Definitions
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Structures on 2-categories
A lax-idempotent 2-monad, also called a Kock–Zöberlein or KZ 2-monad, encodes a certain kind of property-like structure that a category, or more generally an object of a 2-category, can carry.
Lax-idempotent 2-monads have occasionally also been called KZ monads in the literature, but this terminology may be confusing, as it is inconsistent with terminology of lax-idempotent -monads: a 1-monad may be viewed as a 2-monad on a locally-discrete 2-category, in which case lax-idempotence is equivalent to idempotence.
The archetypal examples are given by 2-monads on Cat that take a category to the free cocompletion of under a given class of colimits – then an algebra is a category with all such colimits, which are of course essentially unique. Moreover, given thus-cocomplete categories and , a functor , and a diagram in , there is a unique arrow given by the universal property of the colimit. It is this property that lax-idempotence generalizes.
A 2-monad on a 2-category is called lax-idempotent if given any two (strict) -algebras , and a morphism , there exists a unique 2-cell making a lax morphism of -algebras:
Dually, a 2-monad is called colax-idempotent if gives rise to a colax -morphism :
A 2-monad is lax idempotent if and only if, for every object , there is an adjunction with invertible counit or an adjunction with invertible counit (with either adjunction implying the other).
An extensive list of equivalent conditions is given on the page of lax-idempotent 2-adjunctions.
A 2-monad as above, with unit , is lax-idempotent if and only if for any -algebra there is a 2-cell such that are the unit and counit of an adjunction .
(Adapted from Kelly–Lack). The multiplication is a -algebra on , and is a morphism from the underlying object of to that of . So there is a unique making into a lax -morphism. Set . The triangle equalities then require that:
is equal to . The composite makes a lax -morphism from to (paste with the identity square ). But , and also makes this into a lax -morphism, so by uniqueness .
is equal to . But this follows directly from the unit coherence condition for the lax -morphism .
Conversely, suppose , algebras on and are given. Take to be the mate of with respect to the adjunctions and , which is given in this case by pasting with , so we have that . The mate of in turn is given by , which because mates correspond bijectively is equal to . So satisfies the unit condition.
Consider the diagrams expressing the multiplication condition: because (and the same for ), their boundaries are equal, so we have 2-cells . Their mates under the adjunction are given by pasting with . One is pasted with , and the other is given by composing with the identity (and then pasting with ), but because this is also equal to . The two original 2-cells are hence equal, because their mates are equal, and so is indeed a lax -morphism.
Since ‘s multiplication makes itself into a (generalized) -algebra, the above implies (and in fact is implied by) the requirement that there exist a modification making . Conversely, given an algebra , the 2-cell is given by .
A different but equivalent condition is that there be a modification such that and ; and given as above, is given by .
These various conditions can also be regarded as ways to say that the Eilenberg-Moore adjunction for is a lax-idempotent 2-adjunction. Thus, is a lax-idempotent 2-monad exactly when this 2-adjunction is lax-idempotent, and therefore also just when it is the 2-monad induced by some lax-idempotent 2-adjunction.
Dually, for to be colax-idempotent, it is necessary and sufficient that any of the following hold.
For any -algebra there is a 2-cell such that .
There is a modification making .
There is a modification such that and .
Theorem gives a necessary condition for an object to admit a -algebra structure, namely that admit a left adjoint with identity counit. In the case of pseudo algebras, this necessary condition is also sufficient.
To give a pseudo -algebra structure on an object is equivalently to give a left adjoint to with invertible counit.
In particular, an object admits at most one pseudo -algebra structure, up to unique isomorphism. Thus, -algebra structure is property-like structure.
In many cases it is interesting to consider the pseudo -algebras for which the algebra structure has a further left adjoint, forming an adjoint triple. Algebras of this sort are sometimes called continuous algebras.
As mentioned above, the standard examples of lax-idempotent 2-monads are those on whose algebras are categories with all colimits of a specified class. In this case, the 2-monad is a free cocompletion operation. Dually, there are colax-idempotent 2-monads which adjoin limits of a specified class. A converse is given by (PowerCattaniWinskel), who show that a 2-monad is a monad for free cocompletions if and only if it is lax-idempotent and the unit is dense (plus a coherence condition).
Another important example of a colax-idempotent 2-monad is the monad on that takes to the projection out of the comma category. The algebras for this monad are Grothendieck fibrations over ; see also fibration in a 2-category. The monad is lax-idempotent, and its algebras are opfibrations.
This latter is actually a special case of a general situation. If is a (2-)monad relative to which one can define generalized multicategories, then often it induces a lax-idempotent 2-monad on the 2-category of such generalized multicategories (aka “virtual -algebras”), such that (pseudo) -algebras are equivalent to (pseudo) -algebras. When is the 2-monad whose algebras are strict 2-functors and whose pseudo algebras are pseudofunctors , then a virtual -algebra is a category over , and it is a pseudo -algebra just when it is an opfibration. Similarly, there is a lax-idempotent 2-monad on the 2-category of multicategories whose pseudo algebras are monoidal categories, and so on.
pseudo-distributive laws involving lax-idempotent 2-monads have an especially nice form; see (Marmolejo) and (Walker).
For ordinary 1-monads there exists a presentation due to Manes as “Kleisli triples” with primary data a family of unit morphisms and lifts avoiding the iteration of the endofunctor. A similar presentation exists for lax-idempotent 2-monads as shown in Marmolejo-Wood (2012). It is shown then in Walker (2017) that provided the units of this presentation are fully faithful (a reflection of the fully-faithfulness of the Yoneda embedding) (almost) all the axioms of a Yoneda structure are satisfied. In cases where size plays no role like e.g. the ideal completion of posets the two concepts coincide. For further details see at Yoneda structure or Walker (2017).
Classical references are
Max Kelly, Steve Lack, On property-like structures, TAC 3(9), 1997. (abstract)
Anders Kock, Monads for which structures are adjoint to units , Aarhus Preprint 1972/73 No. 35. (pdf)
Anders Kock, Monads for which structures are adjoint to units, JPAA 104:41–59, 1995.
Ross Street, Fibrations and Yoneda’s lemma in a 2-category, Lecture Notes in Mathematics, Vol. 420, 1974, pp. 104–133. [doi:10.1007/BFb0063102]
Ross Street, Fibrations in Bicategories , Cah. Top. Géom. Diff. XXI no.2 (1980). (numdam)
Volker Zöberlein, Doctrines on 2-categories , Math. Zeitschrift 148 (1976) pp.267-279. (gdz)
Francisco Marmolejo, Doctrines whose structure forms a fully faithful adjoint string, Theory and Applications of Categories 3 (1997), 23–44. (TAC)
Textbook accounts:
Peter Johnstone, Sketches of an elephant vol.1 , Oxford UP 2004. (B1.1.11, pp.250-54)
Marta Bunge, Jonathon Funk, Singular coverings of Toposes , Springer Heidelberg 2006. (pp.79ff)
See also:
Marta Bunge, Tightly Bounded Completions , TAC 28 no.8 (2013) pp.213-240. (abstract)
Marta Bunge, Jonathon Funk, On a bicomma object condition for KZ-doctrines, JPAA 143 (1999) 69-105 [doi:10.1016/S0022-4049(98)00108-X]
A. J. Power, G. L. Cattani, G. Winskel, A representation result for free cocompletions, JPAA 151:273–286, 2000 doi
Their distributive laws come into focus in
Francisco Marmolejo, Distributive laws for pseudomonads, Theory and Applications of Categories, 5 5 (1999) 81-147 [tac]
Francisco Marmolejo, Richard J. Wood, Kan extensions and lax idempotent pseudomonads , TAC 26 no.1 (2012) pp.1-19. (abstract)
Charles Walker, Distributive Laws via Admissibility, arXiv
The relation to Yoneda structures:
The logical-syntactical side:
Discussion of the adjoint functor theorem:
Last revised on October 17, 2024 at 08:40:02. See the history of this page for a list of all contributions to it.