The n-Category Café

Let’s start in a setting that’s hopefully comfortable: a forgetful functor $U\colon C\to S$. We want to think of $S$ as a category of “unstructured” set-like objects (or perhaps algebraic objects), and $C$ as a category of “spaces” over it. That is, an object of $C$ should be thought of as an object of $S$ equipped with some sort of “space-structure”, “topology”, or “cohesion”.

For instance, if $S$ is sets, then $C$ could be topological spaces, convergence spaces, subsequential spaces, locales, the category of sheaves on a site like CartSp or Diff or Top, or Johnstone’s topological topos. We don’t assume that $U$ is faithful in general, but we may as well assume that it is an isofibration.

We say that $C$ has discrete objects if $U$ has a fully faithful left adjoint, and codiscrete objects if $U$ has a fully faithful right adjoint. By abstract nonsense, if $U$ has both adjoints, then one is fully faithful if and only if the other is so. On the other hand, we might also wonder whether $U$ is a Grothendieck fibration or opfibration. In fact, these are closely related.

Theorem 1: Suppose $C$ has a terminal object preserved by $U$. If $U$ is a fibration, then $C$ has codiscrete objects.

Proof: Define $G\colon S\to C$ by $G(a)=$ the result of pulling back the terminal object $1\in C$ along the unique map $a\to 1$ in $S$. It is easy to verify that $U\dashv G$ and that $G$ is fully faithful. $\Box$

Theorem 2: Suppose $C$ has pullbacks preserved by $U$. If $C$ has codiscrete objects, then $U$ is a fibration.

Proof: Assuming $U\dashv G$ with $G$ fully faithful, and given $x\in C$ with $U(x)=b$ and a morphism $f\colon a\to b$ in $S$, consider the pullback $$\array{f^\ast x & \overset{}{\to} & G a \\ \downarrow && \downarrow\\ x & \underset{}{\to} & G b}$$ Since this pullback is preserved by $U$, we have $U(f^\ast x) = a$. (Or at least $U(f^\ast x) \cong a$, and since $U$ is an isofibration we can choose $f^\ast x$ to make this an equality.) The universal property of a cartesian arrow is again easy to verify. $\Box$

Thus, if $C$ has finite limits preserved by $U$, then it has codiscrete objects if and only if it is a fibration. Dually, of course, if $C$ has finite colimits preserved by $U$, then it has discrete objects if and only if it is an opfibration. (More generally, if $C$ is complete or cocomplete and $U$ is continuous or cocontinuous, then we can construct “final lifts of small $U$-structured sinks”.)

Now suppose that $C$ lacks one or both of discrete and codiscrete objects; how can we modify it so that it will have them? One idea is to construct a new category whose objects are explicitly “objects of $S$ equipped with $C$-structure”. Specifically, we consider the category of triples $(a\in S, x\in C, a\to U(x))$ (that is, the comma category of $Id_S$ over $U$).

In this context, we call this category the scone (short for Sierpinski cone) of $C$ over $S$, or $scn_S(C)$. It comes equipped with obvious functors $U'\colon scn_S(C)\to S$ and $i^\ast \colon scn_S(C)\to C$. We also have a functor $i_\ast \colon C\to scn_S(C)$ defined by $i_\ast (x) = (U(x),x,id_{U(x)})$, which is right adjoint to $i^\ast$.

Intuitively, $a$ is a set, $x$ is a space, and $a\to U(x)$ says that each element of $a$ corresponds to a point of $x$. If this morphism is not injective, then our new object has “multiple points that can’t be told apart by the topology”, while if it is not surjective, then our new object “has room for more points in the topology than are actually present”. This suggests the following.

Theorem 3: If $C$ has a terminal object preserved by $U$, then $U'\colon scn_S(C)\to S$ has a fully faithful right adjoint, which takes $a\in S$ to $(a,1,!\colon a\to U(1))$. Thus $scn_S(C)$ has codiscrete objects.

Proof: Easy. $\Box$

Moreover, Theorem 3 is a corollary of Theorem 1, because $U'$ is always a fibration: for $f\colon a\to b$ we have $f^\ast (b, x, b\to U(x)) = (a,x, a\xrightarrow{f} b \to U(x))$. In fact, $U'$ is the free fibration generated by $U$: the category of fibrations over $S$ is 2-monadic over $Cat/S$, and $U\mapsto U'$ is the 2-monad with unit $i_\ast$.

This 2-monad is colax-idempotent, so that $U$ is itself a fibration if and only if $i_\ast$ has a right adjoint $i^!$ that commutes with $U$ and $U'$. Therefore, from theorems 1 and 2, we conclude:

• If $C$ has a terminal object preserved by $U$ and $i_\ast$ has a right adjoint over $S$, then $C$ has codiscrete objects. This is easy to see directly by composition of adjoints, since $U = U' \circ i_\ast$ and $U'$ always has a right adjoint.

• If $C$ has pullbacks preserved by $U$ and codiscrete objects, then $i_\ast$ has a right adjoint over $S$. The adjoint is defined by the pullback $$\array{i^! x & \overset{}{\to} & G(a)\\ \downarrow && \downarrow\\ x & \underset{}{\to} & G U(x) }$$

Thus, if we restrict to the category of lex categories and lex functors over $S$, then we can also regard $U'$ as the free category-with-codiscrete-objects generated by $U$.

Dually, of course, we can consider the “co-scone” which is the free opfibration and the free category-with-discrete-objects. However, we also have the following nice fact.

Theorem 4: If $U$ has a left adjoint, then so does $U'$, which is fully faithful if the left adjoint of $U$ is so. Thus if $C$ has discrete objects, so does $scn_S(C)$.

Proof: Let $F\dashv U$; we define $F'\colon S\to scn_S(C)$ by $F'(a) = (a, F(a), a\to U F(a))$. The universal property is easy to verify. $\Box$

This means that there must be a distributive law relating the scone and the co-scone, enabling us to talk about joint algebras for the two monads. These joint algebras are, of course, functors into $S$ which are both fibrations and opfibrations, or equivalently (in the lex and colex case) those having both discrete and codiscrete objects.

Let’s bring it all together by recalling two important examples. Firstly, suppose that $C$ and $S$ are toposes and $U$ is the direct image part of a geometric morphism (thus it has a left-exact left adjoint). Then by Theorem 4, $U'$ also has a left adjoint, which inherits left-exactness; thus $U'$ is also the direct image part of a geometric morphism. Finally, $i_\ast$ always has a left-exact left adjoint $i^\ast$, so the morphism $i\colon C\to scn_S(C)$ lives in $Topos/S$.

In this case, having codiscrete objects (which then implies also having discrete ones) is called being a local $S$-topos. The fact that $scn_S(C)$ is the free local $S$-topos on $C$ appears in C3.6.5 of Sketches of an Elephant. Theorem 2 implies that for any local $S$-topos, the “global sections” morphism $U\colon C\to S$ is a fibration and opfibration, a useful thing to know. Note that in this case $U$ is not generally faithful.

Secondly, let $S=Set$ and $C=Loc$ be the category of locales, with $U$ the “set of points” functor (also not faithful). Then $scn_S(C)$ is the category of topological systems defined in Steve Vickers’ book Topology via Logic. These are “midway” between topological spaces and locales, having both a frame of opens and a set of points, neither of which is necessarily determined by the other.

There is also a way to recover the usual category of topological spaces. If $U\colon C \to S$ has codiscrete objects, we say that $x\in C$ is concrete if $x\to G U(x)$ is a monomorphism. This is equivalent to saying that $U$ is faithful on morphisms with codomain $x$. Dually, if $C$ has discrete objects, we say $x$ is co-concrete (“ncrete”?) if $F U(x) \to x$ is an epimorphism. This is equivalent to saying that $U$ is faithful on morphisms with domain $x$. Restricting to the concrete or co-concrete objects are two dual ways to force $U$ to become faithful.

In the case of local toposes, it is often the concrete objects which we are interested in. When $C$ is the category of sheaves on a concrete site, then the concrete objects are precisely the concrete sheaves, which form a quasitopos.

On the other hand, in a category of the form $scn_S(C)$, an object $(a,x,a\to U(x))$ is concrete just when $x$ is subterminal. Thus, in this case (such as the category of topological systems), there are very few concrete objects. However, if $C$ has discrete objects, then $(a,x,a\to U(x))$ is co-concrete just when the adjunct map $F(a) \to x$ is an epimorphism. Hence the co-concrete topological systems are precisely the topological spaces.

Finally, we can categorify this picture: the notions of discrete and codiscrete objects, fibration, and scone all make sense for higher categories. When we come to concreteness, we have to choose notions with which to categorify “monic” and “epic”. There is probably a reasonable notion of concrete (∞,1)-sheaf in a local $(\infty,1)$-topos, though it’s perhaps not immediately obvious what notion of “monic” should be used. And Richard Garner’s ionads are (roughly) the co-concrete objects of the scone of the 2-category $Topos$, where “epic” is replaced by geometric surjection.