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A monad with arities is a monad that admits a generalized nerve construction. This allows us to view its algebras as presheaves-with-properties in a canonical way.
This generalized nerve construction also generalizes the construction of the syntactic category of a Lawvere theory.
Let be a category, and a subcategory. As explained at dense functor, for any object of , there is a canonical cocone over the forgetful functor , which we call the canonical -cocone at . The subcategory is called dense if this cocone is colimiting for every object of .
If be a category and is a dense subcategory, then the -nerve functor is given by
A monad on is said to have arities if sends canonical -cocones to colimiting cocones.
The nerve theorem consists of two statements:
I. If is dense in and if is a monad with arities on , then has a dense subcategory given by the free -algebras on objects of .
It follows (?) that the nerve functor is full and faithful. This allows us to view -algebras as presheaves (on ) with a certain property. The second part of the nerve theorem tells us what this property is.
II. Let be the free algebra functor. A presheaf is in the essential image of if and only if the restriction along ,
is in the essential image of .
For now, see the paper of Berger, Melliès, and Weber below…
See the discussion at
The associated paper is
- Mark Weber, Familial 2-functors and parametric right adjoints (2007) (tac)
These ideas are clarified and expanded on in