The forgetful functor [C,Set] --> Set/ob C always has a right adjoint, given by right Kan extension along the inclusion C_0 --> C, where C_0 is the discrete category with the same objects as C. For the same construction in a more general context, see B2.3.16 in `Sketches of an Elephant'. Peter Johnstone On Sat, 19 Dec 2015, Aleks Kissinger wrote:
It's common to describe the category of (directed, multi-) graphs as a functor category Graph := [2, Set], where 2 here is the category with 2 objects and 2 parallel arrows (s & t).
For a pair of sets (V,E), one can construct the indiscrete graph I(V,E) as a graph with vertices V and edges E x V x V, where the source and target maps are just the 2nd and 3rd projection respectively. This gives a right adjoint to the forgetful functor from Graph to pairs of sets. This enables one to construct a category of graphs with a fixed set of vetex/edge labels as a slice over Graph:
Graph / I(Lv, Le)
since a graph hm G --> I(Lv,Le) is the same as a map U(G) --> (Lv,Le), which is just a pair of functions assigning labels to the vertices and edges of G.
This seems to me like a pretty standard trick, which extends to any functor category from a C which is in some sense "suitably acyclic". For instance, consider a category of "partitioned graphs" [3, Set], where 3 has objects (P,V,E) and arrows:
E --s--> V, E --t--> V, and V --p--> P
where, p assigns each of the vertices a partition. For a triple (P,V,E) we can form the indiscrete partitioned graph with:
- partitions P - vertices V x P - edges E x (V x P) x (V x P) - p = pi2, s = pi2, t = pi3
which gives a right-adjoint to the forgetful functor from partitioned graphs to triples of sets. This is clearly an instance of a general recipe, whereby you start with the objects with no arrows out, and work your way backwards, always adding copies of the codomain of every out-arrow. Again one can attach labels to partitioned graphs by slicing:
[3,Set] / I(Lp,Lv,Le)
So, my question: Is the general case a known/studied construction? If so, could someone provide a reference?
Best,
Aleks
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