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Often a 2-monad is best replaced with a KZ monad. For example:

  • $Fun(B,Cat)$ is 2-monadic over $Cat/Ob B$, but KZ over $Cat/B$.

  • $SymMonCat$ is 2-monadic over $Cat$, but KZ over $Cat/Fin_\ast$.

  • The category of Beck-Chevalley bifibrations over $B$ is 2-monadic over $Cat/B$, but KZ over $Cat/Span(B)$ (where it consists of the cocartesian fibrations).

  • (This one is a bit of a stretch.) Finitely-cocomplete categories are 2-monadic over pre-derivators (i.e. functors $FinCat^{coop} \to Cat$), but the KZ forgetful functor to $Cat$ is often easier to use.

In case you've never heard of lax-idempotent (KZ) monads, I've included some background and expanded on the above examples at the end.

(N.B. I am blithely ignoring strictness considerations in this question.)

The pattern: The general pattern I am getting at above is that a 2-monad on one 2-category $C$ can often fruitfully be replaced with a lax-idempotent 2-monad on a differerent 2-category $C'$, with the same algebras. Thus one trades objects of $C$ with certain structure for objects of $C'$ with certain properties. One wins if an object of $C'$ is not much harder to specify than an object of $C$, and the properties in question are easier to check than the structure in question is to specify. Sometimes it's not even feasible to describe the original 2-monad directly, but it is nonetheless one can approach its algebras via the alternative description.

Question: Is there something systematic going on here? For example, is there some kind of adjunction between lax-idempotent 2-monads and all 2-monads, and we sometimes get lucky and find that it induces an equivalence of categories of algebras?

I think that the work of Cruttwell and Shulman, and also of Hermida is somehow relevant here. But it seems that passing from a categorical structure to the "virtual" version gives slightly different KZ replacements than the ones I've listed above. For example, Cruttwell and Shulman's theory takes the 2-monad for symmetric monoidal categories, and yields the adjunction between symmetric monoidal categories and symmetric multicategories -- not the adjunction between symmetric monoidal categories and Gamma-categories. And in their framework, one still needs to work directly with the original 2-monad.

Clubs also seem relevant, but I'm not sure they cover all the cases above.


Here is some background explanation.

What is a lax-idempotent 2-Monad?

There are a number of different ways to approach lax-idempotent 2-monads (sometimes called K(ock-)Z(oberlein) monads). For a flavor, here are some equivalent definitions. A lax-idempotent 2-monad $(T,\eta,\mu)$ on a 2-category $C$ is a 2-monad such that...

  1. ... there are canonically split adjunctions $\eta_{Tc} \dashv \mu_c \dashv T \eta_c$ for each $c \in C$.

  2. ... the simplicial object $T^\bullet : \Delta^{op} \to Fun(C,C)$ extends to a 2-functor $\Delta^{op} \to Fun(C,C)$ (where $\Delta^{op}$ is considered as a full 2-subcategory of $Cat^{op}$).

  3. ... a $T$-algebra $m : Tc \to c$ is equivalent to a split adjunction $m \dashv \eta_c$.

  4. ... One need not even specify the 2-monad structure explicitly. It suffices to give an assignment on objects $c \mapsto Tc$ and unit cells $\eta_c: c \to Tc$ satisfying a certain property -- see Marmolejo and Wood.

As usual in 2-category theory, there are 3 dual notions: (co)lax-idempotent 2-(co)monads.

Examples of lax-idempotent 2-monads:

The key fact, which can be seen from (3) above, is that an object $c \in C$ can admit a $T$-algebra structure in at most one way. That is, lax-idempotent 2-monads encode property-like structure. Replacing a structure with a property is often a good idea in higher category theory, so there are many familiar examples:

  • The "free colimit completion" 2-monad on $Cat$ is lax-idempotent. Likewise, the "free finite coproduct completion", or any other completion under a class of colimits.

We also have the examples alluded to above:

  • Let $B$ be a category. The unstraightening functor $Fun(B,Cat) \to Cat/B$, or equivalently (by the Grothdieck construction) the forgetful functor $OpFib(B) \to Cat/B$ is 2-monadic, inducing a lax-idempotent 2-monad on $Cat/B$, which sends $p : E \to B$ to $B \downarrow p$.

  • The functor $SymMonCat \to Fun(Fin_\ast, Cat)$ sending a symmetric monoidal category to the associated $\Gamma$-category is 2-monadic, inducing a lax-idempotent 2-monad on $Fun(Fin_\ast, Cat)$.

  • Composing the above two examples, one sees (I think -- I should be careful, since monadic functors are not stable under composition!) that $SymMonCat \to Cat/Fin_\ast$ is another example.

  • Let $B$ be a category with pullbacks. Let $BC(B) \subseteq Cat/B$ be the subcategory of functors which are simultaneously cartesian and cocartesian fibrations, and satisfy the Beck-Chevalley condition for all pullback squares. The forgetful functor is 2-monadic, but not lax-idempotent (in fact, it can be obtained from a distributive law between the 2-monad for cocartesian fibrations mentioned above, and the dual one for cartesian fibrations). However, we can encode the same data in terms of a fibration over $Span(B)$, and thus the forgetful functor $BC(B) \to Span(B)$ is lax-idempotent by the fibration example above. Variations are possible with different kinds of "Burnside categories" as the Barwick school likes to call them.

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I don't expect you'll be able to find one context that includes all such examples of "lax-idempotent replacements". But I can think of a couple other general examples that apply whenever your 2-monad equips categories with "algebraic structure".

  1. If $T$ is the Lawvere theory for the algebraic structure, i.e. the free category-with-products containing a model, then algebras are product-preserving functors $T\to \mathrm{Cat}$, and product-preservation is a lax-idempotent (even pseudo-idempotent) property.

  2. If $T$ is the free multicategory (or cartesian multicategory, etc.) containing a model of the algebraic structure, then algebras are multicategory functors $T\to \mathrm{Cat}$, which can be identified with opfibrations of multicategories over $T$, again a lax-idempotent property on multicategories over $T$.

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