Background: Let $\mathbf{Lat}$ be the 2-category of lattices which can be viewed as a subcategory of the 2-cateogry of posets $\mathbf{Pos}$, that is, objects in $\mathbf{Pos}$ that have all finite products and coproducts (a.k.a meets and joins in lattice-speak); we may or may not require 2-morphisms in $\mathbf{Lat}$ to preserve meets and joins (i.e continuous and cocontinuous).

I am considering (what I call) cellular sheaves valued in lattices which are just functors $F: X \rightarrow \mathbf{Lat}$ where $X$ is the face relation poset of a cell complex. In order to do "sheaf theory" with sheaves valued in $\mathbf{Lat}$, it would be nice to have a notion of a coproduct and product in this category. I think product is fairly clear: just use the product partial order; meets and joins are what you think they would be. As far as a coproduct, I am not sure. If anyone has any suggestions? I have heard of a "free product of lattices" but it is not defined in a language I can understand. Not even clear to me that the "free product" that Gratzer defines is unique.

By "sheaf theory", I mainly mean taking limits and colimits over all the stalks. Another property I would like (or like to know doesn't hold) is the existence of all equalizers and (maybe if coproducts exist) coequalizers which would guarantee the existence of all small (co)limits. That would be lovely.

In general, looking for references on lattice theory, "non-abelian" sheaf theory, or anything about categories and lattice theory.

Thanks in advance!

alljoins, not just finite joins. $\endgroup$ – David Roberts Oct 10 '18 at 0:19