-ness, so I think everything I say today will make sense in the 1-categorical world as well. I will assume that $H$ is locally cartesian closed, and we’ll also need some “presentability” conditions that I won’t make precise.

Now, the first observation is that not every reflective subcategory can underlie a reflective subfibration, because the underlying reflective subcategory of any reflective subfibration is an exponential ideal, and hence its reflector must preserve finite products. (This is also true “locally” for each $C_x\subseteq H/x$.) To see why, let $C_x \subseteq H/x$ be a system of reflective subcategories, with the reflector $L_x\colon H/x \to C_x$ commuting with pullback $f^\ast$ along any $f\colon x\to y$. This implies that the right adjoint of $L_x$, namely the inclusion $C_x \hookrightarrow H/x$, commutes with the right adjoint of $f^\ast$, namely dependent product $\Pi_f$. Since the exponential $y^x$ can be computed by pulling $y$ back to $H/x$, then applying $\Pi_x\colon H/x \to H$, this implies that if $y\in C_1$, then so is $y^x$; thus $C_1$ is an exponential ideal. And a standard argument implies that a reflective subcategory is an exponential ideal if and only if its reflector preserves finite products.

What about the converse — does every reflective exponential ideal underlie a reflective subfibration? To answer this, it’s useful to think of reflective subcategories as localizations. Recall that given a class $S$ of morphisms in $H$, an object of $H$ is $S$-local if $(-\circ f)\colon H(y,z) \to H(x,z)$ is an equivalence for all $f\colon x\to y$ in $S$. Under suitable conditions on $S$ and $H$ (such as if $H$ is locally presentable and $S$ is small-generated), the $S$-local objects form a reflective subcategory, called the localization of $H$ at $S$ (the reflector also happens to be the universal functor inverting all maps in $S$). Conversely, any reflective subcategory $C\subseteq H$ is the localization at some $S$; we may take $S$ to be the class $S_C$ of all morphisms inverted by the reflector.

Now we observed that $C\subseteq H$ is an exponential ideal if and only if its reflector preserves finite products, and this is easily seen to be equivalent to saying that $S_C$ is closed under finite products — in other words, if $f\colon x\to y$ lies in $S_C$, then for any $a\in H$ we have that also $a\times f\colon a\times x\to a\times y$ lies in $S_C$. Conversely, by the opposite argument, if $S$ is a class of morphisms which is closed under finite products in this sense, then the $S$-local objects must be an exponential ideal. Thus, a reflective subcategory is an exponential ideal just when it is the localization at some class of morphisms closed under finite products.

In particular, this means that the reflective exponential ideals are the internal localizations, in the following sense. Given any class $S$, say that $z$ is internally $S$-local if for any $f\colon x\to y$ in $S$, the induced map $z^y \to z^x$ is an equivalence in $H$. By the Yoneda lemma, this is equivalent to saying that $H(a,z^y) \to H(a,z^x)$ is an equivalence for all $a$, and therefore also equivalent to saying that $z$ is $\overline{S}$-local in the usual sense, where $\overline{S} = \{ a\times f \;|\; f\in S \}$. Of course, if $S$ is closed under products, then $\overline{S}=S$. Thus, just as every reflective subcategory is a localization, every reflective exponential ideal is an internal localization.

Now, suppose that $S$ is closed under products, and for any $a\in H$, define $a^\ast(S)$ to consist of all morphisms $$\array{ c\times x & \xrightarrow{c\times f} & c\times y\\ & \searrow & \downarrow\\ & & a }$$ in $H/a$, for all $f\colon x\to y$ in $S$ and all morphisms $c\to a$. By assumption on $S$, we have $1^\ast(S) = S$.

Moreover, for any morphism $g\colon a\to b$, the pullback functor $g^\ast\colon H/b \to H/a$ takes $b^\ast(S)$ into $a^\ast(S)$, while its left adjoint $\Sigma_g\colon H/a \to H/b$ takes $a^\ast(S)$ into $b^\ast(S)$. By adjunction, the first property implies that if $z\in H/a$ is $a^\ast(S)$-local, then $\Pi_g(z) \in H/b$ is $b^\ast(S)$-local, while the second property implies that if $x\in H/b$ is $b^\ast(S)$-local, then $g^\ast x$ is $a^\ast(S)$-local.

Suppose $H$ and $S$ are such that the localization of $H/a$ at $a^\ast(S)$ exists, for all $a$; call it $C_a$. Then since pullback preserves local objects, we have a pullback-stable system of reflective subcategories of slice categories; to see that it is a reflective subfibration, we need to check that the reflectors commute with pullback. But for this it suffices to show that the right adjoints of the reflectors — namely, the inclusions $C_a \subseteq H/a$ — commute with the right adjoints of pullback — namely the dependent product functors $\Pi_g$ — and this follows from the above observation that $\Pi_g$ preserves local objects. Finally, $C_1 = C$ is the localization at $S$, since $1^\ast(S) = S$.

Hence, modulo size considerations, a reflective subcategory underlies a reflective subfibration if and only if it is an exponential ideal. And to extend a reflective exponential ideal $C$ to a reflective subfibration, we just need to pick a suitable $S$, closed under finite products, such that $C$ is localization at $S$. (I presume that in general, different choices of $S$ for the same $C$ can produce different reflective subfibrations, but I don’t know any examples.)

This might also be a good point at which to mention that localization can also be performed quite naturally in the internal logic, using a higher inductive type; see this post.

What about extensions to a stable factorization system? It’s natural to expect that we’ll need a “left exactness” condition on the reflector that lies in between preserving finite products and preserving finite limits, and it turns out that the appropriate such condition is called having stable units. This condition on a reflective subcategory $C\subseteq H$ was first isolated by Cassidy, Hébert, and Kelly; it has the following equivalent characterizations:

1. For every $x\in C$, the reflective subcategory $C/x \subseteq H/x$ is an exponential ideal.
2. The reflector $L$ preserves all pullbacks over an object of $C$.
3. Every pullback of a unit $x \to L x$ of the reflection is inverted by $L$.

The second condition says exactly that the reflector of $C/x \subseteq H/x$ preserves products, so the equivalence of the first and second conditions is by the standard argument. And clearly the second implies the third; the converse is less obvious but true. Note that since $1\in C$ always, having stable units implies that $L$ preserves products, while of course if $L$ is left exact then it preserves all pullbacks and hence has stable units.

I remarked above that for any reflective subfibration, the reflective subcategory $C_x \subseteq H/x$ is an exponential ideal. Thus, if it happens that $C_x = C/x$, then the reflector for $C = C_1$ has stable units. However, asking for the inclusion $C_x \subseteq C/x$ means asking that the composite of an object $y\to x$ of $C_x$ with an object $x\to 1$ of $C_1$ lies in $C_1$, which is a special case of the condition for our reflective subfibration to be a stable factorization system. In this case, the other inclusion $C/x \subseteq C_x$ also follows by a cancellability property of the right class $M$ in a factorization system. Thus, for any stable factorization system $(E,M)$, the underlying reflective subcategory $M/1$ has stable units.

For the converse, let’s think again in terms of localizations. We saw above that a reflective subcategory is an exponential ideal if and only if it is the localization at some class $S$ closed under products. I claim that analogously, a reflective subcategory has stable units if and only if it is the localization at some class $S$ which is stable under pullback.

Suppose first that $S$ is stable under pullback, let $y$ and $z$ be $S$-local, and suppose given maps $x\to z$ and $y\to z$; we want to show that the domain of the local exponential $(y\to z)^{(x\to z)}$ (an object of $H/z$) is $S$-local; call this object $w$. Since $z$ is $S$-local, to show this, it will suffice to show that $w\to z$ is right orthogonal to $S$. However, by definition of $w$, a commutative square $$\array{a & \overset{}{\to} & w\\ ^f\downarrow && \downarrow\\ b& \underset{}{\to} & z}$$ (with $f\colon a\to b$ in $S$) is equivalent to a commutative square $$\array{a\times_z x & \overset{}{\to} & y\\ \downarrow && \downarrow\\ b\times_z x & \underset{}{\to} & z}$$ and likewise a lift in the first square is equivalent to a lift in the second. But since $S$ is stable under pullback, the map $f\times_z x \colon a\times_z x \to b\times_z x$ lies in $S$, and thus is left orthogonal to $y\to z$ (since both $y$ and $z$ are $S$-local). Thus the second square always has a lift, and hence so does the first. Thus, the $S$-local objects have stable units.

Conversely, suppose $C\subseteq H$ is a reflective subcategory with stable units, and let $T_C$ denote the class of all pullbacks of units $x \to L x$. Then $T_C$ is pullback-stable by construction. Since $C$ has stable units, we have $T_C\subseteq S_C$ (with $S_C$ the class of all morphisms inverted by $L$, as above), so all objects of $C$ are $T_C$-local. Moreover, $C$ is also the localization at $T_C$; for if $z$ is $T_C$-local, then it sees the unit $z\to L z$ as an isomorphism; hence $z\cong L z$ and thus $z\in C$.

Therefore, a reflective subcategory has stable units if and only if it is the localization at some pullback-stable $S$. Note, though, that the class $S_C$ of all maps inverted by a reflector with stable units need not be pullback-stable. An easy counterexample is the subcategory of subterminal objects in $Set$ (or any regular category). Subterminal objects are always an exponential ideal, and pullbacks over a subterminal object coincide with products, so the reflector for this subcategory has stable units. (Alternatively, it underlies the stable factorization system (regular epi, mono), thus has stable units by the argument above.) However, in a classical $Set$, the maps inverted by the reflector consist of all maps between inhabited sets together with the identity map of the empty set, and this class is not stable under pullback (the pullback of $1\to 2$ along the other $1\to 2$ is $0\to 1$).

In fact, $S_C$ is pullback-stable precisely when the reflector of $C$ is left exact. (This fine distinction confused me for quite a while, especially because Prop. 6.2.1.2 in the paper edition of Higher Topos Theory seems to claim that the localization at any pullback-stable class is left exact. That assertion seems to have been removed from the most recent online edition, however.)

With this understanding of stable units in mind, we can address the question of extending a reflective subcategory to a stable factorization system. Suppose $S$ is pullback-stable and that $C$ is the localization at $S$, hence has stable units. For any $a\in H$, let $S_a$ denote the class of morphisms $$\array { x & \xrightarrow{f} & y \\ & \searrow & \downarrow\\ & & a}$$ with $f\in S$. Since $S$ is stable under pullback, the same arguments as in the previous case apply to show that if we take $C_a$ to be the localization at $S_a$ (again, ignoring size issues), we obtain a reflective subfibration. Moreover, an object $x\to a$ of $H/a$ lies in $C_a$ just when it is right orthogonal to all morphisms of $S$, and morphisms with this property are clearly closed under composition. Thus, this reflective subfibration is actually a stable factorization system.

Hence, modulo size considerations, a reflective subcategory underlies a stable factorization system if and only if it has stable units. And to extend a reflective subcategory $C$ with stable units to a stable factorization system, we just need to pick a suitable pullback-stable $S$ such that $C$ is localization at $S$.

As a particular case, we can apply these results to the discrete objects in a cohesive topos (or $(\infty,1)$-topos). The left adjoint $\Pi$ to the “discrete objects” functor makes these into a reflective subcategory, which is assumed to preserve finite products; thus the discrete objects are an exponential ideal. Moreover, it is certainly true in the 1-topos case, and conjecturally so in the $(\infty,1)$-topos case, that this reflective subcategory also has stable units. Therefore, it can be extended to some reflective subfibration or stable factorization system, and so axiomatized in the internal type theory directly (without requiring the sharp-escape route).

In other words, sorry Urs, I still can’t seem to make up my mind about the right way to axiomatize cohesion! This argument implies that we can axiomatize $\Pi$ as well as $\sharp$ without requiring $\sharp Type$, and we pleasingly get the finite-product-preservation of $\Pi$ for free as we remarked earlier. Where I went wrong earlier was focusing on the localization/stabilization factorization system rather than just looking for some stable factorization system. However, I still don’t see any way to describe $\flat$ in this way; I can’t see any reason for the reflective subfibration of discrete objects to also be coreflective, let alone fiberwise equivalent to the reflective subfibration of codiscrete objects. Maybe we should do $\Pi$ directly in this way and use sharp-escape for $\flat$?