Lawvere theory

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#### Categorical algebra
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#### Higher algebra
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# Contents
* table of contents
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## Idea

The notion of _Lawvere theory_ is a joint generalization of the notions of [[group]], [[ring]], [[associative algebra]], etc.

In his [[Functorial Semantics of Algebraic Theories|1963 doctoral dissertation]], Bill Lawvere introduced a new categorical method for doing [[universal algebra]], alternative to the usual way of presenting an algebraic concept by means of its logical [[signature (in logic)|signature]] (with generating operations satisfying equational axioms). The rough idea is to define an [[algebraic theory]] as a [[category]] with finite [[product]]s and possessing a "generic algebra" (e.g., a generic [[group]]), and then define a [[model]] of that [[theory]] (e.g., a group) as a product-preserving [[functor]] out of that [[category]]. This type of category is what is nowadays called a _Lawvere algebraic theory_, or just Lawvere theory.

## Definition
There are two related definitions of Lawvere theory in the literature. Both definitions are widely used and both have a substantial history.
+-- {: .un_defn}
###### Definition A

A **Lawvere [[theory]]** or **finite-product theory** is (equivalently encoded by its [[syntactic category]] which is) a [[category]] $T$ with finite [[products]] in which every [[object]] is [[isomorphism|isomorphic]] to a finite cartesian power $x^n = x \times x \times \cdots \times x$ of a distinguished object $x$ (called the _generic object_ for the theory $T$).

A _[[homomorphism]]_ of such theories $T \to T'$ is a product-preserving [[functor]].

=--

+-- {: .un_defn}
###### Definition B

Let $\mathbb{F}$ be the category of natural numbers and functions between them (a [[skeleton]] of the [[FinSet|category of finite sets]]). This category has canonical chosen [[coproducts]].

A **Lawvere [[theory]]** is a [[category]] $T$ equipped with an [[identity-on-objects functor|identity-on-objects]] strictly [[power]]-preserving functor $J:\mathbb{F}^{\mathrm{op}} \to T$. (Power preservation is here equivalent to product preservation.)
=--

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

Clearly, if $T$ is part of a structure in the form of Definition B, then $T$ is of the form in Definition A, taking $x=J(1)$. See further remarks below.

For $T$ a Lawvere theory, we are to think of the [[hom-set]] $T(n,1) := T(x^n, x)$ as the set of $n$-ary operations definable in the theory. For instance for $T$ the theory of [[abelian group]]s, $T(2,1)$ includes operations like $+ \colon (x, y) \mapsto x + y$, $- \colon (x, y) \mapsto x-y$, and $(x, y) \mapsto 2 x - 3 y$. For $0$-ary or nullary operations, we have $T(0,1) = \{0\}$.

=--

+-- {: .un_defn}
###### Definition

A **[[model]]** of $T$ -- an **[[algebra over a Lawvere theory|algebra over the Lawvere theory]]** or simply **$T$-algebra** -- is a product-preserving [[functor]]

$$
  A : T \to Set
  \,,
$$

and _homomorphism of $T$-algebras_ is a [[natural transformation]] between such functors.

=--

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

Such a functor picks out a single [[set]] $U(A) := A(1)$ and picks for every $n$-ary abstract operation $\phi \in T(n,1)$ an actual $n$-ary operation on the elements of this set

$$
  A(\phi) : U(A)^n \to U(A)
$$

such that these operations are all compatible with each other in the way governed by the composition rules of morphisms in $T$.

When working with Definition B of Lawvere theory, it would be normal to require that $A$ _strictly_ preserves products.
=--

### Remarks

1. From some perspectives, Definition B violates the [[principle of equivalence]], although from another perspective it should be regarded as a basic structural definition that is equivalent in content to an [[clone|abstract clone]], or equivalently a [[relative monad]] over $\mathbb{F}$.
On the other hand, given a structure of the form Definition A, we may choose a power of $x$ for each natural number, and thus derive an [[essentially surjective functor|essentially surjective]] product-preserving functor $\mathbb{F}^{op} \to T$, and then we can always factor this as an identity-on-objects functor to reach Definition B.

1. Some people use 'finite-product theory' to mean any (small) category with [[finite products]], reserving 'Lawvere theory' to refer to finite product theories with the property that every object is isomorphic to a product of finitely many copies of a given object $x$. Finite-product theories $C$ can be regarded as a special case of [multisorted Lawvere theories](https://ncatlab.org/toddtrimble/published/multisorted+Lawvere+theories) (see below) where the set of sorts is $Ob(C)$ itself. Some, but not all, the above discussion generalizes to this case.

1. As _finite-product theories_, Lawvere theories are at one end of a spectrum of [[theory|theories]] of differing logical strengths. For example, there are left exact theories, [[regular theory|regular theories]], geometric theories, and so on, which require for their interpretation categories of differing degrees of strength in their [[internal logic]]. See also [[classifying topos]].

1. If $C$ is a category with finite products, then a group (object) in $C$ may be defined as a product-preserving functor $T_{Grp} \to C$. For example, a topological group may be identified with a functor $T_{Grp} \to Top$, and a Lie group with a product-preserving functor $T_{Grp} \to Man$ into the category of smooth [[manifold]]s. An analogous statement holds for any finitary algebraic theory, when formulated in terms of its Lawvere theory $T$.

### Variations

* A **multisorted** or multityped Lawvere theory for a given set of sorts $S$ is a category with finite products $C$ together with a function $i: S \to Ob(C)$ such that every object of $C$ is isomorphic to a finite product of objects of the form $i(S)$. An example is the theory for ring-module pairs, which may be regarded as a two-sorted theory in which one sort is interpreted as a ring and the other as a module over that ring.

* An [[infinitary Lawvere theory]] allows for infinitary operations.  An example is the theory of [[suplattices]], where we have, for every [[cardinal number]] $n$, an operation to take the [[supremum]] of $n$ elements.  While Lawvere theories correspond to finitary monads on $Set$, infinitary Lawvere theories correspond to arbitrary monads.

* A [[Fermat theory]] is a Lawvere theory equipped with a notion of differentiation.

* A finite-product theory can also be presented without including all the products of the basic types as actual objects.  This yields the notion of [[cartesian multicategory]].

## Examples

### The theory of sets {#TheoryOfSets}

The tautological example of a Lawvere theory is the algebraic theory of _no_ operations. This is also called the **theory of equality**.

Its syntactic category is the category $\mathcal{S}$ on objects $n \in \mathbb{N}$ with morphisms generated by the projections $\pi_i : n \to 1$. This is the [[opposite category]] of the category [[FinSet]]

$$
  \mathcal{S} \simeq FinSet^{op}
  \,.
$$

This is the [[initial object]] in the category of Lawvere theories.

An algebra over this theory is just a bare [[set]]:

$$
  \mathcal{S} Alg \simeq Set
  \,.
$$

For $T$ any Lawvere theory, there is a canonical morphism $\mathcal{S} \to T$. On categories of algebras this induces the functor

$$
  U_T : T Alg \to \mathcal{S}Alg \simeq Set
  \,.
$$

This sends each algebra to its _underlying set_ . For more on this see the section [Free T-algebras](#FreeAlgebras) below.

### The theory of groups

We consider here the theory of [[group]]s (defined however you like). To get the corresponding Lawvere theory $T$, let $F(n)$ (for any natural number $n \geq 0$) be a free group on $n$ generators, and define the Lawvere theory $T_{Grp}$ to be the category [[opposite category|opposite]] to the category of free groups $F(n)$ and group homomorphisms. The generic object $x$ of $T_{Grp}$ is taken to be $F(1)$.

The category of free groups has finite coproducts since $F(m) + F(n) \cong F(m+n)$ (in other words, the inclusion

$$FreeGroup \hookrightarrow Group$$

creates coproducts in $FreeGroup$), so $T_{Grp}$ has finite products, and we have $F(n) = x^n$ in $T_{Grp}$. Any group $G$ defines a product-preserving functor

$$T_{Grp} = FreeGroup^{op} \hookrightarrow Group^{op} \stackrel{\hom(-, G)}{\to} Set$$

since contravariant hom-functors take coproducts to products. Thus any group gives a model of $T_{Grp}$.

The other direction is more interesting. Let

$$M: T_{Grp} \to Set$$

be a model of $T_{Grp}$, i.e., a product-preserving functor. We will define a group structure on $G = M(x)$, the underlying set of the group.

To understand this, let's consider how group multiplication would be defined. The idea is that $x$ in $T_{Grp}$ is a "generic group", so we first need to understand how multiplication works there. The idea is that the product
in the generic group
$$m: x^2 \to x^1$$
corresponds to a homomorphism
$$F(1) \to F(2)$$
which by freeness corresponds to an element $1 \to F(2)$, and the element we are after should be the product $a b$ of the generators $a, b$ of the free group $F(2) = F(a, b)$. The generators $a, b$ themselves correspond to the two coproduct inclusions
$$i_1: F(1) \to F(1) + F(1) = F(2) \qquad i_2: F(1) \to F(1) + F(1) = F(2)$$

Then, since $M$ is assumed to [[exact functor|preserve products]], we obtain a map
$$G \times G = M(x) \times M(x) \cong M(x^2) \stackrel{M(m)}{\to} M(x) = G$$
and this defines the group multiplication on $G$. The group identity and group inversion on $G$ are defined by following similar recipes.

It may be checked that the notion of homomorphism of $T_{Grp}$-models (as defined above) coincides with the usual notion of group homomorphism. In summary, the category of groups is equivalent to the category of models of $T_{Grp}$.

In particular, any hom-functor

$$\hom_{T_{Grp}}(x^n, -): T_{Grp} \to Set$$

preserves products, and so defines a group. This group is precisely the free group on $n$ generators, and a little thought shows that the $n$ generators correspond to the natural transformations
$$\hom_{T_{Grp}}(x, -) \to \hom_{T_{Grp}}(x^n, -)$$
induced by the $n$ projection maps $x^n \to x$.

All of the discussion above for the case of groups generalizes to any finitary [[algebraic theory]] (i.e., any single-sorted theory whose signature consists of function symbols of finite arity, subject to universally quantified equational axioms). In summary:

* The Lawvere theory $T$ is the category opposite to the category of free algebras on finitely many generators,

* The category of algebras is in turn equivalent to the category of product-preserving functors $T \to Set$, and

* The free algebras are retrieved as the representable functors $T \to Set$.

As discussed in the article on [[operad|operads]], the notion of Lawvere theory may also be formulated in terms of operads relative to the theory of [[cartesian monoidal category|cartesian monoidal categories]].

### Other examples
 {#OtherExamples}

Most of the standard structures that are considered in [[algebra]] indeed are models of algebraic/Lawvere theories in the precise sense. The following list gives a few familiar examples and a few not so familiar ones, but there are many more. Beware though that there are also some familiar examples that seem to be algebraic but are not, these we discuss [below](#SomeNonExamples).

* For $k$ a [[field]], the category of _free $k$-associative algebras_ is the (syntactic category of the) theory of ordinary [[associative algebras]] over $k$.

* for $G$ a fixed [[group]], then $G$-[[actions]] ([[permutation representations]]) are an example of algebras over a Lawvere theory

* for $R$ a fixed [[ring]], then $R$-[[modules]] are an example

* for $k$ a [[field]], then [[Lie algebras]] over $k$ are an example

* The category of say [[distributive lattices]] is the category of algebras of a Lawvere theory. So is the category of [[Heyting algebras]].

* The category [[CartSp]] is the (syntactic category of the) theory of [[smooth algebras]] (as used in [[synthetic differential geometry]]). This is also a [[Fermat theory]].

* The terminal Lawvere theory has exactly one morphism $f \colon m \to n$ for each $m,n$. Since there is a unique morphism between each pair of objects, each object is in fact isomorphic. Thus, this theory is a terminal category. Algebras of the terminal Lawvere theory are terminal sets, singletons.

### Characterization of examples
 {#CharacterizationOfExamples}

If $T$ is any [[theory]] given by a [[signature]] consisting of finitary [[operations]] (but no [[relations]]) on a single [[sort]], and a set of [[axioms]] all of which are [[universal quantifier|universally quantified]] [[equations]] between terms, then a model of $T$ can be described as an algebra of a Lawvere theory.

This includes most cases arising in a typical undergraduate course in modern algebra, as the examples [above](#OtherExamples) suggest.

There are also well-known criteria for a category of single-sorted structures $C$, with underlying set-functor $U: C \to Set$, to be the category of algebras of a Lawvere theory.

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

A [[concrete category]] $U \colon C \to Set$ is a category of algebras over a Lawvere theory precisely if $U$

1. is [[monadic functor|monadic]]

1. is _finitary_ in that it preserves [[filtered colimits]].

=--

Another characterization is:

+-- {: .num_example #BirkhoffHSP}
###### Theorem
**([[Birkhoff's HSP theorem]])**

Suppose given a [[language]] $L$ generated by a set of (single-sorted) finitary operations, and a [[class]] $C$ of [[structures]] for $L$. Then $C$ is the class of models for a set of universally quantified equations between terms of $L$ if and only if

1. (H) The class is closed under homomorphic [[images]],

1. (S) The class is closed under subalgebras,

1. (P) The class is closed under taking products.

=--

### Some non-examples
 {#SomeNonExamples}

Here are some notable examples of mathematical structures that look algebraic, but are not models of an algebraic theory in the present sense:

* The class of [[fields]] is not the class of algebras of a Lawvere theory.

* Neither is the class of [[integral domains]].

This might seem obvious since multiplicative inversion in fields is not a global operation, or otherwise the cancellation law of multiplication in integral domains is not a universally quantified axiom (since we have to make an exception of $0$). But one should be careful that there isn't some sneaky alternative axiomatization for these structures which counters these objections!

The first clause in Theorem \ref{RecognizingAConcreteCategoryAsAlgebrasOverALawvereTheory} already rules out fields, since $U$ in that case will create [[limits]] in $C$, but the category [[Field]] of fields does not even have [[products]]. Similarly for [[integral domains]].

The second clause in Theorem \ref{RecognizingAConcreteCategoryAsAlgebrasOverALawvereTheory} suggests another type of non-example:

* The category $SLat$ of [[sup-lattices]] is not the category of algebras of a (finitary) Lawvere theory.

There is a whole class of infinitary sup-operations for sup-lattices (one for every arity = cardinal), but again one may wonder how one rules out any alternative finitary axiomatizations. But this is fairly clear by invoking the second clause and considering the following example: if $U: SLat \to Set$ created filtered colimits, then the countable copower $\mathbb{N} \cdot \mathbf{2}$ of 2-element sup-lattices (which turns out to be the power set $P(\mathbb{N})$ with its usual order) would be the filtered colimit (in fact a union) over finite subsets $S$ of finite copowers $S \cdot \mathbf{2}$, hence a countable union of finite sup-lattices, which is clearly impossible.

## Properties

### Coequalizers of $T$-algebras

+-- {: .un_def}
###### Definition (congruence)
{#congruence}

Let $T$ be a Lawvere theory and $A$ a $T$-algebra. A **[[congruence]]**  on $A$  is an [[equivalence relation]] on the set $A(1)$ such that whenever for all $n \in \mathbb{N}$ whenever any $(a_i \in A(1))_{i=1}^n$ and $(b_i \in A(1))_{i=1}^n$ are pairwise eqivalent, $a_i \sim b_i$, then also for every operation $f \in T(n,1)$ the results are equivalent: $f(a_1, \cdot, a_n) \sim f(b_1, \cdots , b_n)$.

=--

+-- {: .un_lemma}
###### Observation/Definition

For $A$ a $T$-algebra and for every [[relation]] $R \subset A(1) \times A(1)$ there is a smallest [congruence](#congruence) on $A$ containing $R$. Write $\langle R \rangle \subset A(1) \times A(1)$ for this smallest congruence.

=--

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

For $A$ a $T$-algebra and $C$ a congruence on $A$, the relation $A/C(f) \subset (A/C)^n \times A/C$ induced by $A(f)$ for each $f : \in T(n,1)$ are functions and define a $T$-algebra structure on $A(1)/C$.

=--

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

For $f,g : A \to B$ two morphisms of $T$-algebras, the canonical morphism
$B \to B/\langle \f(a)\sim g(a) | a \in A(1)\rangle$ is the [[coequalizer]] of $f$ and $g$.

=--

### Free $T$-algebras and underlying sets {#FreeAlgebras}

Write $\mathcal{S}$ for the (syntactic category of the) algebraic theory of sets (described [above](#TheoryOfSets)). Then for $T$ any other (syntactic category of a) Lawvere theory, there is a canonical morphism

$$
  i_T :  \mathcal{S} \to T
  \,.
$$

By precomposition with $i_T$ we obtain a corresponding functor on $T$-algebras, which we write

$$
  U_T : T Alg \to Set
$$

and call the **underlying set** functor.

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

The functor $U_T$ has a [[left adjoint]] $F_T : Set \to T Alg$.

=--

This is a standard example of a [[free functor]], called the **free $T$-algebra functor**.

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

For $S \in $ [[Set]], let $F_T(S)$ be the $T$-algebra whose underlying set is the set of formal expressions $\{f(s_1, \cdots, s_n) | f \in T(n,1), s_i \in S\}$ with the evident composition operation.

=--

+-- {: .un_lemma}
###### Observation

For $S = (n) \in$ [[FinSet]] a _finite set_ with $n$ elements , the free $T$-algebra on $S$ is just the [[representable functor|representable]]

$$
  F_T(S) = T(n,-) : T \to Set
  \,.
$$

The adjunction isomorphism

$$
  T Alg(F_T(S), A) \simeq Set(S, U_T(A)) \simeq  (A(1))^n \simeq A(n)
$$

is in this case just the [[Yoneda lemma]].

$$
  T Alg(T(n,-),A) \simeq A(n)
  \,.
$$

Notice that this extends to a functor

$$
  F_T(-) : FinSet \to T Alg
$$

which is the composite

$$
  F_T(-) : FinSet \stackrel{i_T^{op}}{\to} T^{op}
  \stackrel{j}{\to}
  T Alg
$$

of the [[Yoneda embedding]] with the opposite of the canonical functor
$i_T : FinSet^{op} \simeq \mathcal{S} \to T$ from the theory of sets,
described [above](#TheoryOfSets).

More generally, for $S \in Set$ not necessarily finite, let
$Sub(S)$ be the [[poset]] of finite subsets of $S$ and their inclusions.

Then $F_T(S)$ is the [[filtered colimit]] of the representables
corresponding to the finite subsets

$$
  F_T(S) \simeq {\lim_{\to}}_{{n \in Sub(S)}} T(n,-)
  \,.
$$

As discussed [below](#CoLimitsOfTAlgebras), these filtered colimits of $T$-algebras are computed objectwise.

=--

### Adjunctions of relatively free $T$-algebras {#AdjCatAlgs}

The following establishes that more generally any morphism of Lawvere theories leads to an adjunction between their categories of algebras.

Let $T_1$ and $T_2$ be Lawvere theories and $f : T_1 \to T_2$ a morphism. Write
$f^* : T_2 Alg \to T_1 Alg $ for the functor on categories of algebras induced by precomposition with $f$.

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

The functor $f^*$ has a [[left adjoint]] $f_* : T_1 Alg \to T_2 Alg$.

=--

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

Here is an elementary proof:

Let $F_{T_i} \dashv U_{T_i}$ be the two [free algebra/underlying-set adjunctions](#FreeAlgebras). For $A$ a $T_1$-algebra there is a $T_1$-[congruence](#congruence) $\Gamma$ such that

$$
  A \simeq F_{T_1} U_{T_1}(A) / \Gamma
  \,.
$$

Since for any set $S$ we have $U_{T_1} F_{T_1}(S) \subset U_{T_2} F_{T_2}(S)$ it follows that $\Gamma \subset T_{T_2} F_{T_2} U_{T_1}(A)$. For $\langle \Gamma \rangle$ the smallest $T_2$-congruence containing $\Gamma$ we have that $F_{T_2} U_{T_1}(A) / \langle \Gamma \rangle$ is a $T_2$-algebra.

This one checks is  $f_* A$.

Here is a more high-powered way to obtain this using the [[monad]]s $K_i$ whose algebras are $T_i$-algebras:

for $A$ a $T_1$-algebra let $f_* A  := K_2 \circ_{K_1} A$ the the evident reflexive [[coequalizer]]

$$
  K_2 K_1 A \stackrel{\to}{\to} K_2 A \to K_2 \circ_{K_1} A
$$

in $T_2 Alg$.

=--

### Limits and colimits of $T$-algebras {#CoLimitsOfTAlgebras}

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

For $T$ a Lawvere theory, the category $T Alg$ has all small [[limit]]s and [[colimit]]s.

The [[limit]]s and the [[filtered colimit]]s in $T Alg$ are computed pointwise.

=--

The category of algebras $T Alg$ has a [[zero object]] if and only if $T$ has a single nullary operation (Proposition 3.3 of [Wraith](#Wraith)).

## An open problem

A [[Higman's embedding theorem|famous result by G. Higman]] in group theory says that a finitely generated group can be embedded in a finitely presented group precisely if it has a presentation whose defining relations are a recursively enumerable set of words. Clearly, this question can be asked for every similar algebraic theory and it has been in fact conjectured by the group theorist W. Boone that the same result holds more generally for every single-sorted algebraic theory.  While in fact the conjecture is not true for all single-sorted algebraic theories (see [[Boone conjecture]] for further details), as was known by Soviet mathematicians, the general question "In which algebraic categories does (an analogue of) Higman's embedding theorem hold?" is still interesting. In fact, it has been urged by [[F. W. Lawvere]] on several occasions that this [[Boone conjecture]] should be settled by the category theory community.

## Related concepts

* [[algebraic theory]] / [[generalized algebraic theory]] / **Lawvere theory** / [[second order algebraic theory]] / [[2-Lawvere theory]] / [[(∞,1)-algebraic theory]]

* [[internalization]]

* [[enriched Lawvere theory]]

* [[globular theory]]

* [[lambda theory]]

* [[monad]] / [[(∞,1)-monad]]

* [[operad]] / [[(∞,1)-operad]]

* [[PRO]]

* [[PROP]]

* [[C-systems]]

* [[Boone conjecture]]

* [[Freyd category]]

## References

The origin of the categorical formulation of [[algebraic theories]] as Lawvere theories is in

* [[William Lawvere]], *[[Functorial Semantics of Algebraic Theories]]*, Ph.D. thesis Columbia University (1963).  Published with an author's comment and a supplement in: Reprints in Theory and Applications of Categories **5** (2004) pp 1--121. ([tac:tr5](http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html))

The concept was then streamlined in

* [[Samuel Eilenberg]], J. B. Wright, _Automata in General Algebras_ , Information and Control **11** (1967) pp.452-470.

Also still worthwhile reading are the following early papers:

* {#Wraith} [[Gavin Wraith]], _Algebraic Theories_ , Aarhus Lecture Notes Series no.22 (1969). ([pdf, 3,7MB](https://www.cs.ox.ac.uk/people/ohad.kammar/scans/wraith-algebraic-theories.pdf))

* [[Gavin Wraith]], _Algebras over Theories_ , Colloquium Mathematicum **XXIII** no.2 (1971) pp.181-190. ([link](https://eudml.org/doc/264519))

An early textbook treatment was in

* {#Pareigis70} [[Bodo Pareigis]], _Categories and Functors_ , New York: Academic Press 1970. (chap. 3; [link](https://epub.ub.uni-muenchen.de/7244/))

Modern textbook treatments are

* [[Francis Borceux]], _Handbook of categorical algebra 2 -- Categories and structures_ , Encyclopedia of Mathematics and its Applications, Cambridge UP 1994. (chap. 3)

* M. C. Pedicchio,  F. Rovatti, _Algebraic Categories_ , pp.269-310 in Pedicchio, Tholen (eds.), _Categorical Foundations_ , Encyclopedia of Mathematics and its Applications **97**, Cambridge UP 2004.

A recent monograph is

* [[Jiri Adamek|J. Adámek]], [[Jiri Rosicky|J. Rosicky]], [[Enrico Vitale|E. M. Vitale]], _Algebraic Theories - a Categorical Introduction to General Algebra_ , Cambrige UP 2010. ([draft](http://www.iti.cs.tu-bs.de/~adamek/algebraic_theories.pdf))

The concept of an internal algebraic theory in topos theory is dicussed in

* [[Peter Johnstone]], [[Gavin Wraith]], _Algebraic theories in toposes_ , pp.141-242 in LNM **661** Springer Heidelberg 1978.

* [[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977. (Dover reprint 2014; sec. 6.4, pp.190-198)

Comparison with the notion of [[monads]]:

* [[Martin Hyland]], [[John Power]], _The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads_, Electronic Notes in Theor. Comp. Sci. **172** (2007) 437-458 [[doi:10.1016/j.entcs.2007.02.019](https://doi.org/10.1016/j.entcs.2007.02.019), [preprint](https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2007/hp07.pdf)]

An elementary proof of the equivalence between infinitary [[Lawvere theories]] and [[monads]] on the [[category of sets]] is given in Appendix A of

* [[Martin Brandenburg]], Large limit sketches and topological space objects, [arXiv:2106.11115](https://arxiv.org/abs/2106.11115).

[[distributive law|Distributive laws]] for algebraic theories are discussed in

* [[Eugenia Cheng]], _Distributive laws for Lawvere theories_ , arXiv:1112.3076 (2011). ([abstract](http://arxiv.org/abs/1112.3076))

Voevodsky proves an equivalence between Lawvere theories and l-bijective [[C-systems]] here:

* {#Voevodsky} [[Vladimir Voevodsky]],  *Lawvere theories and C-systems*, arXiv:1512.08104 (2015). ([abstract](http://arxiv.org/abs/1512.08104))

Other references are

* M. Jibladze, T. Pirashvili, _Cohomology of Algebraic Theories_ , J. Algebra **137** no.2 (1991) pp.253–296.

* [[Steve Lack]], [[Jiri Rosicky]], _Notions of Lawvere theory_ , arXiv:0810.2578. ([abstract](http://arxiv.org/abs/0810.2578))

* [[Enrico Vitale]], _Localization of Algebraic Categories_ , JPAA **108** (1996) pp.315-320.

* [[Enrico Vitale]], _Localization of Algebraic Categories 2_ , JPAA **133** (1998) pp.317-326.

Lawvere theories, and finitary monads, are identified with certain [[enriched categories]] and the passage from one to the other with a [[cocompletion]], in

* [[Richard Garner]], *Lawvere theories, finitary monads and Cauchy-completion*, 2013, [arxiv](https://arxiv.org/abs/1307.2963)

* [[Richard Garner]] and [[John Power]], *An enriched view on the extended finitary monad--Lawvere theory correspondence*, 2017, [arxiv](https://arxiv.org/abs/1707.08694)

A short introductory/overview note:

* {#keml-diagrams} [[Andreas Nuyts]], _Understanding Universal Algebra Using Kleisli-Eilenberg-Moore-Lawvere Diagrams_, [note](https://anuyts.github.io/files/keml-diagrams.pdf)

[[!redirects lawvere theory]]
[[!redirects Lawvere theory]]
[[!redirects Lawvere theories]]
[[!redirects finitary Lawvere theory]]
[[!redirects finitary Lawvere theories]]

[[!redirects finite-product theory]]
[[!redirects finite-product theories]]
[[!redirects finite product theory]]
[[!redirects finite product theories]]
[[!redirects finite-products theory]]
[[!redirects finite-products theories]]
[[!redirects finite products theory]]
[[!redirects finite products theories]]


lawvere theory > history

< [[lawvere theory]]
[[!redirects Lawvere theory > history]]


Lawvere's fixed point theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Logic
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Various [[diagonal arguments]], such as those found in the proofs of the [[halting theorem]], [[Cantor's theorem]], and [[Gödel]]'s [[incompleteness theorem]], are all instances of the _Lawvere fixed point theorem_ ([Lawvere 69](#Lawvere69)), which says that for any [[cartesian closed category]], if there is a suitable notion of [[epimorphism]] from some [[object]] $A$ to the [[exponential object]]/[[internal hom]] from $A$ into some other object $B$

$$
  A \longrightarrow B^A
$$

then every [[endomorphism]] $f \colon B \to B$ of $B$ has a [[fixed point]].

## Precise statement

Let us say that a map $\phi: X \to Y$ is _point-surjective_ if for every point $q: 1 \to Y$ there exists a point $p: 1 \to X$ that lifts $q$, i.e., $\phi p = q$.

+-- {: .num_theorem}
###### Theorem
**(Lawvere's fixed-point theorem)**
In a cartesian closed category, if there is a point-surjective map $\phi: A \to B^A$, then every morphism $f: B \to B$ has a fixed point $s: 1 \to B$ (so that $f s = s$).
=--

+-- {: .proof}
###### Proof
Given $f: B \to B$, let $q: 1 \to B^A$ name the composite map

$$A \stackrel{\delta}{\to} A \times A \stackrel{\phi \times 1_A}{\to} B^A \times A \stackrel{eval}{\to} B \stackrel{f}{\to} B$$

from $A$ to $B$. In [[lambda calculus]] notation, $q = \lambda a: A. f \phi(a)(a)$. Let $p: 1 \to A$ lift $q$. Then calculate

$$\phi(p)(p) = q(p) = (\lambda a: A. f \phi(a)(a))(p) = f \phi(p)(p)$$

where the last equation is a [[beta-reduction]]. Hence $s \coloneqq \phi(p)(p)$ is a fixed point of $f$.
=--

+-- {: .num_remark}
###### Remark
As pointed out by Lawvere, a hypothesis even weaker than point-surjectivity will do. Namely, $g: X \to Y^A$ is called _weakly point-surjective_ iff for every $f: A \to Y$ there is $x: 1 \to X$ such that for every $a: 1 \to A$, we have $g(x)(a) = f(a)$. (This is weaker because $g(x) = f$ cannot be inferred from $g(x)(a) = f(a)$ for all _[[global elements]]_ $a$.)
=--

+-- {: .num_remark}
###### Remark
The statement need not hold if "(weakly) point-surjective" is replaced by "epimorphism". For example, in the cartesian closed category of compactly generated Hausdorff spaces and continuous maps, with $S = S^1$ the circle, the [[Polish space]] $S^\mathbb{N}$ is compactly generated under the product topology; this is the exponential where $\mathbb{N}$ is given the discrete topology. There is a countable dense subspace $i: \mathbb{N} \to S^\mathbb{N}$, but recall that for any full subcategory of the category of Hausdorff spaces, a map $f: X \to Y$ is an epimorphism iff it has a dense [[image]]. On the other hand, there are obvious rotations of $S$ that have no fixed points.
=--

Thus epimorphisms need not be (weakly) point-surjective. Nor are point-surjective maps necessarily epimorphisms; for example, if $U \hookrightarrow V$ is a proper inclusion between proper subobjects of the terminal object $1$ (as may happen in a [[sheaf topos]]), then this is vacuously point-surjective but not an epimorphism.

Point-surjectivity may seem like an inadequate notion of "epimorphism", but it suffices for many purposes. For example,

+-- {: .num_prop}
###### Proposition
**(Cantor's theorem in a topos)**
For any object $X$, there is an epimorphism $f: X \to \Omega^X$ only if the topos is degenerate.
=--

+-- {: .proof}
###### Proof
Suppose there existed such an epi. In a topos, a map $f: X \to Y$ is epi iff the [[image|direct image]] map $\exists_f: \Omega^X \to \Omega^Y$ retracts the [[preimage|inverse image]] map $\Omega^f: \Omega^Y \to \Omega^X$, i.e., $\exists_f \circ \Omega^f = 1_{\Omega^Y}$. Putting $Y = \Omega^X$, the supposition implies that $\exists_f: Y \to \Omega^Y$ is a retraction. But retractions are automatically point-surjective.

We then conclude from Lawvere's fixed point theorem that every endomorphism on $\Omega$, in particular the negation $\neg: \Omega \to \Omega$, has a fixed point $p: 1 \to \Omega$. Then $0 = p \wedge \neg p = p \wedge p = p$, whence $\neg 0 = 0$, or "true = false": the topos is degenerate.
=--

Once we have a proposition $p$ with $p = \neg p$, another way to conclude the proof is to apply Lawvere's fixed-point theorem again to the surjection $p \to 0^p$, where $p$ is regarded as a [[subsingleton]] and $0$ is the initial object, so that $0^p = \neg p$.  This gives a fixed point $1\to 0$ of the identity map $0\to 0$, which again makes the topos degenerate.  (The shorter version above is a [[beta-reduction]] of this.)  For a formalization of this argument and a generalization thereof to [[universe types]], see [Escardo18](#Escardo18).

+-- {: .num_remark}
###### Remark
Another version of Lawvere's fixed-point theorem requires only finite products for its statement. Namely, in a category with finite products, suppose $\Phi: A \times A \to B$ is a morphism with the property that for each $g: A \to B$ there exists $a: 1 \to A$ such that $g \lambda = \Phi \circ (a \times 1_A)$, where $\lambda: 1 \times A \stackrel{\sim}{\to} A$ is the projection. Then every map $f: B \to B$ has a fixed point. This version of the theorem is emphasized by [Yanofsky](#Yanofsky03).
=--

+-- {: .num_remark}
###### Remark
Many applications of Lawvere's fixed point theorem are in the form of negated propositions, e.g., there is no epimorphism from a set to its power set, or [[Peano arithmetic]] cannot prove its own [[consistency]]. However, there are positive applications as well, e.g., it implies the existence of [[fixed-point combinators]] in [[untyped lambda calculus]].
=--

## History
 {#History}

In an interview ([Lawvere 07](William+Lawvere#Interview07)) not long after Gödel's 100th birthday, [[William Lawvere]] answered the question

> We have recently celebrated Kurt Gödel's 100th birthday. What do you think about the extra-mathematical publicity around his incompleteness theorem?

by saying (reproduced in [Lawvere 69 reprint, p. 2](#Lawvere69)):

> In 'Diagonal arguments and Cartesian closed categories' ([Lawvere 69](#Lawvere69)) we demystified the incompleteness theorem of Gödel and the truth-definition theory of Tarski by showing that both are consequences of some very simple algebra in the Cartesian-closed setting. It was always hard for many to comprehend how Cantor's mathematical theorem could be re-christened as a "paradox" by Russell and how Gödel's theorem could be so often declared to be the most significant result of the 20th century. There was always the suspicion among scientists that such extra-mathematical publicity movements concealed an agenda for re-establishing belief as a substitute for science. Now, one hundred years after Gödel's birth, the organized attempts to harness his great mathematical work to such an agenda have become explicit.

## Related pages

* [[fixed point]]

* [[Kleene's fixed point theorem]]

* [[Brouwer's fixed point theorem]]

* [[Lefschetz fixed point theorem]]

* [[Cantor's theorem]]

* [[fixed-point combinator]]

* [[incompleteness theorem]]

* [[Löb's theorem]]

## References
 {#References}

The original article:

* {#Lawvere69} [[William Lawvere]], _Diagonal Arguments and Cartesian Closed Categories_, Lecture Notes in Mathematics, **92** (1969) 134-145 [[tac:15](http://tac.mta.ca/tac/reprints/articles/15/tr15abs.html)]

Review and further development:

* {#Yanofsky03} [[Noson Yanofsky]], *A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points*, The Bulletin of Symbolic Logic **9** 3 (2003) 362-386 [[arXiv:math/0305282](http://arxiv.org/abs/math/0305282), [jstor:3109884](https://www.jstor.org/stable/3109884)]

Exposition:

* [[Andrej Bauer]], _On a proof of Cantor's theorem_ ([web](http://math.andrej.com/2007/04/08/on-a-proof-of-cantors-theorem/))

Formalization in [[Agda]]

* {#Escardo18} [[Martin Escardo]], _On Lawvere's Fixed Point Theorem_ (2018) [[agda development](https://www.cs.bham.ac.uk/~mhe/agda-new/LawvereFPT.html)]

The necessary assumptions for Lawvere's account are reduced in various ways in

* {#Roberts} [[David Michael Roberts]], _Substructural fixed-point theorems and the diagonal argument: theme and variations_,  Compositionality **5** issue 8 (2023) doi:[10.32408/compositionality-5-8](https://doi.org/10.32408/compositionality-5-8),
([arXiv:2110.00239](https://arxiv.org/abs/2110.00239)).

[[!redirects Lawvere fixed point theorem]]


Lawvere's reconstruction theorem

## Lawvere's Reconstruction Theorem ##

This is a famous result from William Lawvere's thesis.

A __finite products theory__ or [[Lawvere theory]] $C$ is a category with finite [[products]] where all objects are finite products of copies of a given object $x$.  A __model__ is a functor $F: C \to Set$ preserving finite products, and a morphism of models is a natural transformation between such functors. This gives us a category $Mod(C)$ of models of $C$.

What is going on here?  A model $F$ is really just a set $F(x)$ together with a bunch of $n$-ary operations coming from the morphisms in $C$, satisfying equational laws coming from the equations between morphisms in $C$. Any sort of algebraic gadget that's just a set with a bunch of $n$-ary operations satisfying equations can be described using a theory of this sort. For example: monoids, groups, abelian groups, rings... and so on. We can describe any of these using a suitable algebraic theory, and in each case, the category $Mod(C)$ will be the category of these algebraic gadgets.

There is a functor

$$R: Mod(C) \to Set$$

which carries each model $F$ to the set $F(x)$. We can think of this as a functor which forgets all the operations of our algebraic gadget and remembers only the underlying set.  This should make you desire a [[adjoint functor|left adjoint]]

$$L: Set \to Mod(C)$$

sending each set to the "free" algebraic gadget on this set. Indeed, such a left adjoint exists!

Given this pair of adjoint functors we we can talk about the category of "finitely generated free models" of our theory.  The objects here are objects of $Mod(C)$ of the form $L(S)$ where $S$ is a finite set, and the morphisms are the usual morphisms in $Mod(C)$. Let us call this category $fg Free Mod(C)$.

Here is a way to reconstruct $C$ from its category of finitely generated free models:

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

If $C$ is a Lawvere theory, $C$ is equivalent to $fg Free Mod(C)^{op}$ (i.e., $fg Free Mod(C)$ is equivalent to the opposite of the category $C$).
=--

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

See William F. Lawvere's Ph.D. thesis,  [Functorial Semantics of Algebraic Theories](http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html).
=--

In other words, you can reconstruct a Lawvere theory from its category of finitely generated free algebras in the simplest manner imaginable: just reversing the direction of all the morphisms!

The above all continues to apply in suitable form even when we replace "finite products" throughout by any other structure given by limits; e.g., "finite limits", "arbitrary (small) products", or "arbitrary (small) limits". (Of course, for each case, we must also look at a correspondingly wider class of "free" models than merely those freely generated on finite sets; specifically, we recover the theory from its representable models, comprising one "freely generated" model for each object of $C$). In particular, in the "arbitrary (small) products" case, we can recover the theory from its free models on arbitrary sets. Indeed, in this case $L$ takes the particularly nice definition $L(k)(b) = Hom_C(x^k, b)$, and, in fact, the adjunction between $L$ and $R$ is [[monadic]]; that is, $Mod(C)$ is equivalent to the [[Eilenberg–Moore category]] of the monad $RL(-) = Hom_C(x^{(-)}, x)$, with the functors $R$ and $L$ corresponding under this equivalence to the forgetful and free functors of the Eilenberg--Moore construction. In fact, every monad can be seen to arise uniquely in this way, taking $C$ as the dual of its [[Kleisli category]] (automatically a category with small products generated by a single object, as $Set$ is a category with small coproducts generated by the single object $1$, and the Kleisli category is generated by a left adjoint applied to $Set$); thus, we have a correspondence between monads on $Set$ and categories with small products generated by a single object, both equally well representing single-sorted, infinitary algebraic theories, and, in the former setting, the above recoverability result is just the immediate fact that a monad can be recovered from the forgetful functor on its Eilenberg--Moore category (as, indeed, it can of course be recovered from any adjunction giving rise to it).

(This all also works just as well for inescapably multi-sorted theories (that is, even without the restriction that $C$ be generated by a single object); in this case, we can't help but recognize that there are multiple forgetful functors from $Mod(C)$ to $Set$, one for each object of $C$, each with a corresponding free model on $1$ (by the [[Yoneda lemma]]), and, again, the theory is recovered as the dual of the category of all these models (by the [[Yoneda embedding lemma]]))

[[!redirects Lawvere reconstruction theorem]]
[[!redirects Lawvere's reconstruction theorem]]


Lawvere-Tierney topology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

# Lawvere--Tierney topologies
* table of contents
{: toc}

## Idea

A _Lawvere--Tierney topology_ (or _(local) operator_, or _[[modal logic|modality]]_, also called _[geometric modality](modal+type+theory#GeometricModality)_) on a [[topos]] is a way of saying that something is 'locally' [[true]].  Unlike a [[Grothendieck topology]], this is done directly at the stage of [[logic]], defining a _[[geometric logic]]_.  In fact, it is a generalisation of Grothendieck topology in this sense: If $C$ is a [[small category]], then choosing a Grothendieck topology on $C$ is equivalent to choosing a Lawvere--Tierney topology in the [[presheaf topos]] $\Set^{C^\op}$ on $C$.

The use of "topology" for this and the related Grothendieck concept is regarded by some people as unfortunate; see the [historical note](https://ncatlab.org/nlab/show/Grothendieck+topology#historical_note_on_terminology) at [[Grothendieck topology]] for some reasons why.  A proposed replacement for "Grothendieck topology" is [[coverage|(Grothendieck) coverage]]; see the afore-mentioned historical note for some possible replacements for "Lawvere--Tierney topology."

## Definition

Let $E$ be a [[topos]], with [[subobject classifier]] $\Omega$.

### The closure operator

+-- {: .num_defn #LTTopologyDef}
###### Definition

A __Lawvere--Tierney topology__ in $E$ is ([[internalization|internally]]) a [[closure operator]] given by a [[exact functor|left exact]] [[idempotent monad]] on the internal meet-[[semilattice]] $\Omega$.

This means that: a Lawvere--Tierney topology in $E$ is a [[morphism]]

$$
  j: \Omega \to \Omega
$$

such that

1. $j true = true$, equivalently $\id_\Omega \leq j: \Omega \to \Omega$ ('if $p$ is true, then $p$ is locally true')

   $$
     \array{
       * &\stackrel{true}{\to}& \Omega
       \\
       & {}_{\mathllap{true}}\searrow & \downarrow^{\mathrlap{j}}
       \\
       && \Omega
     }
   $$

1. $j j = j$ ('$p$ is locally locally true iff $p$ is locally true');

   $$
     \array{
       \Omega &\stackrel{j}{\to}& \Omega
       \\
       & {}_{\mathllap{j}}\searrow & \downarrow^{\mathrlap{j}}
       \\
       && \Omega
     }
   $$

1. $j \circ \wedge = \wedge \circ (j \times j)$ ('$p \wedge q$ is locally true iff $p$ and $q$ are each locally true')

   $$
     \array{
       \Omega \times \Omega &\stackrel{\wedge}{\to}& \Omega
       \\
       {}^{\mathllap{j \times j}}\downarrow && \downarrow^{\mathrlap{j}}
       \\
       \Omega \times \Omega &\underset{\wedge}{\to}& \Omega
     }
     \,.
   $$

=--

Here $\leq$ is the internal [[partial order]] on $\Omega$, and $\wedge: \Omega \times \Omega \to \Omega$ is the internal [[meet]].

This appears for instance as ([MacLaneMoerdijk, V 1.](#MacLaneMoerdijk)).

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

By the definition of [[subobject classifier]] $j$ is equivalently a [[subobject]]

$$
  J \hookrightarrow \Omega
$$

satisfying three conditions. This perspective gives the direct relation to [[Grothendieck topologies]], as discussed [below](#RelationToGrothendieckTopology).

=--

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

Equivalently, the third axiom in def. \ref{LTTopologyDef} can be replaced with the (internal) statement that $j$ is order-preserving.

The equivalence amounts to the fact that, within the [[internal logic]] of [[topoi]], one can demonstrate that every [[monad]] on the [[preorder]] of [[truth values]] is in fact [[strong monad|strong]] (a special case of the fact that, for an [[endofunctor]] on some [[closed monoidal category|monoidal closed]] $V$, [[tensorial strength|tensorial strengths]] are the same as $V$-[[enriched category|enrichments]], as described in the article on the former), and therefore _automatically_ preserves finite meets.

Thus, a Lawvere-Tierney topology is the same thing as an internal [[closure operator]] on the preorder $\Omega$ (aka, a [[Moore closure]] on the one-element set), which amounts to the same thing as a natural closure operator on subobjects.

Specifically, given any [[subobject]] inclusion $X \hookrightarrow Y$ in $E$, consider its [[characteristic morphism]] $\chi_X: Y \to \Omega$. Then $j \circ \chi_X$ is another morphism $Y \to \Omega$, which defines another subobject $j_*(X)$ of $Y$, taken as the closure of our original subobject. The elements of $j_*(X)$ are those elements of $Y$ that are 'locally' in $X$.

=--

+-- {: .num_defn #TheClosureOperator}
###### Definition

The **[[closure operator]]** induced by $j$ is the [[natural transformation|transformation]]

$$
  \overline{(-)}_X : Sub(X) \to Sub(X)
$$

on the [[subobject]] lattice of $X \in E$, [[natural transformation|natural]] in $X$, that is given by the [[commuting diagram]]

$$
  \array{
    Hom(X, \Omega) &\stackrel{\simeq}{\to}& Sub(X)
    \\
    {}^{\mathllap{Hom(1,j)}}\downarrow && \downarrow^{\mathrlap{\overline{(-)}}}
    \\
    Hom(X,\Omega) &\stackrel{\simeq}{\to}& Sub(X)
  }
  \,.
$$

=--

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

This means that for $U \hookrightarrow X$ a [[subobject]], with characteristic morphism $char U : X \to \Omega$, its closure is the subobject classified by

$$
  char \overline{U} : X \stackrel{char U}{\to} \Omega \stackrel{j}{\to} \Omega
  \,.
$$

=--

This appears for instance as ([MacLaneMoerdijk, p. 220](
MacLaneMoerdijk)).

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

A morphism $j : \Omega \to \Omega$ is a Lawvere-Tierney topology, def. \ref{LTTopologyDef} precisely if the corresponding closure operator, def. \ref{TheClosureOperator} satisfies for all $X, Y \in E$

1. $A \subset \overline{A}$;

1. $\overline{\overline{A}} = \overline{A}$;

1. $\overline{A \cap B} = \overline{A} \cap \overline{B}$.

=--

This appears as
([MacLaneMoerdijk, V 1., prop 1](#MacLaneMoerdijk)).

### Sheaves

Using Lawvere--Tierney topologies,
the notion of [[sheaf]] and [[sheafification]] generalizes from [[Grothendieck topos|Grothendieck topoi]] to arbitrary topoi.

Let $E$ be a [[topos]] with Lawvere-Tierney topology $j$, def. \ref{LTTopologyDef} and associated closure operator $\overline{(-)} : Sub(-) \to Sub(-)$, def. \ref{TheClosureOperator}.

+-- {: .num_defn #DenseMonos}
###### Definition

A subobject $U \in Sub(X)$ is called **dense** if $\overline{U} = X$.

The corresponding [[monomorphism]] $U \hookrightarrow X$ is called a **[[dense monomorphism]]**.

=--

+-- {: .num_defn #JSheaf}
###### Definition

An object $F \in E$ is called a **$j$-[[sheaf]]** if it is a [[local object]] with respect to the dense monomorphisms.

This means that $F$ is a $j$-sheaf if for every dense $U \hookrightarrow X$ the induced morphism

$$
  Hom(X,F) \to Hom(U,F)
$$

is an [[isomorphism]].

$F$ is a **$j$-[[separated presheaf]]** if this morphism is itself a [[monomorphism]].

=--

This is for instance in ([MacLaneMoerdijk, p. 223](#MacLaneMoerdijk)).

## Properties

### $j$-Sheaf subtoposes
 {#SheafSubtoposes}

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

For $E$ a [[topos]] and $j$ a Lawvere-Tierney topology on $E$, the inclusion

$$
  Sh_j(E) \hookrightarrow E
$$

of [j-sheaves](#JSheaf) is a [[geometric embedding]].

So in particular $Sh_j(E)$ is itself a [[topos]] and the embedding is a [[full and faithful functor]] which has a [[exact functor|left exact]] [[left adjoint]] functor $E \to Sh_j(E)$: this is called the **[[sheafification]]** functor.

=--

This appears for instance as ([MacLaneMoerdijk V 3., theorem 1](#MacLaneMoerdijk)).

### Equivalence with Grothendieck topologies
 {#RelationToGrothendieckTopology}

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

For $C$ a [[small category]] and $E := [C^{op}, Set]$ its [[presheaf topos]], Lawvere--Tierney topologies in $E$ are equivalent to [[Grothendieck topology|Grothendieck topologies]] on $C$.

=--

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

The [[subobject classifier]] in a [[presheaf topos]] is the  presheaf that assigns to $U \in C$ the set of all [[sieve]]s in $C$ on $U$

$$
  \Omega : U \mapsto Sieves_C(U)
  \,.
$$

since we have

$$Sieves_C (U)=Sub_E(y(U))=hom(y(U),\Omega)=\Omega(U)$$

A [[subobject]] $J \hookrightarrow \Omega$ is therefore precisely a choice of a collection of sieves on each object, which is closed under pullback. The proof therefore amounts to checking that the condition that such a collection of sieves is a [[Grothendieck topology]] on $C$ is equivalent to the statement that the characteristic map $j : \Omega \to \Omega$ of $J \hookrightarrow \Omega$ (see remark \ref{LTTopologyAsSubobject}) is a Lawvere-Tierney topology.

=--

Here is more discussion of this point:

Suppose that $C$ is a small site.  Then given a [[subpresheaf]] inclusion $F \hookrightarrow G$ in $\Set^{C^\op}$, an object $X$ of $C$, and an element $f$ of $G(X)$, we say $f$ is locally in $F$ (that is, $f \in j_*(F)(X)$) if and only if, for some [[cover|covering family]] $c = (c_i: U_i \to X)_i$ on $X$, the restriction $c^*(f)$ of $f$ to $c$ is in $F$ (that is, each $c_i^*(f) \in F(U_i)$).  This intuitively defines the "local" modality that is the Lawvere--Tierney topology corresponding to the given Grothendieck topology on $C$.

As a specific example, take the usual Grothendieck topology on [[Top]], given by the usual notion of open cover. Taking real-valued functions on a space defines a presheaf (in fact a sheaf) $G: X \mapsto [X,R]$ on $\Top$; the constant functions form a subpresheaf $F$ of $G$. A real-valued function $f: X \to R$ belongs to $j_*(F)$ iff it is *locally* constant; that is, for some open cover $(U_i)_i$ of the domain $X$, each restriction $f|U_i$ is constant.

To make this precise in terms of the above definition, we need to understand the subobject classifier in $E = Set^{C^{op}}$.  But according to the definition, $\Omega$ is simply the representing object for the functor
$$Sub: E^{op} \to Set$$
which takes an object $F$ of $E$ to the collection of subobjects of $F$, $Sub(F)$. In other words, $Sub(F) \cong \hom_E(F, \Omega)$. Applied to $F = \hom_C(-, c)$, we have then
$$Sub(\hom_C(-, c)) \cong \hom_{Set^{C^{op}}}(\hom_C(-, c), \Omega) \stackrel{Yoneda}{\cong} \Omega(c)$$
In other words, we find that the functor $\Omega: C^{op} \to Set$ is defined by
$$\Omega(c) = \{sieves\,on\,c\}$$

Next, if $J$ is a Grothendieck topology on $C$, then the collection of $J$-covering sieves on $c$ (which we denote by $J(c)$( is a subcollection of all sieves on $c$, and so we have an inclusion
$$J(c) \hookrightarrow \Omega(c)$$
and this inclusion is natural in $c$, by virtue of the first axiom on covering sieves. Thus we have a subobject
$$J \hookrightarrow \Omega$$
and again, by definition of subobject classifier, this subobject corresponds to a uniquely determined element
$$j \in \hom_E(\Omega, \Omega)$$
which is just the Lawvere--Tierney operator $j: \Omega \to \Omega$.

Conversely, any morphism $j:\Omega\to\Omega$ determines a subobject $J$ of $\Omega$, which therefore associates to every object $c$ a set of sieves on $c$.  It is easy to check that the axioms for covering sieves in a Grothendieck topology correspond exactly to the required properties of the operator $j$.

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

For $C$ a [[small site]] and $j$ the Lawvere-Tierney topology on the [[presheaf topos]] $E = [C^{op}, Set]$ given by prop. \ref{RelationToCoveringSieves} the [j-sheaves](#SheafSubtoposes) are precisely the [[sheaves]] in the ordinary sense of Grothendieck topologies.

=--

### Relation to lex reflectors
 {#ClosureOperation}

As discussed there, [[categories of sheaves]] are also characterized as being [[reflective subcategories]] of the given ambient topos

$$
  Sh_j(\mathcal{E})
   \stackrel{\overset{L}{\leftarrow}}{\underset{}{\hookrightarrow}}
  \mathcal{E}
 \,.
$$

Here we discuss explicit translations between the structure given by the [[localization|reflector]] $L$ and the corresponding Lawvere-Tierney topology $j : \Omega \to \Omega$ in a way that makes the relation to [[modal type theory]] and [[monads (in computer science)]] most manifest.

+-- {: .num_defn #ClosureOperatorOfReflection}
###### Definition

Given a reflector $\sharp : \mathcal{E} \stackrel{L}{\to} Sh_j(\mathcal{E}) \hookrightarrow \mathcal{E}$, define for each object $X \in \mathcal{E}$ a **closure operator**,
being a [[functor]] on the [[poset of subobjects]] of $X$

$$
  c_L : Sub(X) \to Sub(X)
  \,,
$$

by sending any [[monomorphism]] $A \hookrightarrow X$ classified by a [[characteristic function]] $\chi_A : X \to \Omega$ to the [[pullback]] $c_L(A)$ in

$$
  \array{
    c_L(A) &\to& \sharp A
    \\
    \downarrow && \downarrow
    \\
    X &\to& \sharp X
  }
  \,,
$$

where $X \to \sharp X$ is the [[unit of an adjunction|adjunction unit]].

=--

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

This is well defined. Moreover, $c_L$
commutes with [[pullback]] ([[change of base]]).

=--

This appears as ([Johnstone, lemma A4.3.2](#Johnstone)).

+-- {: .num_defn }
###### Definition

A family of functors $Sub(X) \to Sub(X)$ for all objects $X$ that commutes with [[change of base]] is called a **[[universal closure operation]]**.

=--

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

Given a [[left exact functor|left exact]] reflector $\sharp$ as above with induced closure operation $c_L$, the corresponding Lawvere-Tierney operator $j : \Omega \to \Omega$ is given as the composite

$$
  j : \Omega \to \sharp \Omega \stackrel{\chi_{\sharp true}}{\to} \Omega
  \,,
$$

where

* $\Omega \to \sharp \Omega$ is the [[unit of an adjunction|adjunction unit]];

* $\chi_{\sharp true} : \sharp \Omega \to \Omega$ is the [[characteristic function]] of the result of applying $\sharp$ to the [[subobject classifier|universal subobject]]

  $$
    (* \stackrel{\sharp true}{\hookrightarrow} \sharp \Omega)
     :=
    \sharp (* \stackrel{true}{\hookrightarrow} \Omega)
  $$

  (which is again a [[monomorphism]] since $\sharp$ preserves [[pullbacks]]).

=--

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

For $A \hookrightarrow X$ any [[subobject]] with [[characteristic function]] $\chi_A : X \to \Omega$, we need to show that we have a [[pullback]] [[diagram]]

$$
  \array{
    c_L(A) &\to& &\to& &\to& *
    \\
    \downarrow && && && \downarrow
    \\
    X &\stackrel{\chi_A}{\to}& \Omega
    &\stackrel{}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
  \,.
$$

The pullback along the rightmost morphism is by definition $# * \to \sharp \Omega$

$$
  \array{
    c_L(A) &\to& &\to& # * = * &\to& *
    \\
    \downarrow && && \downarrow && \downarrow
    \\
    X &\stackrel{\chi_A}{\to}& \Omega
    &\stackrel{}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
  \,.
$$

Moreover, by the [[natural transformation|naturality]] of the [[unit of an adjunction|adjunction unit]] we have a [[commuting diagram]]

$$
  \array{
    X &\to& \sharp X
    \\
    {}^{\mathllap{\chi_A}}\downarrow && \downarrow^{\mathrlap{\sharp \chi_A}}
    \\
    \Omega &\to& \sharp \Omega
  }
  \,.
$$

Using this in the remaining bottom morphism of our would-be pullback square we find that equivalently

$$
  \array{
    c_L(A) &\to& &\to& # * = * &\to& *
    \\
    \downarrow && && \downarrow && \downarrow
    \\
    X &\stackrel{}{\to}& \sharp X
    &\stackrel{\sharp \chi_A}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
$$

needs to be a pullback diagram. Since $\sharp$ preserves pullbacks we have that also the middle square in

$$
  \array{
    c_L(A) &\to& \sharp A &\to& # * = * &\to& *
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    X &\stackrel{}{\to}& \sharp X
    &\stackrel{\sharp \chi_A}{\to}&
    \sharp \Omega
    &\stackrel{}{\to}&
    \Omega
  }
$$

is a pullback. But then also the left square is a pullback, by def. \ref{ClosureOperatorOfReflection}. This finally means, by the [[pasting law]], that also the total rectangle is a pullback.

=--

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

Equivalently, by the [[pasting law]], we have that $j : \Omega \to \Omega$ is the [[characteristic function]] of the $L$-closure, def. \ref{ClosureOperatorOfReflection}, of the universal subobject $* \to \Omega$, because we have a [[pasting diagram]] of [[pullback]] squares

$$
  \array{
    c_L(*) &\to& \sharp * = * &\to & *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \Omega &\to& \sharp \Omega &\stackrel{\chi_{\sharp true}}{\to} & \Omega
  }
  \,.
$$

In this form the statement appears in the proof of ([Johnstone, Theorem A4.3.9](#Johnstone)).

=--

## Enriched generalization

* [[Francis Borceux]], _Algebraic localizations and elementary toposes_, Cah. Top. Géom. Diff. Cat. **21** (1980), no. 4, 393–401. ([MR82g:18002](http://www.ams.org/mathscinet-getitem?mr=606384), [pdf](http://archive.numdam.org/article/CTGDC_1980__21_4_393_0.pdf))

* [[Francis Borceux]], _Sheaves of algebras for a commutative theory_, Ann. Soc. Sci. Bruxelles Sér. I **95** (1981), no. 1, 3–19. ([MR83c:18006](http://www.ams.org/mathscinet-getitem?mr=628032))

Let $\mathcal{C}$ be a small category enriched over $Set^{\mathbf{T}}$ where $\mathbf{T}$ is a [[commutative algebraic theory]]. Then $[\mathcal{C}^{op},\text{Set}^{\mathbf{T}}]$. A $\mathbf{T}$-sieve as an enriched subfunctor of $\mathcal{C}(-,x)\colon\mathcal{C}^{op}\rightarrow\text{Set}^{\mathbf{T}}$. A $\mathbf{T}$-topology is a set $J(x)$ of emptypage$\mathbf{T}$-sieves for every $x$, satisfying some axioms. Borceux defines the notion of a sheaf over such enriched site and proves the existence and exactness of the associated sheaf functor.

He proves that there is an object $\Omega_{\mathbf{T}}$ in $[\mathcal{C}^{op},\text{Set}]$ which classifies subobjects in $[\mathcal{C}^{op},\text{Set}^{\mathbf{T}}]$. Moreover, there is a correspondence between

1. localizations of $[\mathcal{C}^{\text{op}},\text{Set}^{\mathbf{T}}]$

2. $\mathbf{T}$-topologies on $\mathcal{C}$

3. morphisms $j\colon\Omega_{\mathbf{T}}\rightarrow\Omega_{\mathbf{T}}$ satisfying the Lawvere-Tierney axioms for a topology

## Related concepts

* [[modality]]

* [[modal type theory]]

* [[monad (in programming theory)]]

## References

The notion is introduced as a [[geometric modality]] on p. 3 of

* {#Lawvere} [[William Lawvere]], _Quantifiers and sheaves_, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 ([pdf](http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0329.0334.ocr.pdf))

Detailed discussion of Lawvere-Tierney operators as [[geometric modalities]] is in

* {#Goldblatt} [[Robert Goldblatt]], _Grothendieck topology as geometric modality_, Mathematical Logic Quarterly, Volume 27, Issue 31-35, pages 495–529, (1981)

Textbook accounts include section V.1 of

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_,

(the notion of sheaves in section V.3, the sheafification functor in section V.3 and the relation to Grothendieck topologies in section V.4);

and section A4.4 of

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]]_

Discussion in [[homotopy type theory]] is in

* {#QuirinTabareau16} Kevin Quirin, Nicolas Tabareau, _Lawvere-Tierney sheafification in Homotopy Type Theory_, Journal of Formalized Reasoning, Vol 9, No 2, (2016) ([web](https://jfr.unibo.it/article/view/6232))

The additive version of Lawvere-Tierney modular closure operators on categories of modules (as an additive version of a presheaf topos) -- equivalent to [[Gabriel topology]] (=$Ab$-enriched Grothendieck topology) has been studied (and the equivalence proven) several years before Lawvere and Tierney in

* [[Kim Lin Chew]], _Closure operations in the study of rings of quotients_, Bull. Math. Soc. Nanyang Univ. (1965) 1-20.

[[!redirects sheafification in a Lawvere-Tierney topos]]
[[!redirects sheafification in a Lawvere–Tierney topos]]
[[!redirects sheafification in a Lawvere--Tierney topos]]
[[!redirects Lawvere-Tierney topology]]
[[!redirects Lawvere–Tierney topology]]
[[!redirects Lawvere--Tierney topology]]
[[!redirects Lawvere-Tierney topologies]]
[[!redirects Lawvere–Tierney topologies]]
[[!redirects Lawvere--Tierney topologies]]

[[!redirects Lawvere-Tierney operator]]
[[!redirects Lawvere–Tierney operator]]
[[!redirects Lawvere--Tierney operator]]
[[!redirects Lawvere-Tierney operators]]
[[!redirects Lawvere–Tierney operators]]
[[!redirects Lawvere--Tierney operators]]

[[!redirects local modality]]
[[!redirects local modalities]]


lax (∞,1)-colimit

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The refinement of the concept of [[lax colimits]] from [[category theory]] to [[(infinity,1)-category theory]].

## $(\infty,1)Cat$-valued diagrams

In the special case of functors $f : C \to (\infty,1)Cat$, lax (co)limits can be given by the [[(∞,1)-end]] and coend.

$$
  laxcolim(f)   \simeq \int^{c \in C} (C_{c/})^{op} \times f(c)
$$
$$
  oplaxcolim(f)   \simeq \int^{c \in C} C_{c/} \times f(c)
$$
$$
  laxlim(f)  \simeq \int_{c \in C} Fun(C_{/c}, f(c))
$$
$$
  oplaxlim(f)  \simeq \int_{c \in C} Fun((C_{/c})^{op}, f(c))
$$

where $C_{/\bullet} : C \to (\infty,1) Cat$ and $C_{\bullet/} : C^{op} \to (\infty,1)Cat$ are the functors sending a morphism $c \to c'$ of $C$ to the composition functors $C_{/c} \to C_{/c'}$ and $C_{c'/} \to C_{c/}$.

Take care to note that lax colimits correspond to _oplax_ cones, just as in the 2-categorical case.

These operations also have simple descriptions in terms of fibrations

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

Let $f : C \to (\infty,1)Cat$. Then

* $laxlim(f) \simeq Fun_C(C, el_C(f))$
* $oplaxlim(f) \simeq Fun_{C^{op}}(C^{op}, \overline{el}_{C^{op}}(f))$
* $laxcolim(f) \simeq \overline{el}_{C^\op}(f)$
* $oplaxcolim(f) \simeq el_C(f)$

where $el$ and $\overline{el}$ are the covariant and contravariant [[(∞,1)-Grothendieck construction]], and $Fun_K$ is the hom-category functor on $(\infty,1)Cat_{/K}$.

=--

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

For the case of lax limits (and dually oplax limits), we can compute
$$
  \begin{aligned}
  Map(A, laxlim(f))
  &\simeq \int_{c \in C} Map(A, Fun(C_{/c}, f(c)))
  \simeq \int_{c \in C} Map(A \times C_{/c}, f(c))
  \\&\simeq Nat(A \times C_{/\bullet}, f)
  \simeq Map_C^{cocart}(A \otimes C^{[1]}, el_C(f))
  \\&\simeq Map_C(A \times C, el_C(f))
  \simeq Map(A, Fun_C(C, el_C(f)))
  \end{aligned}
$$
using the fact $A \times C^{[1]} \to C : (a,t) \mapsto t_1$
is the free [[cocartesian fibration]] generated by $A \times C \to C$.

For the case of oplax colimits (and dually lax colimits), we can observe $oplaxcolim$, $el_C$, and the forgetful functor $(\infty,1)Cat_{/C}^{cocart} \to (\infty,1)Cat$ preserve colimits and both sides of the isomorphism send $[1] \times C(c, -) \mapsto [1] \times C_{c/}$. Thus, they agree on the smallest full subcategory of $(\infty,1)Cat^C$
containing these functors and closed under small colimits, which is the entirety of $(\infty,1)Cat^C$.

=--

It's worth noting morphisms in $(\infty,1)Cat_{/C}$ between cocartesian fibrations can be thought of as lax transformations, so this agrees with the intuition that $laxlim(f)$ should be the category of lax natural transformations $1 \Rightarrow f$ and similar.

## Examples

* Let $t : C \to (\infty,1)Cat$ be the terminal functor. Then $laxcolim(t) \simeq C^{op}$ and $oplaxcolim(t) \simeq C$,

* If $J : [1] \to (\infty,1)Cat$ is the diagram depicting a functor $f : A \to B$, then the lax limits are the comma categories $laxlim(J) \simeq (f \downarrow B)$ and $oplaxlim(J) \simeq (B \downarrow f)$. Dually, the lax colimits are mapping cylinders: $laxcolim(J) \simeq ([1] \times A) \amalg_{\{0\} \times A} B$ and $oplaxcolim(J) \simeq ([1] \times A ) \amalg_{\{1\} \times A} B$.

## Related concepts

* [[(infinity,1)-limit]]

## References

* {#GepnerHaugsengNikolaus15} [[David Gepner]], [[Rune Haugseng]], [[Thomas Nikolaus]], _Lax colimits and free fibrations in $\infty$-categories_ ([arXiv:1501.02161](http://arxiv.org/abs/1501.02161))

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_

[[!redirects lax (∞,1)-colimits]]

[[!redirects lax (∞,1)-limit]]
[[!redirects lax (∞,1)-limits]]

[[!redirects lax (infinity,1)-colimit]]
[[!redirects lax (infinity,1)-colimits]]

[[!redirects lax (infinity,1)-limit]]
[[!redirects lax (infinity,1)-limits]]


lax 2-adjunction

#Contents#
* table of contents
{:toc}

## Idea ##

A _lax 2-adjunction_ is an [[adjunction]] 'up to adjointness'.  A 1-categorical adjunction is given by a [[natural isomorphism]] of [[hom-set]]s

$$\phi_{A,B} : D(F A,B) \cong C(A,G B).$$

A [[strict 2-adjunction]] is analogous, where $C,D$ are instead [[strict 2-category|strict 2-categories]], $F,G$ are [[strict 2-functor|strict 2-functors]], and $\phi$ is a strictly 2-natural isomorphism of hom-categories, while a [[biadjunction]] or [[pseudoadjunction]] allows all of these to be "weak up to isomorphism".

A __lax 2-adjunction__ is instead given by a (suitably natural) pair of families of functors

$$\hat\phi_{A,B} : D(F A,B) \overset{\to}{\leftarrow} C(A,G B) :
\check\phi_{A,B} $$

such that for each $A, B$ we have an *adjunction* $\hat\phi_{A,B} \dashv \check\phi_{A,B}$.  Independently of this laxness aspect, we can make a choice of weakness or strictness for the categories, functors, and natural transformations involved; thus one might speak of "lax biadjunctions", "lax pseudoadjunctions", and so on.

It is not entirely clear that the term "lax 2-adjunction" is appropriate; in particular it is not an instance of the general notion of [[lax algebras]], unlike nearly all other usages of the word "lax".  Other terms that have been used for the same or similar notions are "transcendental quasi-adjunction" [(Gray)](#Gray) and "local adjunction" [(Betti-Power)](#BP).

## Definition ##

Just as the notion of [[adjunction]] makes sense not just in $Cat$ but in any 2-category, lax 2-adjunctions can be defined in any (possibly weak) [[3-category]] when the above is suitably reformulated by using the 2-categorical [[Yoneda lemma]].  So, given 0-cells $C,D$ and 1-cells $F:C\to D$, $G:D\to C$ in any 3-category, we say $F$ is __lax left adjoint__ to $G$ if we have

* 2-cells $\eta:C\Rightarrow G F$ and $\epsilon:F G \Rightarrow D$, and

* instead of strict triangle equalities, 3-cells (called _triangulators_)
  $s,t$

  [[lax-adj-mdfs.png:pic]]

  satisfying the _swallowtail identities_:

  [[lax-adj-swlwtl-eta.png:pic]]

  and

  [[lax-adj-swlwtl-eps.png:pic]]

  (where if the 3-category is weak then coherences might need to be inserted to make these make sense.

When interpreted in a [[strict 3-category]] $2Cat$, the [[Gray-category]] $Gray$, or the [[tricategory]] $Bicat$, this yields three levels of weakness of "lax 2-adjunction" according to whether the categories, functors, and/or transformations are strict or pseudo.

## Further generalizations

The following remarks are incomplete.

One can additionally generalize to allow $\eta$ and $\epsilon$ to be [[lax natural transformations]].  [Gray's](#Gray) original notion of "transcendental quasi-adjunction" did this, with the 2-categories and 2-functors being strict.  Lax transformations do not live in any 3-category as usually understood, but Gray's particular case can be done in the "lax Gray-category" $Gray_\ell$.

Note also that there is no Yoneda lemma for lax transformations (though there is for pseudo ones).  So in order for $\hat\phi$ to be determined by $\eta$, $\hat\phi$ must be pseudo natural in $B$, and likewise for $\check\phi$ to be determined by $\epsilon$, it must be pseudo natural in $A$.  My guess is that laxity in the remaining variable will correspond to laxity of $\eta$ and $\epsilon$, and $s,t$ will come from the adjunction $\check\phi\dashv\hat\phi$.  Another option is to require $\eta$ and $\epsilon$ to be pseudo natural on [[wide subcategories]] that include their components; see [[lax F-adjunction]] for an abstract context for such a definition.

One might additionally consider allowing the functors to be lax, as in [(Betti-Power)](#BP) [(Seely)](#Seely).  But lax functors are even harder to incorporate in a 3-category-like structure than lax transformations, since one can't even whisker any sort of transformation by a lax functor.  Seely also seems to say that $\hat\phi$ and $\check\phi$ are both strict in their _first_ coordinate, rather than one in the first and one in the second as seems (to me) to be necessary for a Yoneda restatement.

## Related concepts

* [[lax-idempotent 2-adjunction]]
* [[lax F-adjunction]]

## References ##

* [[Gray-adjointness-for-2-categories]] referred to a lax 2-adjunction as a _transcendental quasi-adjunction_.
 {#Gray}

* Renato Betti and John Power, _On Local Adjointness of Distributive Bicategories_
 {#BP}

* Seely, _Modelling computations: a 2-categorical framework_, [pdf](http://www.math.mcgill.ca/~rags/WkAdj/LICS.pdf)
 {#Seely}

[[!redirects lax 2-adjunctions]]


lax algebra for a 2-monad

# Lax algebras for a $2$-monad
* table of contents
{: toc}

## Idea

If $T$ is a [[2-monad]] on a [[2-category]] $\mathcal{K}$, then in addition to strict (if $T$ and $\mathcal{K}$ are strict) $T$-[[algebra for a monad|algebras]], which satisfy their laws strictly, and [[pseudoalgebra for a 2-monad|pseudo]] $T$-algebras, which satisfy laws up to isomorphism, one can consider also *lax* and *colax* algebras, which satisfy laws only up to a transformation in one direction or the other.

## Definition

If $T$ is a 2-monad on $\mathcal{K}$, a **lax $T$-algebra** in $\mathcal{K}$ consists of an object $A$, a morphism $T A \overset{a}{\to} A$, and (not necessarily invertible) 2-cells
$$\array{T^2A & \overset{T a}{\to} & T A\\
  ^m\downarrow & \Downarrow & \downarrow^a\\
  T A& \underset{a}{\to} & A}$$
and
$$\array{ A & & \overset{id}{\to} & & A\\
  & _i \searrow & \Downarrow & \nearrow_a \\
  & & T A}$$
satisfying suitable axioms.  (Here $m$ is the multiplication of $T$ and $i$ is the unit.)

Of course, in a **colax $T$-algebra** (also called an **oplax $T$-algebra**) the transformations go the other way.  The official way to remember which is lax and which is colax is that a lax $T$-algebra structure on $A$ is a [[lax morphism|lax M-morphism]] $T \to \langle A,A\rangle$, where $T$ is the 2-monad on the 2-category $[\mathcal{K},\mathcal{K}]$ of (some) endofunctors of $\mathcal{K}$ whose algebras are 2-monads, and $\langle A,A\rangle$ is the [[codensity monad]] of $A$, i.e. the right [[Kan extension]] of $1\overset{A}{\to} \mathcal{K}$ along itself.

If the transformations are invertible, then $A$ is a [[pseudoalgebra for a 2-monad|pseudo-algebra]].

## Related pages

* [[pseudoalgebra for a 2-monad]]

* [[lax monoidal category]]

[[!redirects lax algebra for a 2-monad]]
[[!redirects lax algebras for a 2-monad]]
[[!redirects lax algebras for 2-monads]]
[[!redirects lax algebra]]
[[!redirects lax algebras]]

[[!redirects colax algebra for a 2-monad]]
[[!redirects colax algebras for a 2-monad]]
[[!redirects colax algebras for 2-monads]]
[[!redirects colax algebra]]
[[!redirects colax algebras]]

[[!redirects oplax algebra for a 2-monad]]
[[!redirects oplax algebras for a 2-monad]]
[[!redirects oplax algebras for 2-monads]]
[[!redirects oplax algebra]]
[[!redirects oplax algebras]]


lax biend

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Enriched category theory
+-- {: .hide}
[[!include enriched category theory contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A **lax biend** is, together with the related notion of a **pseudobiend**, one of the analogues of [[end]]s in [[2-category theory]].

## Definition

Similar to how [[limits]] are defined as [[representing objects]] of the [[functor]] $\mathrm{Cones}_{(-)}(D)$ of [[cones]] over a [[diagram]] $D$, lax biends are [[representing objects]] of a pseudofunctor $\mathsf{LaxWedges}_{(-)}(D)$. Below we define all objects involved, arriving at the definition of a lax biend in [Section 2.4](#LaxBiends).

### 2.1 Preliminaries
In this section we recall some facts and constructions in the setting of [[bicategories]].

### 2.2 Lax Wedges

Let $\mathcal{C}$ and $\mathcal{I}$ be [[bicategories]], $D\colon\mathcal{I}^\mathrm{op}\times\mathcal{I}\longrightarrow\mathcal{C}$ be a [[pseudofunctor]], and $X$ be an object of $\mathcal{C}$.

A **lax wedge** is a [[lax extranatural transformation]] $\theta\colon\Delta_{X}\overset{\bullet}{\Rightarrow}D$ from the constant pseudofunctor associated to $X$ to $D$.

### 2.3 Functoriality of Lax Wedges

【...】

### 2.4 Lax Biends {#LaxBiends}

【...】

## References

* {#Fosco} [[Fosco Loregian]], _Coend Calculus_ ([arXiv](http://arxiv.org/abs/1501.02503)).

[[!redirects lax biends]]


Lax equation

#Contents#
* table of contents
{:toc}

## Idea

The *Lax equation* is a general kind of [[equation]] controlling [[integrable systems]].

__Lax equation__ is a linear [[ordinary differential equation]] of the form
$$
\frac{d L}{d t} = [M, L]
$$
for $n\times n$-matrix-valued function $L = L(t)$, where $M$ is also a $n\times n$ matrix. The pair $(L,M)$ is also called a __Lax pair__.

The Lax equation is the compatibility condition for the system
$$
\lambda \psi = L \psi
$$
$$
\frac{d \psi}{d t} = M\psi
$$
where $\psi = \psi(t)$ is a vector function which is an eigenvector for $L$ with eigenvalue $\lambda$. To see this make a derivative of $L\psi$ and use the Leibniz rule.
$$
M L \psi = M \lambda \psi = \lambda M \psi = \lambda \frac{d \psi}{d t} = \frac{d (\lambda\psi)}{d t} = \frac{d (L\psi)}{d t} = \frac{d L}{d t} \psi + L \frac{d \psi}{d t} = \frac{d L}{d t} \psi + L M\psi
$$

## References

* [[Peter Lax]], *Integrals of nonlinear equation of evolution and solitary waves*, Commun. on Pure and Applied Mathematics __21__ 5 (1968) 467-490 [[doi:10.1002/cpa.3160210503](http://dx.doi.org/10.1002/cpa.3160210503)]

[[!redirects Lax equations]]
[[!redirects Lax pair]]


lax extranatural transformation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Enriched category theory
+-- {: .hide}
[[!include enriched category theory contents]]
=--
#### Limits and colimits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A **lax extranatural transformation** is to a [[lax transformation]] as an [[extranatural transformation]] is to a [[natural transformation]].

## Definition

Let $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, and $\mathcal{D}$ be bicategories and $P\colon\mathcal{A}\times\mathcal{B}^\mathsf{op}\times\mathcal{B}\to\mathcal{D}$ and $Q\colon\mathcal{A}\times\mathcal{C}^\mathsf{op}\times\mathcal{C}\to\mathcal{D}$ be pseudofunctors (we are going to use the invertibility of their lax functoriality constraints below). The following is [ [Corner 2017, Definition 2.2] ](#Corner), but laxified:

+-- {: .num_defn}
###### Definition

A **lax extranatural transformation** $P\overset{\bullet}{\Rightarrow}Q$ from $P$ to $Q$ consists of

1. _Components._ For each $A\in\mathrm{Obj}(\mathcal{B}$ and each $C\in\mathrm{Obj}(C)$, a lax transformation
   $$\alpha_{(-),B,C}\colon P^B_{(-),B}\Rightarrow Q^{C}_{(-),C},$$
   called the **component** of $\alpha$ at $(B,C)$.

2. _Lax Extranaturality Constraints I._ For each $1$-morphism $g\colon B\longrightarrow B'$ of $\mathcal{B}$, a $2$-morphism
   $$\alpha_{g}\colon\alpha_{A,B',C}\circ P^{\mathsf{id}_{B'}}_{\mathsf{id}_{A},g}\Rightarrow\alpha_{A,B,C}\circ P^{g}_{\mathsf{id}_{A},\mathsf{id}_{B}}$$
   of $\mathcal{D}$ as in the diagram

   \begin{imagefromfile}
       "file_name": "lax_extranaturality_1.svg",
       "width":     "250"
   \end{imagefromfile}

   called the **lax extranaturality constraint of $\alpha$ at $g$**.

3. _Lax Extranaturality Constraints II._ For each $1$-morphism $h\colon C\longrightarrow C'$ of $\mathcal{D}$, a $2$-morphism
   $$\alpha_{h}\colon Q^{\biid_{C}}_{\biid_{A},h}\circ\alpha_{A,B,C}\Rightarrow Q^{h}_{\biid_{A},\biid_{C'}}\circ\alpha_{A,B,C'}$$
   of $\mathcal{D}$ as in the diagram

   \begin{imagefromfile}
       "file_name": "lax_extranaturality_2.svg",
       "width":     "250"
   \end{imagefromfile}

   called the **lax extranaturality constraint of $\alpha$ at $h$**.

【...】

=--

## References

* {#Corner} Alexander Corner, _A Universal Characterisation of Codescent Objects_ ([arXiv](http://arxiv.org/abs/1709.01332)).


lax F-adjunction

# Lax $\mathcal{F}$-adjunctions

* table of contents
{: toc}

## Idea

The notion of [[adjunction]] or [[2-adjunction]] can be "laxified" in many ways; as discussed at [[lax 2-adjunction]], one can make the triangle identities hold only up to a noninvertible cell, make the unit and counit only lax natural, and even consider making the functors only lax.  Of these, lax naturality of the unit and counit is problematic because lax natural transformations do not satisfy a Yoneda lemma.  However, this becomes less of a problem if the lax transformations restrict to pseudo ones on certain well-behaved subcategories.  An abstract context to define this is that of [[F-categories]].

## Definition

Let $K,L$ be [[F-categories]] (strict for simplicity).  A **lax $\mathcal{F}$-adjunction** between them consists of:

* $\mathcal{F}$-functors $F:K\to L$ and $G:L\to K$ (strict for simplicity).
* [[pseudo/lax F-natural transformations]] $\eta : Id_K \to G F$ and $\epsilon : F G \to Id_L$.
* The triangle identities for $\eta,\epsilon$ hold up to isomorphism (which can be made coherent).

If $\epsilon$ is fully pseudo, we call it a **right semi-lax $\mathcal{F}$-adjunction**.  Dually, we have left semi-lax, right semi-oplax, left semi-oplax, and so on.  (We could also consider allowing the triangle identities or functors to be lax, but we will not.)

## Properties

The following theorem seems to fail for 2-adjunctions involving lax transformations; the fact that it holds for lax $\mathcal{F}$-adjunctions thus means that they are significantly better-behaved and more interesting.  It was first observed (without the terminology of $\mathcal{F}$-categories) by [Johnstone](#JohnstonePP).

+-- {: .un_theorem}
###### Theorem
If $F:K\to L$ is an $\mathcal{F}$-functor and $G,G'$ are two lax right $\mathcal{F}$-adjoints for it, then $G\simeq G'$.
=--
+-- {: .proof}
###### Proof
As usual, we consider the composites
$$ G \xrightarrow{\eta' G} G' F G \xrightarrow{G' \epsilon} G' \qquad G' \xrightarrow{\eta G'} G F G' \xrightarrow{G \epsilon'} G. $$
The usual proof of uniqueness of adjoints applies to show that these are pseudo-inverses, but we do need the fact that these unit and counit are pseudo/lax rather than merely lax.  For the proof requires using naturality squares to commute units and counits past each other, and since their components are tight and they are all pseudo-natural on tight morphisms these naturality squares commute up to isomorphism; if they only commuted laxly then the proof would fail.

Note that *a priori* these composites are themselves also only pseudo/lax $\mathcal{F}$-natural.  However, any lax natural transformation that is (pseudo) invertible is actually pseudo natural, so in fact these composites are a pseudo/pseudo $\mathcal{F}-$natural equivalence $G\simeq G'$.
=--

## Examples

### Fibrations in a 2-category

In [Johnstone](#JohnstonePP) it is shown that a [[fibration in a 2-category]] $K$ can equivalently be defined as a morphism $p:E\to B$ with the following property.  Let $K\swarrow B$ denote the [[oplax slice 2-category]], whose objects are morphisms $f:A\to B$ and whose morphisms are pairs $(g:A\to A', \alpha : f' g \Rightarrow f)$.  We make $K\swarrow B$ into an $\mathcal{F}$-category by declaring $(g,\alpha)$ to be tight if $\alpha$ is an isomorphism.

Now, composing with $p:E\to B$ induces an $\mathcal{F}$-functor $\Sigma_p : K\swarrow E \to K\swarrow B$.

+-- {: .un_theorem}
###### Theorem
$p$ is a [[fibration in a 2-category|fibration]] (in the [[Street fibration|pseudo sense]]) if and only if $\Sigma_p$ has a right semi-lax $\mathcal{F}$-adjoint.
=--

## References

* [[Peter Johnstone]], *Fibrations and partial products in a 2-category*
 {#JohnstonePP}

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lax F-natural transformation

# Lax $\mathcal{F}$-natural transformations

* table of contents
{: toc}

## Definition

Let $K$ and $L$ be [[F-categories]] (strict for simplicity), with 2-categories $K_\tau,L_\tau$ of tight morphisms and $K_\lambda,L_\lambda$ of loose morphisms.  Let $F,G:K\to L$ be $\mathcal{F}$-functors (also strict for simplicity).

A **pseudo/lax $\mathcal{F}$-natural transformation** $\alpha : F \to G$ consists of:

* A [[lax natural transformation]] $\alpha_\lambda : F_\lambda \to G_\lambda$,
* The 1-morphism components of $\alpha_\lambda$ are tight, and
* The 2-morphism naturality constraint of $\alpha_\lambda$ at any tight arrow of $K$ is an isomorphism.

In particular, $\alpha_\lambda$ restricts to a pseudo natural transformation $\alpha_\tau : F_\tau \to G_\tau$.

Similarly, an $\mathcal{F}$-natural transformation can be **strict/lax**, **pseudo/oplax**, and so on.

## Examples

* If $K$ and $L$ are chordate (all morphisms are tight), then a pseudo/lax transformation is simply a pseudo transformation.

* If $L$ is chordate but $K$ is inchordate (only identities are tight), then a pseudo/lax transformation is simply a lax transformation.

* If $L$ is inchordate, there are no nonidentity pseudo/lax transformations.

## Applications

* Lax $\mathcal{F}$-natural transformations appear in the notion of [[lax F-adjunction]].

[[!redirects lax F-natural transformation]]
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lax functor

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

# Contents
* automatic table of contents goes here
{:toc}

## Idea

A **lax functor** or **lax $n$-[[n-functor|functor]]** is a morphism of $n$-[[n-category|categories]] that is allowed to have structural cells -- compositors, associators, etc -- that need not be invertible (not even [[weak inverse|weakly]]).

This is to distinguish from [[pseudofunctor]] for which all these cells are required to be [[equivalences]].

This means that the definition of lax functor involves a choice of orientation of these structural cells which is not visible for pseudofunctors. The choice is such that cells map composites of images to images of composite. With the opposite choice one speaks of an **oplax** (or sometimes **colax**) functor**.

The term lax functor can be used for $n$-functors $F : C \to D$ whose domain $C$ is an ordinary [[category]] (regarded as an $n$-category with only trivial higher morphisms), while the codomain $D$ is often taken to be a [[2-category]].

A **normal lax functor** (sometimes called *strictly unitary*) preserves identities strictly.

There exist a similar concept for [[double category|double]] and [[multiple categories]].

## Definition

See the definition at [[pseudofunctor]], and let the [[natural isomorphisms]] in that definition be merely [[natural transformations]].

## Properties

One of the most important properties is that lax functors compose associatively and unitally: we have a category $\mathbf{BiCat}$ with [[bicategories]] as objects and lax functors as morphisms.
If we add [[icons]] as 2-cells, this becomes a [[2-category]].

## Examples

* Any [[lax monoidal functor]] gives an example. In fact [[monoidal categories]] can be presented as bicategories with one object (see [[delooping]]), and thus a lax functor $\mathbf{B}M \to \mathbf{B} N$ corresponds to a lax monoidal functor $M \to N$.

* For $D$ a [[bicategory]], lax functors $F : {*} \to D$ from the [[point]] category to $D$ are equivalent to [[monad]]s in $D$.

  The compositor of the lax functor is the monad product, the unitor is the monad unit.

  * So in particular for $V$ a [[monoidal category]] and $\mathbf{B}V$ its one-object [[delooping]] [[bicategory]], lax functors ${*} \to \mathbf{B}V$ are equivalent to [[monoid]]s in $V$.

* Similarly, oplax functors ${*} \to D$ are equivalent to [[comonad]]s in $D$.

* If $C$ is the [[codiscrete category]] on a set $S$, and $D$ is a [[bicategory]], lax functors $F : C \to D$ are the same as categories enriched in $D$ having $S$ as their set of objects.

   * In particular, if $C = {*}$, then this example reduces to the first one.

   * Another special case arises when $D = \mathbf{B}V$ for some monoidal category $V$.  Then lax functors $F : C \to D$ are the same as categories enriched in the monoidal category $V$.

* It makes sense to ask that a functor is lax _and_ oplax in a compatible way such that ${*} \to D$ yields [[Frobenius algebra|Frobenius]] monads.

  This is of relevance in [[conformal field theory]] where Frobenius algebra objects in [[modular tensor category|modular tensor categories]] and bimodules over them play a central role.

  Some old remarks on this case are in [Note on lax functors and bimodules](http://www.math.uni-hamburg.de/home/schreiber/LaxFunc.pdf).

  This relation between lax-oplax functors and [[conformal field theory]] was developed in detail in

  * Liang Kong, Ingo Runkel, _Cardy algebras and sewing constraints, I_ ([arXiv](http://arxiv.org/abs/0807.3356))

  A general discussion of lax-oplax functors is in section 2.1 there.

+--{.query}
Isn't it odd not to require any extra condition at all on the coherence morphisms? I would have expected a definition where they are required to be split epi, or require that the codomain be a subobject of the domain. Is there a name for something like that?

[[Mike Shulman]]: One could certainly add that as a condition, but I don't think I've ever heard of anyone having a use for it, or giving it a name.  The interesting examples listed above (and others) don't use any such condition.
=--

## See also
* [[lax double functor]]

## References

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Section 4.1 of: _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))

* R. 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 )

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