Welcome to MathOverflow

Question

6

1

Fix a field $K$ and consider a finite directed graph $\Gamma$ where multiple edges between a pair of vertices are allowed so long as the total number of edges is finite. Associate to each vertex $v$ a finite-dimensional $K$-vector space $K_v$, and to each edge $(u,v)$ a linear map $f_{uv}:K_u \to K_v$. There is no commutativity requirement on the composition along different paths between the same two vertices. Those familiar with representation theory will immediately recognize this as a (nice and finite) quiver representation.

Now, $\Gamma$ may also be interpreted as a finite diagram in the category of vector spaces with linear maps as morphisms and as such, it has a (co)limit. Here is my question:

Is there software which takes such a finite quiver representation $\Gamma$ as input and computes a basis for $\lim(\Gamma)$ as a vector space along with all the matrices $\lim(\Gamma) \to K_v$ expressed in that basis for each vertex $v$ of $\Gamma$? And similarly, is there software to compute a basis for $\text{colim}(\Gamma)$ along with the matrices $v \to \text{colim}(\Gamma)$?

In principle everything is finite-dimensional linear algebra: the computations should be tractable and I'm hoping someone has written and tested this already. The types of $\Gamma$s that I'm be most interested in are zig-zags, $$ \Gamma = v_1 \leftarrow v_2 \to v_3 \leftarrow v_4 \to \cdots \leftarrow v_n $$ and also "claw graphs" consisting of a distinguished vertex $u$, other vertices $v_1,\ldots,v_n$ and a single edge $(u,v_k)$ for each $1 \leq k \leq n$.

flag	asked Mar 14 at 16:57 Vidit Nanda 5,915●1●10●44

1

since you are dealing with quivers with no relations, wouldn't you get the direct sum over all the vertices, quotiented by the images of all the morphisms? For example, for a claw with three arrows you'd get the fibred sum on three terms. Maybe this is wrong, or maybe you've already figured it out but want a more concrete description of a basis. – Jacob Bell Mar 14 at 17:18

2

Jacob, I'd like the concrete description and the incoming maps expressed as matrices which respect the colimit's basis. I can compute all of this by hand and therefore probably code it, but computing limits and colimits in vect is such a basic operation that I was hoping someone else had already done it a decade ago. – Vidit Nanda Mar 14 at 17:22

1

In principle, this can be done as follows: Every colimit is built up out of coproducts and cokernels. For your quivers, induct on the size. Bases of coproducts are easy. As for cokernels, one has to find a basis of the image, using Gaussian elimination for example, and then extend this basis to the whole space. The added vectors give a basis for the cokernel. I also wonder if anyone has implemented this yet ... – Martin Brandenburg Mar 14 at 17:57

I presume this is all doable, by someone who can actually do it, which obviously isn't me. But the real question here is: why do you care about the colimit? – Jacob Bell Mar 15 at 0:08

(Co)limit computations for diagrams of Vector Spaces

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.