Para Construction

The Para construction turns a category acted on by parameters into one whose morphisms carry their own parameter object, giving the parametrisation half of learning as a category. Three claims stand behind it: that Para is not merely a construction but a pseudomonad (and, once parameters copy, a grothendieck-construction); that a base action lifts through it by a reverse swap; and that these moves are floors of a single tower of actegory 2-categories rather than a list of independent gadgets.

Parametrised morphisms

An -actegory is a category equipped with an action of a monoidal . Over it, the Para bicategory keeps the objects of and lets each morphism carry a parameter drawn from .

Definition. The Para bicategory

has the objects of . A 1-cell is a pair of a parameter and a parametrised map; a 2-cell is a reparametrisation with . Composition tensors parameters,

modulo the actegory associator, and inserts the unit parameter.

Pseudomonad and Grothendieck construction

Para acts not just on a single actegory but on the whole 2-category of them. Write for the 2-category of -actegories, strong functors, and their transformations.

Proposition. Para is a pseudomonad on

is again an -actegory, and is a pseudofunctor carrying a multiplication and a unit. The multiplication flattens a nested parameter by tensoring the two, and the unit inserts the unit parameter. Subject to the unitor and associator coherences, is a pseudomonad. A parametrised category is therefore an algebra-like object over it rather than an ad-hoc gadget, which is the sense in which “parametrisation” is one structured operation.

Remark. Copyable parameters make it a Grothendieck construction

If carries a comonoid structure, so that parameters admit copy and discard, each gives a writer comonad on , whose coKleisli maps are exactly the parametrised maps . Assembling these into an indexing appears to recover as a grothendieck-construction, so that attaching a parameter is fibring a category of parameters over the base. The comonoid hypothesis does real work, since without it the coKleisli story is only heuristic.

Lifting the base action by the reverse swap

For the tower below to hold, a base -action must survive Para. It does, provided the acting object can be slid past the Para parameter. Lift the action to by acting on the underlying map while leaving the parameter untouched,

where must move through to reach , the job of the reverse swap

built from the actegory associator and the braiding of .

Proposition. Para(F) is a morphism of -actegories

The assignment is a monoid map . The reverse swap makes the -action commute past the Para parameter, so the unit and multiplication of act coherently. Consequently, for a strong the lifted is a morphism of -actegories, the compatibility lemma the transport tower depends on.

The actegory transport tower

The separate moves of taking states, adding parameters, and adding a backward leg line up as one tower, each floor built by a functor that transports the actegory structure of the floor below:

, , and the section are not just operations on single actegories but functors of these 2-categories, so structure propagates one floor at a time.

Proposition. The propagation lemmas

Four transport lemmas do the lifting. (i) If is a markov-category, then is defined for every -actegory . (ii) If is an -actegory, then is again an -actegory. (iii) is a -actegory, taking states of the acting monoidal category too. (iv) A strong monoidal lets one restrict an -action along to an -action. Chaining (i)–(iv) carries a base -actegory up through states, parameters, and lenses without re-proving coherence at each stage.

Remark. A tower, not a list

Presenting these as floors of one tower answers the worry logged in learning as a category economically. The constructions are not independent gadgets to be unified by hand but successive images of one another under structure-preserving functors. The restriction step (iv) is the joint that lets an action defined upstairs be viewed downstairs, and it is the place to check that nothing is silently strictified.

Open questions

A balanced tensor of actegories, and Lens the same way?

The reverse swap looks like one leg of a bimodule structure, and a balanced tensor of -actegories that quotients by it would “resemble a coend”. Two things are open: whether such a tensor of actegories can be defined so that and its action are its shadow, and whether the backward-leg passage , the cornering of optics, can be produced by the same lifting recipe rather than by a separate construction.

References

• M. Capucci and B. Gavranovic, Actegories for the Working Mathematician (arXiv:2203.16351)
• G. Cruttwell, B. Gavranovic, N. Ghani, P. Wilson, F. Zanasi, Categorical Foundations of Gradient-Based Learning (ESOP 2022)