Re: Indiscrete objects in a functor category

[Peter Johnstone <ptj@dpmms.cam.ac.uk>]

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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