The urge to combine 1- and 2-morphisms in slicing a 2-category.

Question

Suppose that $C$ is a 2-category, perhaps $C=\rm{Cat}$, the 2-category of small categories, functors, and natural transformations. Let $T$ be an object in $C$.

I form the new 1-category whose objects are morphisms $f\colon A\rightarrow T$ in $C$, and in which a morphism from $f$ to some $f'\colon A'\rightarrow T$ consists of a pair $(\phi,\phi^\sharp)$ where $\phi\colon A\rightarrow A'$ is a 1-morphism in $C$ and $\phi^\sharp\colon\phi\circ f'\rightarrow f$ is a 2-morphism between arrows $A\rightarrow T$. Call this new category the $(C\Uparrow T)$. An obvious variation comes about by reversing the direction of the 2-morphism, i.e. we could take $\phi^\sharp\colon f\rightarrow\phi\circ f'$; perhaps I might call this variation $(C\Downarrow T)$.

What is the high-brow way to refer to these strange slice-categories? How do you locate them within a good understanding of 2-categories? Where are the properties of such things discussed? What is the relation between these strange slices and the usual 2-categorical slices?

Thanks!

Answer 1

The second definition looks like the 'lax comma category' $C // T$, where a morphism $f \to f'$ is given by a 2-cell $f \to f'\phi$. The defining universal property is the same as for comma objects, except that the 2-cells in the squares are lax natural transformations. Your first definition should be the oplax version.

See Kelly, On clubs and doctrines, LNM 420, or Gray, Adjointness For 2-Categories, LNM 391, who calls these '2-comma categories'.

In more detail, Gray's 2-comma categories come from (strict, I think) 2-functors $A \overset{F}{\rightarrow} K \overset{G}{\leftarrow} B$. An object is a 1-cell $FA \to GB$, a morphism is a square with a 2-cell in, and a 2-cell is given by a pair of 2-cells in $K$ that fit into a commuting cylinder (it's pretty obvious if you draw a picture). In your example, (what I've called) $C // T$ has 2-cells $(\phi,\phi^\sharp) \Rightarrow (\psi,\psi^\sharp)$ given by 2-cells $\alpha \colon \phi \Rightarrow \psi$ such that $\psi^\sharp \circ f'\alpha = \phi^\sharp$. (Again, pictures make it much clearer!) So your slices are actually 2-categories, coming from $C \overset{1}{\rightarrow} C \overset{T}{\leftarrow} \bullet$.

Answer 2

Interesting question!

If C is a 1-category then the overcategory C/T can be described as the lax (or oplax, I forget) limit of the diagram • → C in the 2-category Cat, where the arrow is given by the object T of C. The lax limit means we ask for a universal limit cone on the diagram where the triangle is filled with a noninvertible 2-morphism. We can adapt this definition to any object C of any 2-category equipped with a map T from the terminal object.

When C is a 2-category, I believe we need to ask for a "very lax" limit, in which the triangle is not even filled by a (noninvertible) natural transformation, but only a lax (or oplax, depending on which of your constructions you want) natural transformation. As far as I can see, there is no way to perform your constructions starting only with 2Cat as a 3-category and the data of C and T; we need the extra structure of the (op)lax natural transformations in 2Cat. Moreover, there is no 3-category of 2-categories, functors, and lax natural transformations in which to take a lax limit and obtain your constructions. So, these "lax" overcategories are still rather mysterious to me.

I assume by "the usual 2-categorical slices" you mean to require the 2-morphism between $\phi \circ f'$ and $f$ to be invertible, which is the (op?)lax limit of the diagram • → C in 2Cat.

Answer 3

For what it's worth, the construction you describe features prominently in https://arxiv.org/abs/0807.4146, though that paper does not use 2-categorical language.

More generally, given a 2-category $C$ (which I usually assume is pivotal, though maybe that's not necessary here), one can construct a 1-category $D$ whose objects are 1-morphisms $f:a\to b$, and whose morphisms are "rectangles": the domain $f:a\to b$ along the bottom, the range $f':a'\to b'$ along the top, additional 1-morphisms (of $C$) $g:a\to a'$ and $h:b\to b'$ along the right and left sides, and a 2-morphism of $C$ filling in the rectangle. Composition in $D$ is given by stacking the rectangles vertically. I like to think of the pair $(g, h)$ as the (bi)grading of the morphisms of $D$. What you describe is the case where we restrict $h$ to be an identity 1-morphism (of $C$). In the paper linked to above we put an inner product on $D$ and complete it to a von Neumann algebra (in fact, a factor).

In response to David's comment below:

Modulo some details, a planar algebra is equivalent to a pivotal 2-category whose 2-morphisms are vector spaces and whose 1-morphisms are finitely generated. The standard example is constructed from a pair of factors (irreducible von Neumann algebras) $N\subset M$. From this data we construct a 2-category whose objects are $N$ and $M$, whose 1-morphisms are generated by the two bimodules $_N M_M$ and $_M M_N$, and whose 2-morphisms are intertwinors.
(So for example the 1-morphisms are $M\otimes_N M\otimes_N\cdots\otimes_N M$, thought of as either an $N$-$N$ or $N$-$M$ or $M$-$N$ or $M$-$M$ bimodule.) You can think of the usual planar algebra definition as axiomatizing the "string diagrams" you would draw for this 2-category.

The diagrams in the paper I referred to are rotated 90 degrees from my explanation above. The left and right sides of the rectangles in the paper correspond to the $f$ and $f'$ of your (David's) original question. The tops of the rectangles corresponds to your $\phi$, and the interiors of the rectangles correspond to your $\phi^\sharp$.