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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 $n$-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 $T$ on Cat that take a category $C$ to the free cocompletion $T C$ of $C$ under a given class of colimits – then an algebra $T C \to C$ is a category $C$ with all such colimits, which are of course essentially unique. Moreover, given thus-cocomplete categories $C$ and $D$, a functor $F \colon C \to D$, and a diagram $S$ in $C$, there is a unique arrow $colim T F S \to F(colim S)$ given by the universal property of the colimit. It is this property that lax-idempotence generalizes.
A 2-monad $T$ on a 2-category $K$ is called lax-idempotent if given any two (strict) $T$-algebras $a \colon T A \to A$, $b \colon T B \to B$ and a morphism $f \colon A \to B$, there exists a unique 2-cell $\bar f \colon b \circ T f \Rightarrow f \circ a$ making $(f,\bar f)$ a lax morphism of $T$-algebras:
Dually, a 2-monad $T$ is called colax-idempotent if $f \colon A \to B$ gives rise to a colax $T$-morphism $(f,\tilde f)$:
A 2-monad $(T, \mu, \eta)$ is lax idempotent if and only if, for every object $A$, there is an adjunction $T \eta_A \dashv \mu_A$ with invertible counit or an adjunction $\mu_A \dashv \eta_{T A}$ 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 $T$ as above, with unit $\eta: 1 \to T$, is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta_A \circ a$ such that $(\theta_a ,1_{1_A})$ are the unit and counit of an adjunction $a \dashv \eta_A$.
(Adapted from Kelly–Lack). The multiplication $\mu_A \colon T^2 A \to T A$ is a $T$-algebra on $T A$, and $\eta_A \colon A \to T A$ is a morphism from the underlying object of $a$ to that of $\mu_A$. So there is a unique $\bar\eta_A \colon \mu_A \circ T \eta_A = 1_{T A} \Rightarrow \eta_A \circ a$ making $\eta_A$ into a lax $T$-morphism. Set $\theta_a = \bar\eta_A$. The triangle equalities then require that:
$a \bar\eta_A \colon a \Rightarrow a \circ \eta_A \circ a = a$ is equal to $1_a$. The composite $a \circ \bar\eta_A$ makes $a \circ \eta_A$ a lax $T$-morphism from $a$ to $a$ (paste $\bar\eta_A$ with the identity square $a \circ \mu_A = a \circ T a$). But $a \circ \eta_A = 1_A$, and $1_a$ also makes this into a lax $T$-morphism, so by uniqueness $a \bar\eta_A = 1_a$.
$\bar\eta_A \eta_A \colon \eta_A \Rightarrow \eta_A \circ a \circ \eta_A = \eta_A$ is equal to $1_{\eta_A}$. But this follows directly from the unit coherence condition for the lax $T$-morphism $\bar\eta_A$.
Conversely, suppose $\theta_a$, algebras $a,b$ on $A,B$ and $f \colon A \to B$ are given. Take $\bar f$ to be the mate of $1_f \colon b \circ T f \circ \eta A = f \Rightarrow f$ with respect to the adjunctions $a \dashv \eta_A$ and $1 \dashv 1$, which is given in this case by pasting with $\theta_a$, so we have that $\bar f = b \circ T f \circ \theta_a$. The mate of $\bar f$ in turn is given by $\bar f \circ \eta_A$, which because mates correspond bijectively is equal to $1_f$. So $\bar f$ satisfies the unit condition.
Consider the diagrams expressing the multiplication condition: because $a \circ \mu_A = a \circ T a$ (and the same for $b$), their boundaries are equal, so we have 2-cells $\alpha, \beta \colon b \circ T b \circ T^2 f \Rightarrow f \circ a \circ T a$. Their mates under the adjunction $(T\theta_a, 1) \colon T a \dashv T\eta_A$ are given by pasting with $T \eta_A$. One is $\bar f$ pasted with $T \bar f \circ T \eta_A = T(f \circ \eta_A) = T 1_f = 1_{T f}$, and the other is given by composing $T \eta_A$ with the identity $\mu_B \circ T^2 f = T f \circ \mu_A$ (and then pasting with $\bar f$), but because $\mu_A \circ T \eta_A = 1_{T A}$ this is also equal to $1_{T f}$. The two original 2-cells are hence equal, because their mates are equal, and so $\bar f$ is indeed a lax $T$-morphism.
Since $T$‘s multiplication $\mu$ makes $T$ itself into a (generalized) $T$-algebra, the above implies (and in fact is implied by) the requirement that there exist a modification $\ell \colon 1_{T^2} \to \eta T \circ \mu$ making $(\ell,1) \colon \mu \dashv \eta T$. Conversely, given an algebra $a \colon T A \to A$, the 2-cell $\theta_a$ is given by $T a \circ \ell_A \circ T \eta_A$.
A different but equivalent condition is that there be a modification $d \colon T \eta \to \eta T$ such that $d \eta = 1$ and $\mu d = 1$; and given $\ell$ as above, $d$ is given by $\ell \circ T \eta$.
These various conditions can also be regarded as ways to say that the Eilenberg-Moore adjunction for $T$ is a lax-idempotent 2-adjunction. Thus, $T$ 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 $T$ to be colax-idempotent, it is necessary and sufficient that any of the following hold.
For any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\zeta_a \colon \eta_A \circ a \Rightarrow 1$ such that $(1,\zeta_a) \colon \eta_A \dashv a$.
There is a modification $m \colon \mu \circ \eta T \to 1$ making $(1,m) \colon \eta T \dashv \mu$.
There is a modification $e \colon \eta T \to T\eta$ such that $e\eta = 1$ and $\mu e = 1$.
Theorem gives a necessary condition for an object $A$ to admit a $T$-algebra structure, namely that $\eta_A : A \to T A$ admit a left adjoint with identity counit. In the case of pseudo algebras, this necessary condition is also sufficient.
To give a pseudo $T$-algebra structure on an object $A$ is equivalently to give a left adjoint to $\eta_A : A\to T A$ with invertible counit.
In particular, an object admits at most one pseudo $T$-algebra structure, up to unique isomorphism. Thus, $T$-algebra structure is property-like structure.
In many cases it is interesting to consider the pseudo $T$-algebras for which the algebra structure $T A \to A$ 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 $Cat$ 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 $\eta$ is dense (plus a coherence condition).
Another important example of a colax-idempotent 2-monad is the monad on $Cat/B$ that takes $p \colon E \to B$ to the projection $B/p \to p$ out of the comma category. The algebras for this monad are Grothendieck fibrations over $B$; see also fibration in a 2-category. The monad $p \mapsto p/B$ is lax-idempotent, and its algebras are opfibrations.
This latter is actually a special case of a general situation. If $T$ is a (2-)monad relative to which one can define generalized multicategories, then often it induces a lax-idempotent 2-monad $\tilde{T}$ on the 2-category of such generalized multicategories (aka “virtual $T$-algebras”), such that (pseudo) $\tilde{T}$-algebras are equivalent to (pseudo) $T$-algebras. When $T$ is the 2-monad whose algebras are strict 2-functors $B\to Cat$ and whose pseudo algebras are pseudofunctors $B\to Cat$, then a virtual $T$-algebra is a category over $B$, and it is a pseudo $\tilde{T}$-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 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)
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 14, 2023 at 06:51:14. See the history of this page for a list of all contributions to it.