# Categorical nomenclature

I have a category $C$ and will build a new category $X_C$ out of it as follows. I hope this is a standard construction and that I can find somewhere in the literature its definition and properties worked out. Since I do not know the correct search string despite trying MacLane's book and the nlab, I ask here.

Each object of $X_C$ is a sub-collection $(v_{ij}:c_i \to c_j)$ of morphisms in $C$. The morphisms in $X_C$ between $(v_{ij}:c_i \to c_j)$ and $(w_{k\ell}: d_k \to d_\ell)$ are collections of morphisms $x_{ik} : c_i \to d_k$ of $C$ so that for any quadruple $(i,j,k,\ell)$ the obvious commutation relations hold, namely:

$$x_{j\ell}u_{ij} = w_{k\ell}x_{ik} \text{ as morphisms }c_i \to d_\ell.$$

What is this $X_C$ called? Thinking topologically, it looks like the "join" of the morphism diagrams, but google says nothing useful about "join categories" and the like.

The motivation is as follows: my category $C$ consists of objects where each morphism represents a distance between source and target. Under certain compatibility conditions on morphisms (the obvious commutation relations), I can find a "witness" object within the desired distance of all objects in a collection. The next step is to see how these witnesses evolve as $C$ itself is transformed. For this I need a way to map compatible object collections to other compatible object collections.

• By "sub-collection", am I correct to understand that $\operatorname{Obj}(X_C) = \mathcal{P}(\operatorname{Mor}(C))$, where $\mathcal P$ denotes the "power set" operation? Nov 26, 2012 at 5:06

It looks like the free coproduct completion of the arrow category of $C$ to me.

• I would agree or possibly the category of families of objects from the arrow category as in Janelidze's terminology for his Galois theory, and again when handling operads (perhaps)! Nov 26, 2012 at 11:46

I'm not sure if this is what you mean, but here are some terms that might come in handy to guide the search. If the collection $\{v_{ij}\}$ is always of the same shape, say with $i,j$ ranging over a fixed set, and is closed under composition, then what you're describing sounds like a functor category.

More precisely, such a category consists of diagrams $D$ of shape $J$. Here, $J$ is a category itself, with objects $i,j,\ldots$, and $D \colon J \to C$ is a functor. Explicitly, you can describe $D$ by giving all the objects $D(i)$ and all the morphisms $D(f)\colon D(i)\to D(j)$ for $f \colon i \to j$ in $J$. Your notation $c_i=D(i)$ suggests there is only one morphism $c_i \to c_j$. In that case $J$ has only one morphism $(i \leq j) \colon i \to j$, i.e. is a preorder $(J,\leq)$ regarded as a category; then $v_{ij} = D(i \leq j)$. Morphisms $D \to E$ are natural transformations. This category is usually denoted $C^J$.

The "witness" you mention at the end then sounds like a (co)limit of a diagram.

• But I guess that his $J$ can be any pre-order, or actually, any collection of disjoint arrows. What I mean: Let $I$ denote the interval category $a \to b$, with two objects and one non-trivial arrow. If I under stand Pinying correctly, then his $J$ can be any disjoint union of $I$'s. Unless his category $C$ has a distinguished object (e.g. a zero-object) I do not see how to resolve this in a natural way.
– jmc
Nov 26, 2012 at 5:56
• @Johan: I'm not sure I follow what you mean. A preorder is a category that has at most one morphism between any two objects. There can also be no arrow. In other words, a disjoint union of preorders is again a preorder. Nov 26, 2012 at 6:16
• @Chris, sorry, my fault. I was thinking of a directed set. Then my only remaining remark is that Pinying probably does not want a fixed $J$. And the only way (that I see) to get away with a fixed $J$ (very large) is by mapping unneeded arrows to some distinguished object of $C$.
– jmc
Nov 26, 2012 at 10:11