The n-Category Café

If we’re working in some category other than $Set$, the easiest analogues of ‘one-to-one’ and ‘onto’ are the concepts of ‘monic” and ‘epic’. Here’s one way to define these:

• A morphism $f : a \to b$ is monic if for every object $x$, the function $$\array{ hom(x,a) &\to& hom(x,b) \\ g &\mapsto & f \circ g }$$ is one-to-one.
• A morphism $f : a \to b$ is epic if for every object $x$, the function $$\array{ hom(b,x) &\to& hom(a,x) \\ g &\mapsto & g \circ f }$$ is one-to-one.

In the category of sets, ‘monic’ reduces to ‘one-to-one’, while ‘epic’ reduces to ‘onto’. But note that they’re both defined using the concept of ‘one-to-one’!

What happens if we categorify all this? When we go from functions to functors, the dynamic duo of concepts ‘one-to-one’ and ‘onto’ grows to a trio: ‘faithful’, ‘full’ and ‘essentially surjective’. Our paper explains why this happens and how the pattern keeps on going when we climb up the ladder of $n$-categories. This is a great story, first worked out by James Dolan and Toby Bartels. This story even explains why ‘onto’ can be defined in terms of ‘one-to-one’ as above. I won’t retell this story here.

What if we mimic the concepts of ‘monic’ and ‘epic’ at this categorified level? If we’re working in some 2-category other than $Cat$, we get three concepts that are defined a bit like ‘monic’:

• A morphism $f : a \to b$ is left faithful if for every object $x$, the functor $$\array{ hom(x,a) &\to& hom(x,b) \\ g &\mapsto & f \circ g }$$ is faithful.
• A morphism $f : a \to b$ is left full if for every object $x$, the functor $$\array{ hom(x,a) &\to& hom(x,b) \\ g &\mapsto & f \circ g }$$ is full.
• A morphism $f : a \to b$ is left essentially surjective if for every object $x$, the functor $$\array{ hom(x,a) &\to& hom(x,b) \\ g &\mapsto & f \circ g }$$ is essentially surjective.

In $Cat$ these reduce to the right things… but only if you’re clever:

Theorem. A morphism in $Cat$ is just a functor, and:

• Such a morphism is left faithful if and only if it is faithful.
• It is left faithful and left full if and only if it is faithful and full.
• It is left faithful, left full and left essentially surjective if and only if it is faithful, full and essentially surjective.

Note how we stack these concepts on top of each other. There’s a good conceptual reason for this — part of the great story I’m not telling (because Mike and I already told it). And we need to stack them to get a pretty theorem, since it’s not true that a functor is left full iff it is full. At least that’s what Jonas Frey told me — can you think of a counterexample?

Anyway, Mike also toyed with three other concepts, defined a bit like ‘epic’:

• A morphism $f : a \to b$ is right faithful if for every object $x$, the functor $$\array{ hom(b,x) &\to& hom(a,x) \\ g &\mapsto & g \circ f }$$ is faithful.
• A morphism $f : a \to b$ is right full if for every object $x$, the functor $$\array{ hom(b,x) &\to& hom(a,x) \\ g &\mapsto & g \circ f }$$ is full.
• A morphism $f : a \to b$ is right essentially surjective if for every object $x$, the functor $$\array{ hom(b,x) &\to& hom(a,x) \\ g &\mapsto & g \circ f }$$ is essentially surjective.

These behave in a more quirky manner. Mike showed:

Theorem. A morphism in $Cat$ is just a functor, and:

• Such a morphism is right faithful if it is essentially surjective.
• It is right faithful and right full if it is essentially surjective and full.
• It is right faithful, right full and right essentially surjective if and only if it is essentially surjective, full and faithful.

Note: now, two times out of three we’re saying ‘if’ instead of ‘if and only if’! What about the converses?

This is the question that Jonas Frey’s paper settles.