lax monoidal category

# Lax monoidal categories

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## Idea

A *lax monoidal category* is a [[monoidal category]] in which the associativity and unitality isomorphisms are replaced by not-necessarily-invertible transformations.  An *oplax monoidal category* is similar except that the transformations go in the other direction.

## Overview of variations

For ordinary monoidal categories, the [[biased]] and unbiased definitions coincide up to equivalence (though this is a nontrivial coherence theorem), but in the lax and oplax cases this is no longer true.  Moreover, in the biased cases we can make independent choices of the directions of various of the morphisms.  This yields the following variations (in all cases we omit the coherence axioms for now):

* An **unbiased lax monoidal category** has $n$-ary tensor products $(a_1\otimes \cdots\otimes a_n)$ for all $n\ge 0$, including a 0-ary unit $I = ()$, and generalized associativity maps such as

  $$((a\otimes b\otimes c) \otimes () \otimes (d\otimes e)) \to (a\otimes b\otimes c\otimes d\otimes e)$$

  and a unit map $a\to (a)$.  It is **(strictly) normal** if the later is an isomorphism (identity).  It is a special case of a [[lax algebra for a 2-monad]].

* Dually, an **unbiased oplax monoidal category** has generalized associativity maps in the opposite direction

  $$(a\otimes b\otimes c\otimes d\otimes e) \to ((a\otimes b\otimes c) \otimes () \otimes (d\otimes e))$$

  and a unit map $(a) \to a$.  It is a special case of an oplax algebra for a 2-monad.

* A **left-biased lax monoidal category** has a binary tensor product $a\otimes b$, a unit $I$, and constraints

  $$(a\otimes b)\otimes c\to a \otimes (b\otimes c) \qquad I \otimes a\to a \qquad a\otimes I\to a $$

* A **right-biased lax monoidal category** has a binary tensor product $a\otimes b$, a unit $I$, and constraints

  $$a \otimes (b\otimes c) \to (a\otimes b)\otimes c \qquad I \otimes a\to a \qquad a\otimes I\to a $$

* A **left-biased oplax monoidal category** has a binary tensor product $a\otimes b$, a unit $I$, and constraints

  $$(a\otimes b)\otimes c\to a \otimes (b\otimes c) \qquad a\to I \otimes a \qquad a\to a\otimes I $$

* A **right-biased oplax monoidal category** has a binary tensor product $a\otimes b$, a unit $I$, and constraints

  $$a \otimes (b\otimes c) \to (a\otimes b)\otimes c \qquad a\to I \otimes a \qquad a\to  a\otimes I $$

* A **left-skew monoidal category** has a binary tensor product $a\otimes b$, a unit $I$, and constraints

  $$(a\otimes b)\otimes c\to a \otimes (b\otimes c) \qquad  I \otimes a\to a \qquad a\to a\otimes I $$

* A **right-skew monoidal category** has a binary tensor product $a\otimes b$, a unit $I$, and constraints

  $$a \otimes (b\otimes c) \to (a\otimes b)\otimes c \qquad a\to I \otimes a \qquad a\otimes I\to a $$

Terminologically, note that we use "right" and "left" to indicate the direction of the biased associator, while the direction of the unitors is indicated by "lax" or "oplax" (if they are in the same direction) or "skew" (if they are in different directions).

The relationship between these concepts is summarised in the following table.

| If $V$ is a ___ monoidal category then... | $V^{rev}$ is... | $V^{op}$ is... | $V^{rev,op}$ is... |
|-----------------------|-------|------|--------------|
| left-biased lax | right-biased lax | right-biased oplax | left-biased oplax |
| right-biased lax | left-biased lax | left-biased oplax | right-biased oplax |
| left-biased oplax | right-biased oplax | right-biased lax | left-biased lax |
| right-biased oplax | left-biased oplax | left-biased lax | right-biased lax |
| left-skew | right-skew | right-skew | left-skew |
| right-skew | left-skew | left-skew | right-skew |

In other words, [[reverse monoidal category|$rev$]] and [[opposite category|$op$]] always swap direction; whilst $op$ interchanges lax and oplax (and has no effect on skew-monoidal categories).

There are two other possibilities, in which the definitions of left-skew and right-skew monoidal category are modified so that the associator is reversed, but they seem not to have been studied.

This use of "lax" and "oplax" in the biased case is justified by the fact that *biased (op)lax monoidal categories are a special case of unbiased ones*, with the $n$-ary tensor products defined in terms of the binary one by left- or right-associativity.  The unbiased structures arising in this way can be characterized as those in which certain generalized associativities are identities (or, up to equivalence, isomorphisms).

## Definitions

The above definitions are complete except for the coherence axioms.  In the unbiased case these can be deduced from the general notion of lax algebra.  In the biased cases, the axioms are roughly the same as those in MacLane's original definition of monoidal category: one pentagon, three unit triangles, and one unit-unit axiom.  (Kelly's later reduction of these to one pentagon and one unit triangle relies on invertibility of the constraints.)  In all cases it is possible to orient all of these axioms so as to make sense.

## Related notions

First note that all kinds of lax monoidal categories can be generalized to **lax monoids** in a [[monoidal bicategory]].  Using this generalization, we can make connections to many other kinds of monoidal structures:

* Oplax monoidal categories (i.e. oplax monoids in [[Cat]]) can be identified with [[multicategories]] that are "weakly representable".

* Multicategories themselves are precisely lax monoids in [[Span]], and also normal lax monoids in [[Prof]].

* We obtain various kinds of lax [[promonoidal category]] by working in [[Prof]] instead of [[Cat]].

* In particular, non-unital [[closed categories]] can be defined as biased lax monoids in [[Prof]] whose multiplication is corepresentable in the first argument and whose left unitor is invertible.  If the identity is also representable, we obtain a (unital) closed category.  The identification of biased lax monoids with particular unbiased ones thereby specializes to the identification of closed categories with closed multicategories.

* Just as the [[delooping]] of a [[monoidal category]] is a [[bicategory]], so too is the delooping of a skew-monoidal category a **skew-bicategory**.

## References

* [[John Bourke]], [[Stephen Lack]], _Skew monoidal categories and skew multicategories_, Journal of Algebra **506** (2018), 237-266. [[arXiv:1708.06088](https://arxiv.org/abs/1708.06088), [doi:10.1016/j.jalgebra.2018.02.039](https://doi.org/10.1016/j.jalgebra.2018.02.039)]

Skew monoidal closed structures on the category of [[Gray categories]] are considered in:

* [[John Bourke]], [[Gabriele Lobbia]], _A skew approach to enrichment for Gray-categories_, Advances in Mathematics volume 434 (2023). [[arXiv:2212.12358](https://arxiv.org/abs/2212.12358), [doi.org/10.1016/j.aim.2023.109327](https://doi.org/10.1016/j.aim.2023.109327)]

Lax monoidal categories are called **multitensors** in:

* [[Michael Batanin]] and [[Mark Weber]], _Algebras of higher operads as enriched categories_, Applied Categorical Structures 19 (2011): 93-135.

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lax morphism

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# Lax morphisms
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## Idea

In general, if $A$ and $B$ are [[categories]] (or, more generally, any category-like things, such as [[objects]] of some [[2-category]]) equipped with [[algebraic structure]], a *lax morphism* $f\colon A\to B$ is one which "preserves" the algebraic structure only up to a not-necessarily invertible transformation.

Of course, this transformation goes in one particular direction; a *colax morphism* is one where the transformation goes in the other direction.  The case of 2-monads, below, provides an almost universally applicable way to decide which direction is "lax" and which is "colax".

## Definition

### For algebras of 2-monads

Let $T$ be a [[2-monad]] on a 2-category $K$, and let $A$ and $B$ be (strict, pseudo, or even lax or colax) $T$-algebras.  A **lax $T$-morphism** $f\colon A\to B$ is a morphism in $K$ together with a [[2-cell]]

$$\array{ T A & \overset{T f}{\to} & T B\\
^{a} \downarrow & \swArrow & \downarrow^{b}\\
A & \underset{f}{\to} & B}$$

satisfying some axioms.

If the 2-cell goes in the other direction, then we say $f$ is a **colax $T$-morphism** (or **oplax $T$-morphism**).  Equivalently, a colax $T$-morphism is a lax $T^{co}$-morphism, where $T^{co}$ is the induced 2-monad on the 2-cell dual $K^{co}$ (see [[opposite 2-category]]).

If the 2-cell is invertible, we call $f$ a **pseudo** or **strong** $T$-morphism.

### For coalgebras of 2-comonads

Let $W$ be a 2-comonad on $K$, i.e. a 2-monad on the 1-cell dual $K^{op}$, and let $C$ and $D$ be $W$-coalgebras.  A **lax $W$-morphism** $f\colon C\to D$ is a morphism in $K$ together with a 2-cell

$$\array{ C & \overset{f}{\to} & D\\
^{c} \downarrow & \swArrow & \downarrow^{d}\\
W C & \underset{W f}{\to} & W D}$$

satisfying some axioms.

Note that a lax morphism of algebras for the 2-comonad $W$ is a *colax* morphism of algebras for the 2-monad $W^{op}$.  The reason we choose to call this direction for coalgebras "lax" is that if $T$ is a 2-monad with a right [[adjoint functor|adjoint]] $T^*$, then $T^*$ automatically becomes a 2-comonad such that $T^*$-coalgebras are the same as $T$-algebras, and with the above definition, lax $T$-morphisms coincide with lax $T^*$-morphisms.

## Examples

* A [[lax monoidal functor]] is a lax morphism for the 2-monad on [[Cat]] whose algebras are [[monoidal categories]].  Similarly, an [[oplax monoidal functor]] is a colax morphism for this 2-monad.

* A [[lax natural transformation]] between [[2-functors]] $C\to D$ is a lax morphism for the 2-monad on $[ob(C),D]$ whose algebras are 2-functors (which exists if $D$ is cocomplete and $C$ is small).  Similarly, an oplax natural transformation is a colax morphism for this 2-monad.  If $D$ is also complete, then this 2-monad has a right adjoint, which then as usual becomes a 2-comonad whose coalgebras are also 2-functors.  The above conventions for lax morphisms between coalgebras mean that a lax natural transformation is unambiguously "lax" rather than "colax", whether we regard the 2-functors as algebras for a 2-monad or coalgebras for a 2-comonad.

  Some authors have tried to change the traditional meanings of "lax" and "colax" in this case, but the general framework of 2-monads gives a good argument for keeping it this way (even if in this particular case, oplax transformations are more common or useful).

* A [[lax functor]] between 2-categories is a lax morphism for the 2-monad on Cat-graphs whose algebras are 2-categories.

* A [[lax algebra for a 2-monad]] $T$ is a lax morphism $T\to \langle A,A\rangle$ for the 2-monad whose algebras are 2-monads, where $\langle A,A\rangle$ is the [[codensity monad]] of the object $A$.

* If $T$ is a [[lax-idempotent 2-monad]], then (by definition) *every* morphism in the underlying 2-category $K$ between (the objects underlying) $T$-algebras has a unique structure of lax $T$-morphism.  For instance, every functor between categories with (some class of) [[colimits]] is a lax morphism for the 2-monad which assigns those colimits; the unique lax structure map is the canonical comparison $colim (F\circ D) \to F(colim D)$.  Such a morphism is strong/pseudo exactly when it preserves the colimits in question.

* For [[probability monads]] on a [[locally posetal 2-category]], such as the [[Radon monad#the_ordered_case|ordered Radon monad]], the lax morphisms of algebras corresponds to [[concave maps]] or a suitable generalization thereof.

## Categories of lax morphisms

For any 2-monad $T$, there are a 2-categories:

* $T Alg_l$ of $T$-algebras and lax morphisms
* $T Alg_c$ of $T$-algebras and colax morphisms
* $T Alg_p$ (frequently written just $T Alg$) of $T$-algebras and pseudo morphisms
* (if $T$ is strict) $T Alg_s$ of $T$-algebras and strict morphisms

We have obvious 2-functors

$$ \array{ & & & & T Alg_l \\
  & & & \nearrow\\
  T Alg_s & \to & T Alg_p\\
  & & & \searrow\\
 & & & & T Alg_c }$$

which are [[bijective on objects functor|bijective on objects]], [[faithful functor|faithful]] on 1-cells, and [[locally fully faithful 2-functor|locally fully faithful]].

Therefore, we can also assemble a number of [[F-categories]] of $T$-algebras and any suitable pair of types of $T$-morphism: strict+pseudo, strict+lax, strict+colax, pseudo+lax, or pseudo+colax.

If we want to consider both lax and colax $T$-morphisms together, the natural structure is a [[double category]]: there is a straightforward definition of the squares in a double category whose vertical arrows are colax $T$-morphisms and whose horizontal arrows are lax ones.  We could then, if we wish, add some "F-ness" to incorporate pseudo and/or strict morphisms as well.

The 2-category $T Alg_p$ is fairly well-behaved; for strict $T$, it admits all strict [[PIE-limits]] (if the base 2-category does), and therefore all [[2-limits]] (i.e. bilimits).  When $T$ is [[accessible functor|accessible]], $T Alg_p$ admits all 2-colimits as well (but not, in general, many strict 2-colimits).

However, the 2-categories $T Alg_l$ and $T Alg_c$ are not so well-behaved; they do not have many limits or colimits.  But once we enhance them to [[F-categories]], they admit all [[rigged limits]].  All three 2-categories also admit [[lax morphism classifier|morphism classifiers]]; that is, the inclusions $T Alg_s \to T Alg_*$ have left 2-adjoints.

## Related pages

* [[2-monad]]

* [[lax morphism classifier]]

## References

* R. Blackwell, [[Max Kelly]], [[John Power]], _Two-dimensional monad theory_, ([doi](https://doi.org/10.1016/0022-4049%2889%2990160-6))

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lax natural transformation

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## Idea

The notion of  _lax natural transformation_ is a generalization of the notion of [[natural transformation]] from [[category theory]] to [[higher category theory]].

As a natural transformation is a [[morphism]] between two [[functor]]s between [[categories]], a lax natural transformation is a morphism between [[2-functors]] between [[2-categories]]:

Where a natural transformation has a [[commutative square|commuting]] naturality square, a lax natural transformation has a [[2-morphism]] filling that square.  If that 2-morphism is required to be invertible, one speaks of a [[pseudonatural transformation]], and if it is required to be an identity (which implies that the square commutes), then one speaks of a [[strict 2-natural transformation]] (this latter notion only really makes sense for [[strict 2-functors]] between [[strict 2-categories]]).

In the general terminology of [[higher category theory]], a lax natural transformation may equivalently be called a (lax) [[k-transfor|1-transfor]].

## Definitions ##

Given (possibly weak) [[2-categories]] $C,D$ and (possibly [[lax functor|lax]] or oplax) [[2-functors]]
$F, G : C\to D$, a __lax natural transformation__ $\alpha:F\Rightarrow G$
is given by

* for each $A\in C$ a [[1-morphism]] $\alpha_A:F(A)\to G(A)$ in $D$,
as usual

* for each $f:A\to B$ in $C$ a [[2-morphism]] $\alpha_f: G(f) \circ
\alpha_A \Rightarrow \alpha_B \circ F(f)$:

\begin{tikzcd}
F A \ar[r,"F(f)"] \ar[d,"\alpha_A"']\ar[dr,phantom,"\Rightarrow"] & F B \ar[d,"\alpha_B"]\\
G A \ar[r,"G(f)"'] & G B
\end{tikzcd}

such that

* for each $A,B$, the $\alpha_f$ are the components of a
(strict) [[natural transformation]] $\alpha_{A,B}:
(\alpha_A)^* \circ G_{A,B} \dot{\to} (\alpha_B)_* \circ
F_{A,B}$

* the assignment $f\mapsto \alpha_f$ behaves sensibly with
respect to identities and composition (see the
references for details).

An __oplax natural transformation__ is as above, only with the $2$-cells $\alpha_f$ reversed.  This distinction is not entirely consistent in the literature; see the discussion of terminology below.

An (op)lax natural transformation $\alpha$ is a __[[pseudonatural transformation]]__ if each $\alpha_f$ is [[isomorphism|invertible]], and a __strict natural transformation__ or __[[strict 2-natural transformation]]__ if each is an [[identity morphism|identity]].

In all of these cases, the word 'natural' is often dropped for brevity.

## Categories and $n$-categories of lax transformations ##

Lax transformations from $F$ to $G$ and [[modification]]s between them form a [[category]] $Lax(F,G)$; likewise we have $Ps(F,G)$ and $Oplax(F,G)$ consisting of pseudo-natural and oplax transformations, respectively.

Pushing it up a notch, for 2-categories $C$ and $D$ we have hom-2-categories $2Cat_{lax}(C,D)$, $2Cat_{ps}(C,D)$, and $2Cat_{oplax}(C,D)$ whose objects are 2-functors (generally taken to be strong or strict), whose morphisms are lax, pseudo, or oplax transformations respectively, and whose 2-cells are modifications.  These hom-2-categories are the [[internal homs]] for the various version of the [[Gray tensor product]] on $2Cat$.

Finally, there is a [[3-category]] consisting of 2-categories, (strong or strict) 2-functors, pseudo-natural transformations, and modifications.  No laxness is possible at this level (without "laxifying" the notion of 3-category).

## "Lax" versus "oplax" ##

The choice of which direction to call "lax" and which to call "oplax" is not made consistently in the literature.  The convention used above is Benabou's original choice, as well as that of Leinster, Borceaux, and the majority of Australian writing on category theory.  However, some references, such as the [[Elephant]], make the opposite choice.  One or two references use "left lax" and "right lax" instead.

It is arguably the case that the direction we call "oplax" occurs more commonly in practice.  For instance, [[icons]] are a special sort of oplax transformations (although if "lax" and "oplax" were switched, then the acronymic derivation of the word "icon" would no longer work).  Likewise, when monoidal categories are viewed as one-object 2-categories, [[monoidal natural transformation]]s are special oplax transformations (in fact, they are precisely the icons).

On the other hand, the convention used above (besides having a little more weight of tradition) has the advantage that there is a [[2-monad]] whose algebras are 2-functors, and for which [[lax morphism|lax and oplax algebra morphisms]] are precisely lax and oplax transformations, respectively, under this convention.  Thus, for instance, theorems such as [[doctrinal adjunction]] can be applied to lax and oplax transformations without needing to switch back and forth between two different meanings of "lax."

+-- {: .query}
How consistent are different authors in this convention?  Should we include a warning ---or for that matter, a reassurance?  ---Toby

[[Finn Lawler|Finn]]:  I don't know myself, but back in [May](http://ncatlab.org/nlab/show/2009+May+changes) (see end of page) Jonas Frey said 'Johnstone in the elephant and Gurski in his PhD-thesis for example use an other orientation than Leinster and Borceux in their books'.  The one above is Leinster's.  I haven't seen any of the other references.

_Toby_:  OK, I just checked the Elephant, and it is reverse.

_Todd_: My own impression is that Leinster and Borceux are in the minority here, and that a bigger warning may be warranted. I think for instance that Johnstone is on the side of most Australians, who after all have done the most work in this area.

My own convention is that of Street: the arrow orientations in a product of [[parity complex]]es is to follow the usual sign conventions for chain complexes. For example, if I write $\alpha \otimes f$ instead of $\alpha_f$, and $\partial^-$ for 'source' and $\partial^+$ for 'target', and interpreting '+' as union, then for $\varepsilon \in \{-1, 1\}$ we would have

$$\partial^{\varepsilon}(\alpha \otimes f) = \partial^{\varepsilon}(\alpha) \otimes f + \alpha \otimes (-1)^{dim(\alpha)} \partial^\varepsilon f$$

so accordingly, since $dim(\alpha) = 1$, the set of source cells of $\alpha \otimes f$ would be $\{F \otimes f, \alpha \otimes G\}$, if you follow my drift. Street's rule of thumb then is that 'lax' would follow this "normal" order, and 'colax' the opposite.

_Toby_:  I didn\'t want to say anything before, but my own intuition follows Todd\'s logic.

[[Mike Shulman]]: Actually, my impression is that Leinster and Borceaux (specifically, the convention $\alpha_f: G(f) \circ
\alpha_A \Rightarrow \alpha_B \circ F(f)$) are on the side of the Australians.  See for instance p189 of Kelly, "On Clubs and Doctrines" in the Sydney Category Seminar reports.  It's also the one originally used by Benabou.  And the acronymic derivation of the word [[icon]] depends on using this convention.

I also think there is a very good reason to use that convention. Namely, given a small 2-category $C$, there is a [[2-monad]], say $T$, on $Cat^{ob(C)}$, whose algebras are functors $C\to Cat$ and such that a lax $T$-morphism, in the general sense of 2-monad theory, is a lax natural transformation with this convention.  Ideally there would be a general rule which we could apply in any given situation to decide which direction should be called "lax" and which "oplax;" this doesn't quite seem to be the case, but 2-monad theory covers the vast majority of cases, so I think we should stick to it whenever possible.

This isn't just for consistency and predictability, but so that we can apply _theorems_ of 2-monad theory without having to worry about the words being swapped around on us.  For example, [[doctrinal adjunction]] says that given a 2-monad $T$ on a 2-category $K$, two $T$-algebras $A$ and $B$, and an adjunction $f:A\leftrightarrows B:g$ in $K$, to give the left adjoint $f$ the structure of a colax $T$-morphism is the equivalent to giving the right adjoint $g$ the structure of a lax $T$-morphism.  This is one of those insufficiently-well-known facts that is frequently rediscovered in special cases.  For instance, given an adjunction between monoidal categories, to make the left adjoint a colax monoidal functor is the same as to make the right adjoint a lax monoidal functor.  And if I have a lax natural transformation $\alpha:F\to G$ such that each component $\alpha_A$ has a left adjoint, then those left adjoints automatically form an oplax natural transformation.  I think this would be unutterably confusing if "lax natural transformations" were the _oplax_ $T$-morphisms.

[[Todd Trimble]]: I got around finally to consulting Ross Street about this, and Steve Lack also entered the discussion. I think you may be right (Mike) and I wrong about which is more common in Australia; certainly Kelly for example goes along with Bénabou.

However, the feelings about that in Australia may be slightly less sanguine then yours. Ross says he has at times felt this was the wrong direction (see for example Coherence for Tricategories, where the direction goes the other way). For example, the direction for monoidal natural transformations is the reverse of Bénabou's. When Ross asked Steve if his opinion on this were strong or lax, he replied:

"I don't think I'd say I have a strong opinion.

"I agree that the 'other' direction (what has in Australia been called oplax, and which Todd recalls below) seems to be more important (e.g. prevalence of oplax limits/colimits, fact that they generalize monoidal transformations). I also feel that there is probably no perfect system - just as there no perfect system out which things are op and which are co.

"My 'limits for lax morphisms' suggests that there is a mismatch: 2-categories of lax morphisms have oplax limits, but again I think that if this were fixed something else would pop out elsewhere. For instance you can describe oplax naturals as being oplax morphisms somewhere or other.

"If oplax naturals were no longer called oplax naturals then the name icon would become odd (or should that be even odder) but part of the point of naming it was to get away these issues, so I don't think that is too serious." (End of Steve's email)

(Incidentally, I don't know or have forgotten this 'icon' business. What does it stand for, again?)

Ross summarized his feelings by emphasizing the wisdom of being flexible in these matters. Both directions arise and are important.

Ross also mentioned that the lax terminology began with the paper "Two constructions on lax functors", where the transformations in Bénabou's direction were dubbed "left lax", and the ones in the other "right lax". Perhaps we could use this terminology ourselves to avoid confusion, and cite this paper as reference. Your thoughts?

[[Mike Shulman]]: I don't really understand the point of view that "the more important" direction should be called lax and "the less important" direction oplax.  Are limits called "limits" rather than "colimits" because they're "more important"?  I wouldn't say so---it's just that we need to have some convention about which one gets the prefix.  As Ross said, both directions arise and are important, but that doesn't mean we should be inconsistent in naming them.  (Maybe inconsistency isn't what he means by "being flexible," but then I don't know what he means.)  Wouldn't it be confusing if the word "limit" meant either (what we now call) "limit" or "colimit" depending on which one a particular author cared more about at the moment?  And there are so many other *really* poor choices of terminology in mathematics, that I'm surprised people choose to fight a battle about the positioning of an "op."

Speaking for myself, "left lax" and "right lax" would confuse me more.  How do I remember which is which?  Neither one looks particularly "left" or "right" to me.

_Todd_: Well, it's not that I really care to argue about it that much. As a matter of historical record we _do_ have different conventions appearing in the literature, and you seemed to be plumping for one of those conventions earlier. Now I can't tell: it looks as though you're now saying that the convention is arbitrary, but we should stick with the original (which is Benabou's).

Linguistically, 'oplax' would be considered as a term derived from 'lax', so I think a natural tendency for people is to apply the primary term to whatever situation they feel is the more dominant. I think that's what Ross must have meant when he said that he's sometimes felt Benabou's direction is the "wrong" one. (I'm not sure if that applies to limit/colimit; in one sense, colimits could be considered secondary because their definition in the enriched context can be seen as based on hom functors hom(-, c) taking them to ordinary limits in the base of enrichment.)

Perhaps I shouldn't speak for Ross, but I know him a bit, and I think by 'flexible' he means not being too doctrinaire about these things -- his tendency would be to state at the outset of a paper "by lax transformation we mean..." and specify the direction clearly. He's bounced back and forth; apparently he feels the convention is not so strongly established yet that he needs to fix it once and for all, the important thing is to maintain consistency throughout a paper (or talk). But maybe you should ask him next time you see him, if you feel strongly about this.

So I'm guessing that Ross and Steve are tolerant and non-fussed about this particular issue, and they in particular are not the ones who battle over things like 'co' and 'op'.

Anyway, this whole discussion started only to sound a warning to the reader that conventions do differ in the literature. Note that I never said we should change the entry, and the arguments put forth for my own preference were not to start an argument. There are good arguments on both sides.

[[Mike Shulman]]: I'm sorry if I sounded too argumentative; I didn't mean to.  What I meant was that I'm surprised people have tried to change Benabou's original terminology at all.  I believe it was Jaap van Oosten who wrote that "the only thing worse than bad terminology is continually changing terminology."  Benabou did use some terminology quite worthy of being changed, like "morphism" and "homomorphism" for "lax 2-functor" and "strong 2-functor" respectively, but switching the meanings of "lax" and "oplax" seems like engendering a lot of confusion for very little gain, when the only argument in favor of it is that oplax ones are (maybe) more common.  On top of this, the argument about 2-monads is a positive reason to prefer Benabou's choice.

Anyway, in terms of productive work, I expanded the entry somewhat to include a warning to the reader about the different choices that exist and a summary of the reasons for making either choice.  Feel free to improve it.

_Todd_: I think you've certainly made a good start. Thanks for writing this up.
=--

## The Yoneda Lemma ##

Given a bicategory $C$, a lax functor $F:C^{op}\to Cat$ and a $0$-cell $A\in C$, there
are [[adjoint functor]]s

$$ I : Lax(y A,F) \stackrel{\to}{\leftarrow} F(A) : J $$

such that $J \dashv I$.  Here $y A:C \to Cat^{C^{op}}$ is the image of $A$ under the
bicategorical [[Yoneda embedding]].

+-- {: .proof}
###### Proof (sketch)

For $a\in F(A)$, let $J(a):y A\Rightarrow F$ be the lax transformation defined by $J(a)_B(f) = (F f)(a)$.  The components

$$  J(a)_g : F(g) \circ J(a)_X \dot{\to} J(a)_Y \circ g_* $$

for $g:X\to Y$ in $C$ are given by $F$'s comparison map $(F_{g,-})_a$.  The coherence conditions follow from those on $F$ wrt the associator and left unitor of $C$.  For $x:a\to b$ in $F(A)$, $J(x)$ is a modification because $F_{g,f}$ is a natural transformation (i.e. 2-cell in $Cat$).

Let $I(\alpha) = \alpha_A(1_A)$ as usual.  For $m:\alpha\ddot{\to}\beta$ a modification, $I(m)=(m_A)(1_A)$.

The unit $\eta: F A \dot{\to} I J$ at $a\in F A$ is the component $(F_A)_a$ of $F$'s comparison map, which is natural in $a$ by definition.  The counit $\epsilon_\alpha$ is obtained via the usual 1-chasing, and is given in components by

\[ (\epsilon_\alpha)_B(f) = \alpha_B(r_f) \circ (\alpha_f)(1_A) \]

where $r$ is the right unitor of $C$.  Some diagram-chasing confirms that this is indeed natural in everything, and so $\epsilon: J I \dot{\to} Lax(y A,F)$.

The triangle identities may be proved by expanding the definitions above and using the coherence conditions on lax functors and transformations and coherence of $C$.

See Gray for the case [[strict 2-categories]] and strict 2-functors.
=--

+-- {: .query}
[[Finn Lawler|Finn]]:  I don't know how natural this is in $F$ and $A$, but I'll try to figure it out.

Also, if you reverse the definitions of lax and oplax transformations, then you should do the same for functors, and I think in that case $I\dashv J$.

[[Mike Shulman]]: Please don't try to reverse the definitions of lax and oplax functors!  As far as I know everyone is consistent about those, regardless of the choices they make about lax and oplax transformations, so let's not create _additional_ confusion.  Lax functors have comparison maps that go $F g \circ F f\to F(g\circ f)$.
=--

## Related concepts

* [[homotopy]]

* [[transfor]]

  * [[natural transformation]]

    * [[extranatural transformation]], [[dinatural transformation]]

  * [[pseudonatural transformation]]

    * [[pseudo-extranatural transformation]]

  * **lax natural transformation**

  * [[2-dinatural transformation]]

  * [[lax F-natural transformation]]

## References ##

See the references at [[2-category]]. For instance (note slightly outdated terminology)

* Gray, _[[Gray-adjointness-for-2-categories|Adjointness For
2-Categories]]_

The definition is spelled out explicitly in the following, where they are called *lax transformations*:

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

In the following **oplax natural transformations** are defined, but called, simply, "transformations":

* [[Tom Leinster]], _Basic bicategories_,
[arXiv:math/9810017](http://arxiv.org/abs/math/9810017).

* [[Ross Street]], _Two constructions on lax functors_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 13 (1972) no. 3 , p. 217-264 [numdam](http://www.numdam.org/item?id=CTGDC_1972__13_3_217_0)

[[!redirects lax natural transformations]]
[[!redirects lax transformation]]
[[!redirects lax transformations]]
[[!redirects oplax natural transformation]]
[[!redirects oplax natural transformations]]
[[!redirects oplax transformation]]
[[!redirects oplax transformations]]

[[!redirects lax 2-natural transformation]]
[[!redirects lax 2-natural transformations]]
[[!redirects oplax 2-natural transformation]]
[[!redirects oplax 2-natural transformations]]


lax-idempotent 2-adjunction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

* automatic table of contents goes here
{:toc}

## Idea

A [[lax-idempotent 2-monad]] generalizes the notion of [[idempotent monad]] to [[2-categories]] by replacing [[inverse morphism|inverses]] with [[adjoints]].  A **lax-idempotent 2-adjunction** (or **KZ 2-adjunction**) similarly generalizes the notion of [[idempotent adjunction]], and is related to lax-idempotent [[2-monads]] in the same way that [[idempotent adjunctions]] are related to [[idempotent monads]].

## Definition

We will need to use all three kinds of composition in the 3-category $2 Cat$.  We write composition along 0-cells (2-functors, denoted with upper case Latin letters) with juxtaposition.  We write composition along 1-cells (functors, denoted with lower case Greek letters) with a dot; this is of course composition along 0-cells *in* a 2-category.  And we write composition along 2-cells (transformations, denoted with lower case Latin letters) with $\circ$, which is composition along 1-cells *in* a 2-category.

Let $F : C \rightleftarrows D : G$ be a [[2-adjunction]] with unit $\eta: 1_C \to G F$ and counit $\epsilon: F G \to 1_D$.  (For simplicity, we will assume it is a strict 2-adjunction, but the same definitions and proofs work in the pseudo case with some equalities replaced by isomorphisms.)
Write $M = GF$ and $\mu = G \epsilon F : M^2 \to F$ and $C = FG$ and $\delta = F \eta G : C \to C^2$.

The 2-adjunction is said to be **lax-idempotent** if one (hence all) of the following equivalent conditions hold (of which the latter 7 are dual to the former 7; the duality involves reversing 2-cells).

1. The triangle identity $1_F = \epsilon F . F\eta$ is the unit of an adjunction $F\eta \dashv \epsilon F$.

2. The induced equality $1_{G F} = G \epsilon F . G F\eta$ is the unit of an adjunction $G F\eta \dashv G \epsilon F$.

   In monad notation: the induced equality $1_M = \mu . M \eta$ is the unit of an adjunction $M \eta \dashv \mu$.

3. The induced equality $1_{F G} = \epsilon F G . F \eta G$ is the unit of an adjunction $F \eta G \dashv \epsilon F G$.

   In comonad notation: the induced equality $1_C = \epsilon C . \delta$ is the unit of an adjunction $\delta \dashv \epsilon C$.

4. The induced 2-monad $M = G F$ is [[lax-idempotent 2-monad|lax-idempotent]], meaning that every strict Eilenberg-Moore algebra $(A, \alpha)$ has the property that the identity $1_A = \alpha . \eta_A$ is the unit of an adjunction $\alpha \dashv \eta_A$.

5. There is a modification $w : G F \eta \to \eta G F$ such that $w . \eta = 1$ and $(G\epsilon F) . w = 1$.

   In monad notation: there is a modification $w : M \eta \to \eta M$ such that $w . \eta = 1$ and $\mu . w = 1$.

6. There is a modification $w' : G F \eta G \to \eta G F G$ such that $w' . (\eta G) = 1$ and $(G\epsilon F G) . w' = 1$.

7. There is a modification $w'' : F G F \eta \to F \eta G F$ such that $w'' . (F \eta) = 1$ and $(F G \epsilon F) . w'' = 1$.

And the duals (which we will refer to below using a prime, i.e. 1'-7'):

1. The triangle identity $G \epsilon . \eta G = 1_G$ is the counit of an adjunction $G \epsilon \dashv \eta G$.

2. The induced equality $F G \epsilon . F \eta G = 1_{F G}$ is the counit of an adjunction $F G \epsilon \dashv F \eta G$.

   In comonad notation: the induced equality $C \epsilon . \delta = 1_C$ is the counit of an adjunction $C \epsilon \dashv \delta$.

3. The induced equality $G \epsilon F . \eta G F = 1_{G F}$ is the counit of an adjunction $G \epsilon F \dashv \eta G F$.

   In monad notation: the induced equality $\mu . \eta M = 1_M$ is the counit of an adjunction $\mu \dashv \eta M$.

4. The induced 2-comonad $F G$ is colax-idempotent, meaning that every strict Eilenberg-Moore coalgebra $(Z, \zeta)$ has the property that the identity $\epsilon_Z . \zeta = 1_Z$ is the counit of an adjunction $\zeta \dashv \epsilon_Z$.

5. There is a modification $v : \epsilon F G \to F G \epsilon$ such that $\epsilon . v = 1$ and $v . (F\eta G) = 1$.

   In comonad notation: there is a modification $v : \epsilon C \to C \epsilon$ such that $\epsilon . v = 1$ and $v . \delta = 1$.

6. There is a modification $v' : \epsilon F G F \to F G \epsilon F$ such that $(\epsilon G) . v' = 1$ and $v' . (F \eta G F) = 1$.

7. There is a modification $v'' : G \epsilon F G \to G F G \epsilon$ such that $(G \epsilon) . v'' = 1$ and $v'' . (G F\eta G) = 1$.

+-- {: .proof}
###### Proof of equivalence

First of all, note that, by uniqueness of the left/right adjoint, most of the properties listed are (essentially) mere propositions. So we will content ourselves with proving logical equivalence.

We first prove $1 \Rightarrow 2 \Rightarrow 5 \Rightarrow 6 \Rightarrow 1'$, which by duality implies that 1, 2, 5, 6, 1', 2', 5' and 6' are equivalent.

* ($1 \Rightarrow 2$) Trivial by whiskering.

* ($2 \Rightarrow 5$) We have $M \eta \dashv \mu$ with unit $h = 1 : 1_M \to \mu . M \eta$ and counit $e : M \eta . \mu \to 1_{M^2}$. Let $w = e . \eta M : M \eta \to \eta M$. This satisfies the equations:

  * $w . \eta = e . \eta M . \eta = e . M \eta . \eta = (M \eta . h)^{-1} . \eta = 1_{M \eta . \eta}$ by an adjunction law.

  * $\mu . w = \mu . e . \eta M = (h . \mu)^{-1} . \eta M = 1_{\mu . \eta M} = 1_{1_M}$.

* ($5 \Rightarrow 6$) Trivial by whiskering.

* ($6 \Rightarrow 1'$) We take the unit $h : 1_{G F G} \to \eta G . G \epsilon$ of the desired adjunction to be
  $$
  \begin{aligned}
    1
    &\overset{triangle}{=} G F G \epsilon . G F \eta G\\
    &\xrightarrow{G F G \epsilon . w'} G F G \epsilon . \eta G F G\\
    &\overset{naturality}{=} \eta G . G \epsilon
  \end{aligned}
  $$
  This satisfies the required triangle identities:

  * $G \epsilon . h = G \epsilon . G F G \epsilon . w' = G \epsilon . G \epsilon F G . w' = G \epsilon . 1 = 1$,

  * $h . \eta G = G F G \epsilon . w' . \eta G = G F G \epsilon . 1 = 1$.

We now prove $1 \Rightarrow 3 \Rightarrow 5'$, adding $3$ and $3'$ to the set of equivalent properties.

* ($1 \Rightarrow 3$) Trivial by whiskering.

* ($3 \Rightarrow 5'$) We have $\delta \dashv \epsilon C$ with unit $h = 1 : 1_C \to \epsilon C . \delta$ and counit $e : \delta . \epsilon C \to 1_{C^2}$. Let $v = C \epsilon . e$. This satisfies the equations:

  * $\epsilon . v = \epsilon . C \epsilon . e = \epsilon . \epsilon C . e = \epsilon . (h . \epsilon C)^{-1} = 1_{\epsilon . \epsilon C}$.

  * $v . \delta = C \epsilon . e . \delta = C \epsilon . (\delta . h)^{-1} = 1_{1_C}$.

We now prove $5 \Rightarrow 7 \Rightarrow 1$, adding $7$ and $7'$ to the set of equivalent properties.

* ($5 \Rightarrow 7$) Trivial by whiskering.

* ($7 \Rightarrow 1$) We take the counit $e : F \eta . \epsilon F \to 1_{F G F}$ of the desired adjunction to be
  $$
  \begin{aligned}
    F \eta . \epsilon F
    &\overset{naturality}{=} \epsilon F G F . F G F \eta \\
    &\xrightarrow{\epsilon F G F . w''} \epsilon F G F . F \eta G F \\
    &\overset{triangle}{=} 1
  \end{aligned}
  $$
  This satisfies the required triangle identities:

  * $\epsilon F . e = \epsilon F . \epsilon F G F . w'' = \epsilon F . F G \epsilon F . w'' = \epsilon F . 1 = 1$,

  * $e . F \eta = \epsilon F G F . w'' . F \eta = \epsilon F G F . 1 = 1$.

Finally, we prove that $3'$ is equivalent to $4$ (hence dually, $3$ is equivalent to $4'$).

* ($3' \Rightarrow 4$) Let $(A, \alpha)$ be an algebra. Then $\alpha . \eta_A = 1_A$.
  We will carry over the structure of $(MA, \mu_A)$ by conjugating with $\alpha . {-} . \eta_A$.
  For $MA$, we have

  * $e_A = 1_A : \mu_A . \eta_{MA} \to 1_{MA}$,

  * $h_A : 1_{M^2 A} \to \eta_{MA} . \mu_A$.

  Define

  * $h' = M \alpha . h_A . M\eta_A : 1_{MA} = M \alpha . M\eta_A \to M \alpha . \eta_{MA} . \mu_A . M\eta_A = M \alpha . \eta_{MA} = \eta_A . \alpha$.

  This satisfies the adjunction laws:

  * $\alpha . h' = \alpha . M \alpha . h_A . M\eta_A = \alpha . \mu_A . h_A . M\eta_A = \alpha . (e_A . \mu_A)^{-1} . M\eta_A = \alpha . 1 . M\eta_A = 1$.

  * $h' . \eta_A = M \alpha . h_A . M\eta_A . \eta_A = M \alpha . h_A . \eta_{MA} . \eta_A = M \alpha . (\eta_{MA} . e_A)^{-1} . \eta_A = M \alpha . 1 . \eta_A = 1$.

* ($4 \Rightarrow 3'$) This is immediate, since $(MX, \mu_X)$ is always a strict Eilenberg-Moore algebra.

=--

Note that when $C$ and $D$ are locally [[discrete category|discrete]], hence just 1-categories, this reduces to the usual characterization of [[idempotent adjunctions]].

In contrast to that situation, however, the lax-idempotent situation is of interest even when the adjunction is monadic or comonadic.  In the monadic case, the implication $4 \Rightarrow 4'$ means that the induced 2-comonad on the 2-category of algebras for a lax-idempotent 2-monad is colax-idempotent.  Its (pseudo) coalgebras are the [[continuous algebras]] for the original 2-monad.

## References

As "KZ-adjunctions":

* J. Meseguer, U. Montanari, and V. Sassone. *ω-Inductive completion of monoidal categories and infinite petri net computations*. (1993) ([pdf](https://eprints.soton.ac.uk/261872/1/w-ind-compl.pdf))

As "KZ-adjointness" (defined by both (1) *and* (6)):

* [[Marta Bunge]] and [[Jonathon Funk]], *Singular Coverings of Toposes*.

As "lax idempotent 2-adjunctions":

* [[Maria Manuel Clementino]] and [[Fernando Lucatelli Nunes]], §3 of: _Lax comma 2-categories and admissible 2-functors_. [arXiv:2002.03132](https://arxiv.org/abs/2002.03132) (2020).

[[!redirects lax-idempotent 2-adjunction]]
[[!redirects lax-idempotent 2-adjunctions]]
[[!redirects lax idempotent 2-adjunction]]
[[!redirects lax idempotent 2-adjunctions]]
[[!redirects KZ 2-adjunction]]
[[!redirects KZ 2-adjunctions]]
[[!redirects Kock-Zoberlein 2-adjunction]]
[[!redirects Kock-Zoberlein 2-adjunctions]]


lax-idempotent 2-monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **lax-idempotent 2-monad**, also called a **Kock--Zöberlein** or **KZ 2-monad**, encodes a certain kind of [[stuff, structure, property|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 [[idempotent monad|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 [[pseudoalgebra for a 2-monad|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.

## Definition

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:
\[
\array{
T A & \overset{T f}{\to} & T B \\
a \downarrow & \swArrow \bar f & \downarrow b \\
A & \underset{f}{\to} & B
}
\]

Dually, a 2-monad $T$ is called **colax-idempotent** if $f \colon A \to B$ gives rise to a colax $T$-morphism $(f,\tilde f)$:
\[
\array{
T A & \overset{T f}{\to} & T B \\
a \downarrow & \neArrow \tilde f & \downarrow b \\
A & \underset{f}{\to} & B
}
\]

### Equivalent conditions

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-adjunction]]s.

+-- {: .num_theorem #AlgebraAdjoint}
###### Theorem
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]] $a \dashv \eta_A$.
=--

+-- { .proof}
###### Proof
**(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 [[adjunction|triangle equalities]] then require that:

  1.  $a \bar\eta_A \colon a \Rightarrow a \circ \eta_A \circ a = a$       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 $a \bar\eta_A = 1_a$.

  2. $\bar\eta_A \eta_A \colon \eta_A \Rightarrow \eta_A \circ a \circ \eta_A = \eta_A$ 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 category|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 $\mu e = 1$.

## Algebras

Theorem \ref{AlgebraAdjoint} 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.

+--{: .num_theorem #PseudoAlgebras}
###### Theorem
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]].

## Examples

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](#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 $B/p \to B$ 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.

## Properties

* [[pseudo-distributive laws]] involving lax-idempotent 2-monads have an especially nice form; see [(Marmolejo)](#MarmolejoDL) and [(Walker)](#WalkerDL).

* 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](#MW12)). It is shown then in Walker [(2017)](#WalkerYS) 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)](#WalkerYS).

## Related concepts

* [[idempotent monad]]
* [[2-monad]]
* [[stuff, structure, property]]
* [[property-like structure]]
* [[continuous algebra]]
* [[free cocompletion]]
* [[lax-idempotent 2-adjunction]]

## References

Classical references are

* [[Max Kelly]], [[Steve Lack]], _On property-like structures_, TAC 3(9), 1997. ([abstract](http://www.tac.mta.ca/tac/volumes/1997/n9/3-09abs.html))

* [[Anders Kock]], _Monads for which structures are adjoint to units_ , Aarhus Preprint 1972/73 No. 35. ([pdf](http://home.imf.au.dk/kock/msau1.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](https://doi.org/10.1007/BFb0063102)]

* [[Ross Street]], _Fibrations in Bicategories_ , Cah. Top. Géom. Diff. **XXI** no.2 (1980). ([numdam](http://www.numdam.org/item?id=CTGDC_1980__21_2_111_0))

* Volker Zöberlein, _Doctrines on 2-categories_ , Math. Zeitschrift **148** (1976) pp.267-279. ([gdz](https://gdz.sub.uni-goettingen.de/id/PPN266833020_0148?tify={%22pages%22:[273],%22view%22:%22info%22}))

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](http://tac.mta.ca/tac/volumes/28/8/28-8abs.html))

* {#BF99} [[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 {#PowerCattaniWinskel} doi

Their distributive laws come into focus in

* {#MarmolejoDL} [[Francisco Marmolejo]], *Distributive laws for pseudomonads*, Theory and Applications of Categories, **5** 5 (1999) 81-147 [[tac](http://tac.mta.ca/tac/volumes/1999/n5/5-05abs.html)]

* {#MW12} [[Francisco Marmolejo]], [[Richard J. Wood]], _Kan extensions and lax idempotent pseudomonads_ , TAC **26** no.1 (2012) pp.1-19. ([abstract](http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html))

* Charles Walker, *Distributive Laws via Admissibility*, [arXiv](https://arxiv.org/abs/1706.09575)
 {#WalkerDL}

The relation to [[Yoneda structures]]:

* Charles Walker, *Yoneda Structures and KZ Doctrines*, [arxiv](https://arxiv.org/abs/1703.08693)
 {#WalkerYS}

The logical-syntactical side:

*  [[Jiri Adamek]], Lurdes Sousa, *KZ-monadic categories and their logic*, [tac](http://tac.mta.ca/tac/volumes/32/10/32-10abs.html)

Discussion of the [[adjoint functor theorem]]:

* {#ALL23} [[Nathanael Arkor]], [[Ivan Di Liberti]], [[Fosco Loregian]], _Adjoint functor theorems for lax-idempotent pseudomonads_, [[arXiv:2306.10389](https://arxiv.org/abs/2306.10389)]

[[!redirects KZ monad]]
[[!redirects KZ-monad]]
[[!redirects KZ 2-monad]]
[[!redirects KZ doctrine]]
[[!redirects KZ-doctrine]]

[[!redirects KZ monads]]
[[!redirects KZ-monads]]
[[!redirects KZ 2-monads]]
[[!redirects KZ doctrines]]
[[!redirects KZ-doctrines]]

[[!redirects lax-idempotent 2-monad]]
[[!redirects lax-idempotent 2-monads]]
[[!redirects lax-idempotent monad]]
[[!redirects lax-idempotent monads]]

[[!redirects colax-idempotent 2-monad]]
[[!redirects colax-idempotent 2-monads]]
[[!redirects colax-idempotent monad]]
[[!redirects colax-idempotent monads]]

[[!redirects lax-idempotent]]
[[!redirects lax-idempotence]]
[[!redirects colax-idempotent]]
[[!redirects colax-idempotence]]

[[!redirects lax idempotent 2-monad]]
[[!redirects lax idempotent 2-monads]]
[[!redirects lax idempotent monad]]
[[!redirects lax idempotent monads]]
[[!redirects lax idempotent pseudomonad]]
[[!redirects lax idempotent pseudomonads]]
[[!redirects lax-idempotent pseudomonad]]
[[!redirects lax-idempotent pseudomonads]]

[[!redirects colax idempotent 2-monad]]
[[!redirects colax idempotent 2-monads]]
[[!redirects colax idempotent monad]]
[[!redirects colax idempotent monads]]
[[!redirects colax idempotent pseudomonad]]
[[!redirects colax idempotent pseudomonads]]
[[!redirects colax-idempotent pseudomonad]]
[[!redirects colax-idempotent pseudomonads]]

[[!redirects lax idempotent]]
[[!redirects lax idempotence]]
[[!redirects colax idempotent]]
[[!redirects colax idempotence]]


layered type theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

\tableofcontents

## Idea

In [[two-level type theory]], there are also two different kinds of judgments, one for the types representing the metatheory, and one for the types representing the object theory. In [[typed predicate logic]], there are two different kinds of judgments, one for [[types]] and one for [[propositions]]. In [[type theory with shapes]], there are three different kinds of judgments, one for cubes, one for topes, and one for types, where the subtheory of cubes and topes together form a predicate logic over a type theory with finite product types. Examples of [[type theory with shapes]] include certain presentations of [[simplicial type theory]] and [[cubical type theory]]. Independently, [[cubical type theory]] could be considered as a type theory over a typed predicate logic with only one type, where the single type in the predicate logic is called the "interval" and the propositions in the predicate logic layer are called "face formulas".

Thus, one could think of the different judgments as forming different **layers** of the type theory, and these type theories as **layered type theory**.

Some layered type theories have [[split contexts]], in which the contexts (and associated judgments) sit in a sequential hierarchy: for example, in two-level type theory, the [[types]] in the object theory can depend upon [[variables]] in the metatheory, but the types in the metatheory cannot depend upon variables in the object theory.

Layered type theories could be contrasted with theories like the usual presentations of [[Martin-Löf type theory]] and [[higher observational type theory]], which only have one layer and thus are considered to be **unlayered type theory**.

## Examples

* [[typed predicate logic]]
* [[two-level type theory]]
* [[type theory with shapes]]
* [[simplicial type theory]]
* [[cubical type theory]]
* Any dependent type theory with an infinite sequential hierarchy of [[Russell universes]] but no separate type judgment. This includes [[book HoTT]], the theory presented in the [[HoTT book]]. The Russell universes are indexed by the [[natural numbers]], which have to be provided as a primitive in a separate layer specifically containing the natural numbers.

## References

The notion of layer of a type theory appears in

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* $[$[arXiv:1705.07442](https://arxiv.org/abs/1705.07442)$]$

when defining [[type theory with shapes]] and [[simplicial type theory]].

The term "layered type theory" also appears in

* César Bardomiano Martínez, *Limits and colimits of synthetic $\infty$-categories* $[$[arXiv:2202.12386](https://arxiv.org/abs/2202.12386)$]$

[[!redirects layer]]
[[!redirects layers]]


layers of foundations

> There used to be here an attempt to organize [[foundations]] by starting with a most foundational foundation and successively adding [[axioms]]. This was discussed [here](http://nforum.mathforge.org/discussion/4347/layers-of-foundations/#Item_1). Then moved out of the $n$Lab to [[schreiber:layers of foundations|here]].


Layla Hormozi

* [Institute page](https://www.bnl.gov/staff/hormozi)

## Selected writings

On [[topological quantum compilation]] and approximating (cf. the [[Solovay-Kitaev theorem]]) given [[quantum gates]] by (i.e. compiling them to) [[quantum circuit|cicuits]] of [[anyon]] [[braid representation|braid]] [[quantum gates|gates]] (generally considered for [[su(2)-anyons]] and here mostly for universal [[Fibonacci anyons]], to some extent also for non-universal [[Majorana anyons]]):

* [[Nicholas E. Bonesteel]], [[Layla Hormozi]], [[Georgios Zikos]], [[Steven H. Simon]], *Braid Topologies for Quantum Computation*, Phys. Rev. Lett. **95** 140503 (2005) $[$[arXiv:quant-ph/0505065](https://arxiv.org/abs/quant-ph/0505065), [doi:10.1103/PhysRevLett.95.140503](https://doi.org/10.1103/PhysRevLett.95.140503)$]$

* [[Layla Hormozi]], [[Georgios Zikos]], [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Topological Quantum Compiling*, Phys. Rev. B **75** 165310 (2007) $[$[arXiv:quant-ph/0610111](https://arxiv.org/abs/quant-ph/0610111), [doi:10.1103/PhysRevB.75.165310](https://doi.org/10.1103/PhysRevB.75.165310)$]$

* [[Layla Hormozi]], [[Nicholas E. Bonesteel]], [[Steven H. Simon]], *Topological Quantum Computing with Read-Rezayi States*, Phys. Rev. Lett. **103** 160501 (2009) $[$[doi:10.1103/PhysRevLett.103.160501](https://doi.org/10.1103/PhysRevLett.103.160501), [arXiv:0903.2239](https://arxiv.org/abs/0903.2239)$]$

and particularly by "weaves":

* [[Steven H. Simon]], [[Nick E. Bonesteel]], [[Michael H. Freedman]], N. Petrovic, [[Layla Hormozi]], *Topological Quantum Computing with Only One Mobile Quasiparticle*, Phys. Rev. Lett. 96 (2006) 070503 ([arXiv:quant-ph/0509175](https://arxiv.org/abs/quant-ph/0509175), [doi:10.1103/PhysRevLett.96.070503](https://doi.org/10.1103/PhysRevLett.96.070503))

category: people


Lazard ring

#Contents#
* table of contents
{:toc}

## Idea

The _Lazard ring_ is a [[commutative ring]] which is

* the universal [[coefficient]] [[ring]] for commutative 1-dimensional [[formal group laws]].

and by [[Quillen's theorem on MU|Quillen's theorem]] also

* the [[cohomology ring]] that the [[complex cobordism cohomology theory]] assigns to the point.

## Definition

The _Lazard ring_ may be presented by generators $a_{i j}$ with $i,j \in \mathbb{N}$

$$
  L = \mathbb{Z}[a_{i j}] / (relations\;1,2,3\;below)
$$

and relations as follows

1. $a_{i j} = a_{j i}$

1. $a_{10} = a_{01} = 1$; $\forall i \neq 1: a_{i 0} = 0$

1. the obvious associativity relations imposed by $\ell(x, \ell(y, z)) = \ell(\ell(x, y), z)$

where we write
$$
  \ell(x,y) = \sum_{i,j} a_{i j} x^i y^j \in L[[x,y]].
$$

In other words, $\ell(x, y)$ becomes the universal 1-dimensional [[formal group law]] as a formal power series in two variables with coefficients in the Lazard ring, Theorem \ref{walking} below.

## Properties

### As classifying ring for formal group laws

+-- {: .num_theorem #walking}
###### Theorem

For any [[ring]] $S$ with [[formal group law]] $g(x,y) \in S[ [x,y] ]$ there is a unique ring [[homomorphism]] $L \to S$ that sends $\ell$ to $g$.

=--

review includes ([Hopkins 99, theorem 2.3, theorem 2.5](#Hopkins99))

+-- {: .num_remark}
###### Remark

Passing to [[formal dual]] [[spectrum of a commutative ring|ring spectra]], this says that $Spec(L)$ is something like the [[moduli space]] for formal groups.
By [[Quillen's theorem on MU]], the lift of $L$ to [[higher algebra]] is the [[E-infinity ring]] [[MU]] and the [[E-infinity ring spectrum]] $Spec(MU)$ is something like the derived moduli stack for formal group laws.

=--

### Lazard's theorem

[[Lazard's theorem]] states:

+-- {: .num_theorem}
###### Theorem

The Lazard ring is [[isomorphism|isomorphic]] to a [[graded ring|graded]] [[polynomial ring]]

$$
  L \simeq \mathbb{Z}[t_1, t_2, \cdots]
$$

with the [[variable]] $t_i$ in degree $2 i$.

=--

review includes ([Hopkins 99, theorem 2.5](#Hopkins99), [Lurie 10, lect 2, theorem 4](#LurieLect2))

### As the complex cobordism cohomology ring

By [[Quillen's theorem on MU]] the Lazard ring is the [[cohomology ring]] of [[complex cobordism cohomology theory]].

+-- {: .num_theorem}
###### Theorem

Let $M P $ denote the peridodic [[complex cobordism cohomology theory]]. Its [[cohomology ring]] $M P(*)$ over the point together with its [[formal group law]] is naturally isomorphic to the universal Lazard ring with its formal group law $(L,\ell)$.

=--

+-- {: .num_remark}
###### Remark

This can be used to make a cohomology theory out of a [[formal group law]] $(R,f(x,y))$. Namely, one can use the classifying map $M P({*}) \to R$
to build the [[tensor product]]

$$
  E^n(X) := M P^n(X) \otimes_{M P({*})} R,
$$
for any $n\in\mathbb{Z}$.
This construction could however break the left exactness condition. However, $E$ built this way will be left exact if the ring morphism $M P({*}) \to R$ is a flat morphism. This is the [[Landweber exactness]] condition (or maybe slightly stronger). See at _[[Landweber exact functor theorem]]_.

=--

## Related entries

* for some context see _[[A Survey of Elliptic Cohomology - cohomology theories]]_

* [[Landweber exact functor theorem]]

* [[Lazard's theorem]]

## References

* [[Michel Lazard]], *Sur les groupes de Lie Formels à un Paramètre*, Bull. Soc. France **83** (1955) [[numdam:BSMF_1955__83__251_0](http://www.numdam.org/item/?id=BSMF_1955__83__251_0)]

* [[Daniel Quillen]], _On the formal group laws of unoriented and complex cobordism theory_, Bull. Amer. Math. Soc. Volume 75, Number 6 (1969), 1293-1298.  ([Euclid](http://projecteuclid.org/euclid.bams/1183530915))

* {#Adams74} [[John Adams]], part II.5 of _[[Stable homotopy and generalised homology]], University of Chicago Press, Chicago, Ill., 1974, Chicago Lectures in Mathematics.

* {#Kochmann96} [[Stanley Kochmann]], section 4.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Hopkins99} [[Mike Hopkins]], _Complex oriented cohomology theories and the language of stacks_, 1999 course notes ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/coctalos.pdf))

* {#LurieLect2} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 2 _Lazard's theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture2.pdf))

* {#LurieLect3} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 3 _Lazard's theorem (continued)_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture3.pdf))

[[!redirects Lazard's ring]]


Lazard's criterion

> This page is about characterization of [[flat modules]]. For the characterization of the *[[Lazard ring]]* (in [[formal group laws]]) see instead at *[[Lazard's theorem]]*.

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

Let $R$ be a [[commutative ring]]. (or maybe any ring?)

+-- {: .num_prop}
###### Proposition
**(Lazard's criterion)**

An $R$-[[module]] is a [[flat module]] precisely if it is a [[filtered colimit]] of [[free modules]].

=--

This is due to ([Lazard (1964)](#Lazard)). See at _[[flat module]]_ for more.

## References

The original article:

* {#Lazard} [[Daniel Lazard]], _Sur les modules plats_ C. R. Acad. Sci. Paris **258** (1964) 6313-6316

Exposition:

* Robert Hines, *Lazard’s theorem (characterizing flatness)* (2016) [[pdf](https://math.colorado.edu/~rohi1040/expository/lazardstheorem.pdf), [[Hines-LazardTheorem.pdf:file]]]

* [[Stacks Project]], *Lazard's theorem* [[tag:058G](https://stacks.math.columbia.edu/tag/058G)]


Lazard's theorem

> This page is about characterizing the [[Lazard ring]] in [[formal group laws]]. For the characterization of [[flat modules]] see instead at *[[Lazard's criterion]]*.

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--

#Content#
* table of contents
{:toc}

## Statement

+-- {: .num_theorem}
###### Theorem

The [[Lazard ring]] is [[isomorphism|isomorphic]] to a [[graded ring|graded]] [[polynomial ring]]

$$
  L \simeq \mathbb{Z}[t_1, t_2, \cdots]
$$

with the [[variable]] $t_i$ in degree $2 i$.

=--

A proof is spelled out in ([Kochman 96, theorem 4.4.9](#Kochman96)). Another proof is indicated in ([Hopkins 99, section 2](#Hopkins99)), worked out in ([Mathew 12](#Mathew12), [Wilson 13](#Wilson13)). Also ([Lurie 10, lecture 2, theorem 4](#LurieLect2)) and ([Lurie 10, lecture 3](#LurieLect3)).

## Related theorem

* [[Quillen's theorem on MU]]

* [[Landweber exact functor theorem]]

* [[Landweber-Novikov theorem]]

## References

The proof is originally due to :

* [[Michel Lazard]], *Sur les groupes de Lie Formels à un Paramètre*, Bull. Soc. France **83** (1955) [[numdam:BSMF_1955__83__251_0](http://www.numdam.org/item/?id=BSMF_1955__83__251_0)]

* A. Fröhlich, _Formal group_,  Lecture Notes in Mathematics **74**, Springer (1968)

Review includes

* [[Frank Adams]], part II.7 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Kochman96} [[Stanley Kochman]], section 4.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Hopkins99} [[Mike Hopkins]], _Complex oriented cohomology theories and the language of stacks_, 1999 course notes [[HopkinsLecture.pdf:file]]

* {#Mathew12} [[Akhil Mathew]], _[Lazard's theorem II](http://amathew.wordpress.com/2012/05/27/lazards-theorem-ii/)_, 2012

* {#LurieLect2} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 2 _Lazard's theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture2.pdf))

* {#LurieLect3} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010,  Lecture 3 _Lazard's theorem (continued)_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture3.pdf))

* {#Wilson13} [[Dylan Wilson]], _Proof of Lazard's theorem_, lecture at [2013 Pre-Talbot Seminar](http://math.harvard.edu/~hirolee/pretalbot2013/), March 2013 ([[DylanWilsonProofOfLazard.pdf:file]])

[[!redirects Lazard theorem]]


Lazarus Fuchs

* [Wikipedia entry](http://en.wikipedia.org/wiki/Lazarus_Fuchs)

## related $n$Lab entries

* [[Fuchsian group]]

* [[Fuchsian function]]

* [[Picard-Fuchs equation]]

category: people


lctvs dot source

The graph on [[topological vector spaces]] was created using [Graphviz](http://www.graphviz.org/).  The source is laid out below, the command to generate the diagram is:

~~~
dot -Tsvg lctvs.dot -o lctvs.svg
~~~

The XHTML comments need to be stripped from the resulting SVG, a simple script to do this is

~~~
perl -i.orig -ne '/<!--/ || print' -f lctvs.svg
~~~

These can, of course, be combined into:

~~~
dot -Tsvg lctvs.dot | perl -ne '<!--/ || print' > lctvs.svg
~~~

The top lines also need to be deleted (everything up to the opening `svg` tag) but this is simple to do upon cut-and-pasting into the nLab.

The source for the diagram follows.  Apart from a few options at the start, it consists of a list of **nodes** followed by a list of connections between the nodes.  The initial word `digraph` indicates that this is a _directed_ graph and so the edges are written as `->`.  For more on the syntax and options see the Graphviz documentation.

~~~
digraph LCTVS {
mode=KK;
splines=true;
overlap=orthoyx;
FD [label="Finite-Dimensional", href="http://ncatlab.org/nlab/show/Finite-Dimensional topological vector space", shape=box];
Hi [label="Hilbert", href="http://ncatlab.org/nlab/show/Hilbert topological vector space", shape=box];
SC [label="Second-Countable", href="http://ncatlab.org/nlab/show/Second-Countable topological vector space", shape=box];
Nu [label="Nuclear", href="http://ncatlab.org/nlab/show/Nuclear topological vector space", shape=box];
Ba [label="Banach", href="http://ncatlab.org/nlab/show/Banach topological vector space", shape=box];
IP [label="Inner-Product", href="http://ncatlab.org/nlab/show/Inner-Product topological vector space", shape=box];
Se [label="Separable", href="http://ncatlab.org/nlab/show/Separable topological vector space", shape=box];
Mo [label="Montel", href="http://ncatlab.org/nlab/show/Montel topological vector space", shape=box];
Sc [label="Schwartz", href="http://ncatlab.org/nlab/show/Schwartz topological vector space", shape=box];
UB [label="Ultrabornological", href="http://ncatlab.org/nlab/show/Ultrabornological topological vector space", shape=box];
Fr [label="Fréchet", href="http://ncatlab.org/nlab/show/Fréchet topological vector space", shape=box];
DF [label="DF", href="http://ncatlab.org/nlab/show/DF topological vector space", shape=box];
No [label="Normed", href="http://ncatlab.org/nlab/show/Normed topological vector space", shape=box];
Re [label="Reflexive", href="http://ncatlab.org/nlab/show/Reflexive topological vector space", shape=box];
Cn [label="Convenient", href="http://ncatlab.org/nlab/show/Convenient topological vector space", shape=box];
Cp [label="Complete", href="http://ncatlab.org/nlab/show/Complete topological vector space", shape=box];
Br [label="Baire", href="http://ncatlab.org/nlab/show/Baire topological vector space", shape=box];
Me [label="Metrisable", href="http://ncatlab.org/nlab/show/Metrisable topological vector space", shape=box];
SR [label="Semi-Reflexive", href="http://ncatlab.org/nlab/show/Semi-Reflexive topological vector space", shape=box];
Bo [label="Bornological", href="http://ncatlab.org/nlab/show/Bornological topological vector space", shape=box];
LC [label="Locally Complete", href="http://ncatlab.org/nlab/show/Locally Complete topological vector space", shape=box];
Bl [label="Barrelled", href="http://ncatlab.org/nlab/show/Barrelled topological vector space", shape=box];
QC [label="Quasi-Complete", href="http://ncatlab.org/nlab/show/Quasi-Complete topological vector space", shape=box];
Pc [label="Paracompact", href="http://ncatlab.org/nlab/show/Paracompact topological vector space", shape=box];
QB [label="Quasi-Barralled", href="http://ncatlab.org/nlab/show/Quasi-Barralled topological vector space", shape=box];
Sp [label="Sequentially Complete", href="http://ncatlab.org/nlab/show/Sequentially Complete topological vector space", shape=box];
Nm [label="Normal", href="http://ncatlab.org/nlab/show/Normal topological vector space", shape=box];
Mk [label="Mackey", href="http://ncatlab.org/nlab/show/Mackey topological vector space", shape=box];
CP [label="Countably Paracompact", href="http://ncatlab.org/nlab/show/Countably Paracompact topological vector space", shape=box];
FD -> Hi;
FD -> SC;
FD -> Nu;
FD -> Mo;
Hi -> Ba;
Hi -> IP;
Hi -> Re;
SC -> Me;
SC -> Se;
Mo -> Re;
Mo -> Pc;
Nu -> Sc;
Ba -> UB;
Ba -> Fr;
Ba -> DF;
Ba -> No;
IP -> No;
Re -> Bl;
Re -> SR;
UB -> Bo;
Fr -> Cn;
Fr -> Cp;
Fr -> Br;
Fr -> Me;
Cn -> Bo;
Cn -> LC;
Cp -> LC;
Cp -> QC;
Br -> Bl;
Me -> Bo;
Me -> Pc;
SR -> QC;
Bo -> QB;
Bl -> QB;
QC -> Sp;
Pc -> Nm;
QB -> Mk;
Nm -> CP;
}
~~~


Le Tu Quoc Thang

* [webpage](https://people.math.gatech.edu/~letu/)

## Selected writings

On [[universal Vassiliev invariants]]:

* [[Le Tu Quoc Thang]], [[Jun Murakami]], _The universal Vassiliev-Kontsevich invariant for framed oriented links_, Compositio Math. 102 (1996), 42–64 ([arXiv:hep-th/9401016](https://arxiv.org/abs/hep-th/9401016))

* [[Le Tu Quoc Thang]], [[Jun Murakami]], _Representation of the category of tangles by Kontsevich's iterated integral_, ([euclid.cmp/1104272488](https://projecteuclid.org/euclid.cmp/1104272488))

* [[Le Tu Quoc Thang]], [[Jun Murakami]], Tomotada Ohtsuki, _On a universal perturbative invariant of 3-manifolds_, Topology Volume 37, Issue 3, May 1998, Pages 539-574 (https://doi.org/10.1016/S0040-9383(97)00035-9)

Computation of the [[perturbative quantization of 3d Chern-Simons theory|perturbative Chern-Simons]]  [[Wilson loop observable]] ([[universal Vassiliev invariant]]) of the [[unknot]] ("[[Wheels theorem]]"):

* [[Dror Bar-Natan]], [[Le Tu Quoc Thang]], [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))

category: people


leaf

#Contents#
* table of contents
{:toc}

## Idea

In [[topology]], given a [[covering space]] $E \to X$, then  the elements of the set $F$ for $E\vert_U \simeq U \times F$ are called the _leaves_ over the covering.

In [[differential geometry]], a  [[foliation]] consists of [[submanifolds]]. Each of them is called a _leaf_ of the foliation.

In [[combinatorics]], the ends of [[trees]] are called "leaves".

## Related concepts

* [[Bohr-Sommerfeld leaf]]
* [[symplectic leaf]]

[[!redirects leaves]]

categgory: disambiguation


leaf of the distribution

An Introduction to Distributions and Foliations

MTH 864 · Spring 2008
In smooth manifold theory, the notion of a tangent space makes it possi-
ble for differentiation to take place on an abstract manifold. In this paper,
the notion of a distribution will be presented which makes it possible for in-
tegration to take place on an abstract manifold. The first section introduces
terminology and builds intuition via an analogy to the concept of integral
curves. The second section presents the Frobenius theorem–one of the foun-
dational results in smooth manifold theory. The concluding section leaves the
reader with foliations and a brief look at their connection to the Frobenius
theorem.
1 Preliminaries
Let M be an m-dimensional manifold, TpM the tangent space to p ∈ M, and
TM the tangent bundle of M. Throughout this work, things are implicitly
smooth. Recall that a vector field on M is a section of TM and can be thought
of as a choice of tangent vector at every point in the manifold. Given a vector
field V , one can consider an associated integral curve.
Definition. An integral curve of a vector field V is a curve γ : (a, b) → M
(a and b real numbers) such that
γ
′
(t) = Vγ(t)
for all t ∈ (a, b).

1Definition. The Lie bracket of the vector fields V and W is [V, W] = V W −
W V .
It is well known that [V, W] itself defines a (natural) vector field. What
we will find below is that “nice” means given any vector fields V , W with
Vp, Wp ∈ Dp for all p in some neighborhood U, it follows that [Vp, Wp] ∈ Dp.
If this condition is satisfied (it is only necessary to check it on the basis vector
fields), then the distribution D is called involutive.
Lemma. If D is an integrable distribution, then D is necessarily involutive.
Proof. Let V and W be local sections of D ⊂ TM defined on some U ⊂ M
and let p ∈ U. Since D is integrable, we have some S an integral manifold
of D containing p. Since V and W are sections of D, we know that V and
W are tangent to S on U. By properties of the naturality of the Lie bracket,
[V, W] is also tangent to S and so [V, W]p ∈ Dp. This holds for all points in
U and so D is involutive.
As an example, consider the 2-dimensional distribution D on R

Turka Tachloo

3 with
basis vector fields

                                       writer:
                                       NADEEM AHMAD RATHER
                                       M.PHIL MATHEMATICS
                                     R/O TURKA TACHLOO
                                  ANANTNAG(JAMMU AND KASHMIR)


Lean

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[Lean](https://leanprover.github.io/) is a [[proof assistant]] based on [[dependent type theory]]. Like [[Coq]] and [[Agda]], it may be used to implement [[homotopy type theory]].

## Formalized mathematics
  {#FormslizedMathematics}

Some [[mathematics]] that has been formalized in Lean (in particular in the [[Xena project]]):

* basic [[synthetic mathematics]] [[homotopy theory]] ([[HoTT]])

  * [github.com/forked-from-1kasper/ground_zero](https://github.com/forked-from-1kasper/ground_zero)

  * [github.com/gebner/hott3](https://github.com/gebner/hott3)

* [[spectral sequences]]

  * [github.com/cmu-phil/Spectral](https://github.com/cmu-phil/Spectral)

* [[perfectoid spaces]]

  * [github.com/leanprover-community/lean-perfectoid-spaces](https://github.com/leanprover-community/lean-perfectoid-spaces)

* [[schemes]]

  * [[The Stacks Project]]

* The main theorem of [[liquid vector spaces]]
  * [github.com/leanprover-community/lean-liquid](https://github.com/leanprover-community/lean-liquid)

## Related concepts

[[!include proof assistants and formalization projects -- list]]

## References

* Lean website ([leanprover.github.io](https://leanprover.github.io))

* [[Leonardo de Moura]], Soonho Kong, [[Jeremy Avigad]], [[Floris van Doorn]], Jakob von Raumer, _The Lean Theorem Prover (system description)_ ([pdf](http://www.andrew.cmu.edu/user/avigad/Papers/lean_system.pdf))

* [[Jeremy Avigad]],  [[Leonardo de Moura]],  Soonho Kong, _Theorem Proving in Lean_ ([pdf](https://leanprover.github.io/tutorial/tutorial.pdf))

* [[Floris van Doorn]], _[The Lean Theorem Prover](http://homotopytypetheory.org/2015/12/02/the-proof-assistant-lean/)_, 2015

* {#LewisMoura16} Robert Y. Lewis, [[Leonardo de Moura]], _Automation and Computation in the Lean Theorem Prover_, contribution to _[HaTT 2016: Hammers for Type Theories](https://hatt2016.inria.fr/)_, 2016 ([pdf](https://hatt2016.inria.fr/files/2016/06/HaTT_2016_paper_7-5.pdf))

* {#Hales18} [[Thomas Hales]], _[A Review of the Lean Theorem Prover](https://jiggerwit.wordpress.com/2018/09/18/a-review-of-the-lean-theorem-prover/)_, September 2018

* [[Kevin Buzzard]], _[Xena project](https://xenaproject.wordpress.com/)_ (2018)

  * _[Installing Lean and mathlib](https://xenaproject.wordpress.com/installing-lean-and-mathlib/)_

  * _[Learning Lean](https://xenaproject.wordpress.com/category/learning-lean/)_

  * _[What does Lean already know?](https://xenaproject.wordpress.com/what-does-lean-already-know/)_

  (on the [[Xena project]])

  * _The rise of formalism in mathematics_, ICM 2022 [yt](https://www.youtube.com/watch?v=SEID4XYFN7o)

* _[Lean community and Mathlib] (https://leanprover-community.github.io/)_

* [[Kyle Miller]], *Doing math the Lean way*, Berkeley Lean Seminar (2020) [[pdf](https://math.berkeley.edu/~kmill/talks/2020-06-26-lean-seminar.pdf#page=22), [[Miller-MathTheLeanWay.pdf:file]]]

Tutorials and workshops:

* *Lean for the Curious Mathematician 2020*, online event (Jul 2020) [[web](https://leanprover-community.github.io/lftcm2020/)]

* *Lean for the Curious Mathematician 2022*, Brown University (Jul 2022) [[web](https://icerm.brown.edu/topical_workshops/tw-22-lean/)]

* *Lean for the Curious Mathematician 2023*, Düsseldorf Univ. (Sep 2023) [[web](https://lftcm2023.github.io)]


Leandro Aolita

* head of [Quantum Algorithms](https://www.tii.ae/quantum/our-research/quantum-algorithms) division at [QRC](https://www.tii.ae/quantum)-[TII](https://www.tii.ae/)

* [institute page](https://www.tii.ae/team/prof-leandro-aolita)

* [GoogleScholar page](https://scholar.google.ca/citations?user=AHJJnvMAAAAJ&hl=en)

## Selected writings

On relation between the [[Kochen-Specker theorem]] and [[Bell's inequalities]]:

* [[Leandro Aolita]], Rodrigo Gallego, Antonio Acín, Andrea Chiuri, Giuseppe Vallone, Paolo Mataloni, Adán Cabello, *Fully nonlocal quantum correlations*, Phys. Rev. A **85** 032107 (2012) [[arXiv:1105.3598](https://arxiv.org/abs/1105.3598), [doi:10.1103/PhysRevA.85.032107](https://doi.org/10.1103/PhysRevA.85.032107)]

On simulation of quantum [[spin chains]] on [[quantum computers]]:

* M. Gluza, M. Kliesch, [[Jens Eisert]], [[Leandro Aolita]], *Fidelity witnesses for fermionic quantum simulations*, Phys. Rev. Lett. **120** 190501 (2018) [[arXiv:1703.03152](https://arxiv.org/abs/1703.03152), [doi:10.1103/PhysRevLett.120.190501](https://doi.org/10.1103/PhysRevLett.120.190501)]

On [[repeat-until-success computing|repeat-until-success algorithms]] for creating "quantum imaginary time evolution" in [[quantum computing]]:

* Thais de Lima Silva, Márcio M. Taddei, Stefano Carrazza, [[Leandro Aolita]], *Fragmented imaginary-time evolution for early-stage quantum signal processors* [[arXiv:2110.13180](https://arxiv.org/abs/2110.13180), [spire:1952417](https://inspirehep.net/literature/1952417)]

Review:

* [[Leandro Aolita]], *Imaginary-time evolution algorithms for intermediate-scale quantum signal processors*, talk at *[Atomtronics@AbuDhabi2021](https://qrcseminar.tii.ae)* (2021) [[video](https://youtu.be/G2eigbPiMKU)]

On [[quantum error correction]] in continuous-variable [[quantum information theory]]:

* Allan D. C. Tosta, Thiago O. Maciel, [[Leandro Aolita]], *Grand Unification of continuous-variable codes* [[arXiv:2206.01751](https://arxiv.org/abs/2206.01751)]

category: people


least common multiple

#Contents#
* table of contents
{:toc}

## Idea

For $a,b \in \mathbb{N}$ two positive [[natural numbers]], their **least common multiple** $LCM(a,b) \in \mathbb{N}$ is the smallest natural number that is divisible by both $a$ and $b$, i.e. such that there exist $n_a, n_n \in \mathbb{N}$ with $n_a \cdot a = n_b \cdot b = LCM(a,b)$.

## Remarks

### In the natural numbers

Spelled out, this means that the least common multiple of $a, b \in \mathbb{N}$, denoted $\mathrm{lcm}(a, b)$, is the element $d \in \mathbb{N}$ uniquely characterized by the following two conditions:

* $a|d$ and $b|d$;

* if $a|c$ and $b|c$, then $d|c$.

One could equivalently equip the natural numbers with a [[function]] $\mathrm{lcm}:\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ which satisfies the two conditions:

* $a|\mathrm{lcm}(a, b)$ and $b|\mathrm{lcm}(a, b)$;

* if $a|c$ and $b|c$, then $\mathrm{lcm}(a, b)|c$.

It is not true that "least" means least with respect to the usual ordering $\leq$. In particular, $1$ is the minimal element with respect to the divisibility ordering, and $\mathrm{lcm}(1, 1) = 1$ according to the definition above. However, if we construe "least" in the sense of $\leq$, since every natural number is a common multiple of $1$ with itself, then $0$ is the $\leq$-least natural number!

Furthermore, it is also less robust, because the notion of $lcm$ is at bottom an [[ideal]]-theoretic notion: the divisibility order on elements of a [[principal ideal domain]] is a [[preorder]] whose posetal collapse is the collection of ideals, ordered oppositely to [[inclusion]]. Thus, in [[ring]]-theoretic contexts where there is no sensible notion of $\leq$, for example in the ring of [[Gaussian integers]], the notion of $\mathrm{lcm}$ still makes perfectly good sense if we use the first formulation above, expressed purely in terms of divisibility.

From the point of view of principal ideals in a [[pid]] or [[Bézout domain]] $R$, the lcm corresponds to taking their [[meet]]: $(\mathrm{lcm}(a, b)) = (a) \cap (b)$.

### In an arbitrary Bézout unique factorization domain

In an arbitrary [[Bézout domain|Bézout]] [[unique factorization domain]] $R$, the least common multiple function is only a [[prelattice]], because the minimum elements with respect to the least common multiple are given by the group of units $R^\times$. One usually takes the [[quotient monoid]] of the multiplicative structure on $R$ by the group of units $R^\times$ to get a [[lattice]]: thus, the least common multiple is a function $\mathrm{lcm}:R \times R \to R/R^\times$. In particular, in a [[discrete field]] $K$, the quotient $K/K^\times$ is the [[boolean domain]] $\mathrm{bool}$ and the lcm is thus a function $\mathrm{lcm}:K \times K \to \mathrm{bool}$, and in a [[Heyting field]] $F$, the quotient $F/F^\times$ is the [[set]] of [[truth values]] $\Omega$ whose divisibility partial order is the [[opposite poset]] of the usual partial order via [[entailment]], and the lcm is thus a function $\mathrm{lcm}:F \times F \to \Omega$, defined as $\mathrm{lcm}(a, b) \coloneqq \mathrm{isInvertible}(a) \wedge \mathrm{isInvertible}(b)$

## Related concepts

* [[greatest common divisor]]

## References

* Wikipedia, _[Least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple)_

[[!redirects lcm]]
[[!redirects LCM]]

[[!redirects smallest common multiple]]
[[!redirects lowest common multiple]]


Lebesgue integration

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A method of [[integration]].

## Related concepts

* [[Riemann integration]]

## References

Named after [[Henri Lebesgue]].

* Wikipedia, _[Lebesgue integration](http://en.wikipedia.org/wiki/Lebesgue_integration)_

[[!redirects Lebesgue integral]]
[[!redirects Lebesgue integrals]]


Lebesgue measure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

# Lebesgue measure
* table of contents
{: toc}

## Idea

The _Lebesgue measure_ is the usual [[measure]] on the [[real line]] (or on any [[Cartesian space]]).  It may be characterised as the [[Radon measure]] which is translation invariant and assigns measure $1$ to the [[unit interval]] (or unit cube).  It generalises (up to a scalar constant) to [[Haar measure]] on any [[locally compact space|locally compact]] [[topological group]].

## History

The Lebesgue measure's origins can be traced to the broader theory of [[Lebesgue integral|Lebesgue integration]].  The original purpose of the latter, in broad terms, was to expand the class of integrable functions in order to give meaning to functions that are _not_ [[Riemann integral|Riemann integrable]].  In order to accomplish this, the basic properties of the concept of the _length_ of an interval must be understood.  This then leads to the need to fully define the concept of _measure_, particularly in relation to sets.  We begin with a lemma and a corollary.

+-- {: .num_lemma}
###### Lemma

Let I be an interval, $I = I_{1} \cup I_{2} \cup \cdots$ where $I_{1}, I_{2}, \ldots$ are disjoint intervals.  Then ${|I|} = \sum_{j=1}^{\infty} {|I_{j}|}$ (interpreted so that $\sum_{j=1}^{\infty} {|I_{j}|}$ can be $+\infty$, either because one of the summands is $+\infty$ or because the series diverges).
=--

+-- {: .num_corollary}
###### Corollary

If I is any interval, then
$$
  {|I|} = inf \left\{\sum_{j=1}^{\infty} {|I_{j}|} : I \subseteq \bigcup_{j=1}^{\infty} I_{j}\right\}
$$
where $\{I_{j}\}$ is any countable covering of I by intervals.
=--

Now suppose $B$ is an arbitrary set of real numbers.  In order for $B$ to be measurable, we must have ${|B|} \leq \sum_{j=1}^{\infty} {|I_{j}|}$ where $\bigcup_{j=1}^{\infty} I_{j}$ is any countable covering of $B$ by intervals.  We must also have ${|B|} \leq \bigcup_{j=1}^{\infty} {|I_{j}|}$ where the infinum is taken over all countable coverings of $B$ by intervals.

## Definition

We define the __Lebesgue [[outer measure]]__ as follows:
$$
  {|B|} = inf \left\{\sum_{j=1}^{\infty} {|I_{j}|} : B \subseteq \bigcup_{j=1}^{\infty} I_{j}\right\}.
$$

The set $B$ is __Lebesgue measurable__ if
$$ {|A|} = {|A \cap B|} + {|A \setminus B|} $$
holds for every set $A$.  Restricting to these sets, Lebesgue outer measure becomes an honest [[measure]].

Note that once the Lebesgue measure is known for open sets, the outer regularity property allows us to find it for all [[Borel set]]s (but also rather more sets).

## See also

* [[Jordan content]]

## References

Named after [[Henri Lebesgue]].

See also

* [Wikipedia](http://en.wikipedia.org/wiki/Lebesgue_measure).
* [nForum](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=1385)

For a very intuitive and readable derivation, see:

* R. Strichartz, _The Way of Analysis_, Jones & Bartlett, 2000.

[[!redirects Lebesgue measure]]
[[!redirects Lebesgue measures]]

[[!redirects Lebesgue-measurable set]]
[[!redirects Lebesgue-measurable sets]]
[[!redirects Lebesgue measurable set]]
[[!redirects Lebesgue measurable sets]]
[[!redirects Lebesgue-measurable subset]]
[[!redirects Lebesgue-measurable subsets]]
[[!redirects Lebesgue measurable subset]]
[[!redirects Lebesgue measurable subsets]]


Lebesgue number lemma

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _Lebesgue number lemma_ makes a statement about properties of [[open covers]] of [[sequentially compact space|sequentially compact]] [[metric spaces]]. It serves as ingredient of the proof that [[sequentially compact metric spaces are equivalently compact metric spaces]].

## Statement

+-- {: .num_prop}
###### Proposition

Assuming [[excluded middle]] and [[countable choice]] then:

If $(X,d)$ is a [[metric space]] which is [[sequentially compact space|sequentially compact]], then for every [[open cover]] $\{U_i \to X\}_{i \in I}$ there exists a [[positive number|positive]] [[real number]] $\delta \in \mathbb{R}$, $\delta \gt 0$, called a _Lebesgue number_ for $X$, such that for every point $x \in X$ there exists an $i \in I$ such that the [[open ball]] around $x$ of [[radius]] $\delta$ is contained in $U_i$:

$$
  \left(
      (X,d) \, \text{sequentially compact}
  \right)
    \;\Rightarrow\;
   \underset{\delta \gt 0}{\exists}
   \left(
     \underset{x \in X}{\forall}
     \left(
       \underset{i \in I}{\exists}
       \left(
         B^\circ_x(\delta) \subset U_i
       \right)
     \right)
   \right)
$$

=--

+-- {: .proof}
###### Proof

Assume that the statement were not true. This would mean that for every $n \in \mathbb{N}$ there exists a point $x_n \in X$ such that for all $i \in I$ the open ball $B^\circ_{x_n}(1/(n+1))$ is not contained in $U_i$. These points $(x_n)_{n \in \mathbb{N}}$ would constitute a [[sequence]] and so by assumption on $X$ there would exist a sub-sequence $(x_{n_k})_{k \in \mathbb{N}}$ which converges to some $x_\infty \in X$. Hence then there would be some $i_\infty \in I$ with $x_\infty \in U_{i_\infty}$, and this, since $U_{i_\infty}$ is open, also a positive real number $\epsilon \gt 0$ with $B^\circ_{x_\infty}(\epsilon) \subset U_{i_\infty}$. By convergence of the sub-sequence $(x_{n_k})_k$ we could now choose a $k \in \mathbb{N}$ such that

$$
  \frac{1}{n_k + 1} \lt \frac{\epsilon}{2}
   \phantom{AAA}
   \text{and}
   \phantom{AAA}
   d(x_{n_k}, x_{\infinity}) \lt \frac{\epsilon}{2}
  \,.
$$

This would imply that

$$
  B^\circ_{x_{n_k}}(1/n_k)
  \subset
  B^\circ_{x_\infty}(\epsilon)
  \subset U_\infty
  \,.
$$

This contradicts the assumption that none of the $U_i$ contains the open ball $B^\circ_{x_{n_k}}(1/n_k)$, and hence we have a  [[proof by contradiction]].

=--

## References

Named after [[Henri Lebesgue]].

* Wikipedia, _[Lebesgue's number lemma](https://en.wikipedia.org/wiki/Lebesgue%27s_number_lemma)_

[[!redirects Lebesgue number lemma]]
[[!redirects Lebesgue's number lemma]]