## The free T-category on a T-graph

### Steve Lack – 27 November 1996

This talk was inspired by the paper R.F.C.Walters, The free category with products on a multigraph, JPAA 62(1989), 205-210. I proposed an answer to the question ``If products are replaced by some other structure borne by a category, what should replace multigraphs?'' I consider a 2-monad T on Cat, and call a category with a T-algebra structure a T-category. Writing Cat^T for the 2-category of T-categories and morphisms strictly preserving this structure, and Cat//T for the ``double comma category'' whose objects are pairs of maps d,c:E-->TV in Cat; I then define a 2-functor from Cat^T to Cat//T, and show that it has a left adjoint if T has a rank. This 2-category Cat//T should be seen as replacing Cat(Mgph) used by Walters; and a T-graph is defined to be an object of Cat//T for which E and V are discrete.
Various extensions were discussed: dealing with non-strict maps, dealing with the non-monadic case, and replacing Cat by an arbitrary locally presentable 2-category.
These T-graphs seems to be the ``correct notion'' in various particular examples, but are difficult to motivate in the generality described above. There seems to be a connection, not yet well understood, with the paper Connected limits, familial representability and Artin gluing of Carboni and Johnstone. The definitions made in this talk seem a reasonable start to the problem, but there is clearly much still to be understood.

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