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Stream: theory: category theory

Topic: Every double category is a double colimit in a canonical way

Patrick Nicodemus (Aug 13 2023 at 01:42):

We can regard 2-category theory as Cat-enriched category theory. In this setting, if $\mathcal{C}$ is a small category, then the weighted colimit of $1 : 1\to \mathbf{Cat}$ with weights in the "presheaf" $\mathcal{C} : 1\to \mathbf{Cat}$ is $\mathcal{C}$ itself. Thus every category can be written in a canonical way as the colimit of the one-object category. (This is well known but I would like a reference for it if anyone can find one)

Anyway, is there anything similar for double categories? Where every double category is in a canonical way a double colimit of the one object double category, the walking vertical arrow, and the walking composition of two vertical morphisms, or something like this. I am studying a functor on double categories which has a 1-categorical right adjoint and so should preserve colimits, and I am looking for an elegant way to state this, that its behavior on any double category is simply determined by gluing together the "primitive" double categories that make it up.

James Deikun (Aug 13 2023 at 12:33):

The statement in $\bold{Cat}$-enriched category theory is that the functor $1 : 1 \to \bold{Cat}$ is dense. You should probably look for a dense generator in $\bold{Cat}$-internal category theory.

Patrick Nicodemus (Aug 13 2023 at 23:05):

Ok, I'll post what I'm thinking so far.

Context/notation: Double categories are composed of two categories $\mathbb{E}_O$, $\mathbb{E}_A$ called "the category of objects" and "the category of vertical arrows", together with domain, codomain, composition, identity functors.
Morphisms in $\mathbb{E}_O$ are called "horizontal arrows." Objects in $\mathbb{E}_A$ are called "vertical arrows." Morphisms in $\mathbb{E}_A$ are called 2-cells.

My guess is that if I work with the 2-category of double categories, double functors, and horizontal natural transformations, I can construct a simple dense 2-functor which would work. So let $\mathbf{Dbl}$ refer to the 2-category of strict double categories, double functors and horizontal natural transformations.

Let $\Delta^0,\Delta^1,\Delta^2$ be the 0-dimensional, 1-dimensional, and 2-dimensional simplexes respectively, viewed as vertical categories (horizontally discrete). For example, the category $\Delta^1_A$ contains three objects: the "walking vertical 1 arrow" $v$ as well as identity vertical arrows for the domain and codomain objects of $v$.
The usual face and degeneracy maps in the simplex category do extend to double functors between these double categories. Thus we can assemble this into a diagram in the category of double categories, let me write $G= \Delta\mid_{[0],[1],[2]}$ for the full subcategory of the usual simplex category $\Delta$ containing $[0], [1], [2]$, and $R : G\to\mathbf{Dbl}$ for the "free vertical category" functor. The diagram $R$ itself does not contain any nontrivial natural transformations, as $G$ has no nontrivial natural transformations.

My working conjecture is that $R$ is dense in the $\mathbf{Cat}$-enriched sense? I still have to work out the details on pen and paper.

If $\mathbb{E} \in \mathbf{Dbl}$ is a double category then the $\mathbf{Cat}$-enriched nerve of $\mathbb{E}$ with respect to the functor $R$ is a $\mathbf{Cat}$-valued presheaf on $R$.
Let us first look at what $R$ looks like when we only consider the underlying presheaf of objects and ignore morphisms. If we look at its category of elements, $\operatorname{El}(N(R)(\mathbb{E}))$ is a 1-category whose objects are the set of double functors $[\Delta_0,\mathbb{D}] \amalg[\Delta_1,\mathbb{D}] \amalg [\Delta_2,\mathbb{D}]$, essentially the set of objects, vertical morphisms and vertical compositions in $\mathbb{D}$. This is fibered over $G$ by a projection functor $\pi: \mathrm{El}(N(R)(\mathbb{E})) \to G$.

This is a first approximation to the problem of writing $\mathbb{E}$ as the colimit of a diagram of simpler objects, the colimit $R\circ \pi$ would have the same objects and vertical morphisms as $\mathbb{E}$ (I think, up to bijection). Moreover we would recover the vertical composition structure of $\mathbb{E}$ and the unit arrows. However our construction so far is incomplete as it ignores the horizontal arrows in $\mathbb{E}$ and the 2-cells, i.e., it neglects the fact that $\mathbb{E}_O$ and $\mathbb{E}_A$ are categories and not just sets. This is why we need to work 2-categorically.