Yoneda extension

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#### Yoneda lemma
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[[!include Yoneda lemma - contents]]
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#### Limits and colimits
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[[!include infinity-limits - contents]]
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#Contents#
* table of contents
{:toc}

##Idea##

The _Yoneda extension_ of a [[functor]] $F : C \to D$ is a universal [[extension]] ([[Kan extension]]) along the [[Yoneda embedding]] $Y \colon C \to [C^{op},Set]$ of its [[domain]] category to a functor
$$
  \tilde F \colon [C^{op}, Set] \to D
  \,.
$$

The Yoneda extension exhibits the [[presheaf]] category $PSh(C)$ as the [[free cocompletion]] of $C$.

##Definition##

For $C$ a [[small category]] and $F \colon C \to D$ a [[functor]], its **Yoneda extension**

$$
  \tilde F \;\colon\; [C^{op},Set] \to D
$$

is the left [[Kan extension]]  $Lan_Y F \;\colon\; [C^{op}, Set] \to D$ of $F$ along the [[Yoneda embedding]] $Y$:

$$
  \tilde F \;\coloneqq\; Lan_Y F
  \,.
$$

### Remarks ###

Often it is of interest to Yoneda extend not $F \;\colon\; C \to D$
itself, but the composition $Y \circ F \;\colon\; C \to [D^{op}, Set]$ to get a functor entirely between presheaf categories

$$
  \hat F \;\coloneqq\; \tilde{Y \circ F}
  \;\colon\;
  [C^{op},Set] \to [D^{op}, Set]
  \,.
$$

This is in fact a [[left adjoint]] to the restriction functor $F^\ast \;\colon \; [D^{op}, Set] \to [C^{op}, Set]$ which maps $H \mapsto H \circ F$.  This is relevant, for instance, to [[restriction and extension of sheaves]].

### Formula ###

Recalling the general formula for the left [[Kan extension]] of a functor $F \;\colon\; C \to D$ through a functor $p : C  \to C'$

$$
  (Lan F)(c') \simeq \colim_{(p(c) \to c') \in (p,c')} F(c)
$$

one finds for the Yoneda extension the formula

$$
  \begin{aligned}
    \tilde F (A)
    & \;\coloneqq\; (Lan F)(A)
    \\
    & \simeq
    \colim_{(Y(U) \to A) \in (Y,A) } F(U)
  \end{aligned}
  \,.
$$

(Recall the notation for the [[comma category]] $(Y,A) \;\coloneqq\; (Y, const_A)$ whose objects are pairs $(U \in C, (Y(U) \to A) \in [C^{op}, Set] )$.

For the full extension $\hat F : [C^{op}, Set] \to [D^{op}. Set]$ this yields

$$
  \begin{aligned}
    \hat F(A)(V)
    &=
    (\colim_{(Y(U) \to A) \in (Y,A) } F(U))(V)
    \\
    &\simeq
    \colim_{(Y(U) \to A) \in (Y,A) } F(U)(V)
    \\
    &\simeq
    \colim_{(Y(U) \to A) \in (Y,A) } Hom_{D}(V,F(U))
  \end{aligned}
  \,.
$$

Here the first step is from above, the second uses that colimits in presheaf categories are computed objectwise and the last one is again using the [[Yoneda lemma]].

## Properties ##

* The restriction of the Yoneda extension to $C$ coincides with the original functor:
$
  \tilde F \circ Y \simeq F
$.

* The Yoneda extension commutes with small colimits in $C$, i.e. for $\alpha : A \to C$ a [[diagram]], we have $\tilde F (colim (Y \circ \alpha)) \simeq colim F \circ \alpha$ .

* Moreover, $\tilde F$ is defined up to [[isomorphism]] by these two properties.

## Generalizations ##

* [[(infinity,1)-Yoneda extension]]


Yoneda lemma

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###Context###
#### Category theory
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[[!include category theory - contents]]
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#### Yoneda lemma
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[[!include Yoneda lemma - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The **Yoneda lemma** says that the [[set]] of [[morphisms]] from a [[representable presheaf]] $y(c)$ into an arbitrary [[presheaf]] $X$ is in [[natural bijection]] with the set $X(c)$ assigned by $X$ to the representing [[object]] $c$.

The Yoneda lemma is an elementary but deep and central result in [[category theory]] and in particular in [[sheaf and topos theory]]. It is essential background behind the central concepts of _[[representable functors]]_, _[[universal constructions]]_, and _[[universal elements]]_.

## Statement and proof
  {#StatementOfYonedaLemma}

### Classical

+-- {: .num_defn #FunctorUnderlyingTheYonedaEmbedding}
###### Definition
**([[functor]] underlying the [[Yoneda embedding]])

For $\mathcal{C}$ a [[locally small category]] we write

$$
  [C^{op}, Set] \coloneqq Func(C^{op}, Set)
$$

for the [[functor category]] out of the [[opposite category]] of $\mathcal{C}$ into [[Set]].

This is also called the _[[category of presheaves]]_ on $\mathcal{C}$.
Other notation used for it includes $Set^{C^{op}}$ or $Hom(C^{op},Set))$.

There is a [[functor]]

$$
  \array{
    C &\overset{y}{\longrightarrow}& [C^op,Set]
    \\
    c &\mapsto& Hom_{\mathcal{C}}(-,c)
  }
$$

(called the _[[Yoneda embedding]]_ for reasons explained below) from $\mathcal{C}$ to its [[category of presheaves]], which sends each [[object]] to the [[hom-functor]] into that object, also called the [[representable presheaf|presheaf represented]] by $c$.

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+-- {: .num_remark}
###### Remark
**([[Yoneda embedding]] is [[adjunct]] of [[hom-functor]])**

The Yoneda embedding functor $y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set]$ from Def. \ref{FunctorUnderlyingTheYonedaEmbedding} is equivalently the [[adjunct]] of the [[hom-functor]]

$$
  Hom_{\mathcal{C}}
  \;\colon\;
  \mathcal{C}^{op} \times \mathcal{C}
  \longrightarrow
  Set
$$

under the [[product category]]/[[functor category]] [[adjoint functor|adjunction]]

$$
  Hom(C^{op} \times C, Set)
    \stackrel{\simeq}{\to}
  Hom(C, [C^{op}, Set])
$$

in the [[closed monoidal category|closed]] [[monoidal category|symmetric monoidal category]] of categories.

=--

+-- {: .num_prop #YonedaLemma}
###### Proposition
**([[Yoneda lemma]])**

Let $\mathcal{C}$ be a [[locally small category]], with [[category of presheaves]] denoted $[\mathcal{C}^{op},Set]$, according to Def. \ref{FunctorUnderlyingTheYonedaEmbedding}.

For $X \in [\mathcal{C}^{op}, Set]$ any [[presheaf]], there is a canonical [[isomorphism]]

$$
  Hom_{[C^op,Set]}(y(c),X) \;\simeq\; X(c)
$$

between the [[hom-set]] of [[presheaf]] [[homomorphisms]] from the [[representable presheaf]] $y(c)$ to $X$, and the value of $X$ at $c$.

=--

This is the standard notation used mostly in pure [[category theory]] and [[enriched category theory]]. In other parts of the literature it is customary to denote the presheaf represented by $c$ as $h_c$. In that case the above is often written

$$
  Hom(h_c, X) \simeq X(c)
$$

or

$$
  Nat(h_c, X) \simeq X(c)
$$

to emphasize that the morphisms of presheaves are [[natural transformation|natural transformations]] of the corresponding functors.

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

The proof is by chasing the element $Id_c \in C(c, c)$ around both legs of a [[naturality square]] for a [[natural transformation]] $\eta: C(-, c) \to X$ (hence a homomorphism of presheaves):

$$
  \array{
     C(c, c)
       & \stackrel{\eta_c}{\to} &
     X(c) & & & & Id_c & \mapsto & \eta_c(Id_c) & \stackrel{def}{=} & \xi \\
 _\mathllap{C(f, c)} \downarrow & & \downarrow _\mathrlap{X(f)} & & & & \downarrow & & \downarrow _\mathrlap{X(f)} & & \\
C(b, c) & \underset{\eta_b}{\to} & X(b) & & & & f & \mapsto & \eta_b(f) & &
}
$$

What this diagram shows is that the entire transformation $\eta: C(-, c) \to X$ is completely determined from the single value $\xi \coloneqq \eta_c(Id_c) \in X(c)$, because for each object $b$ of $C$, the component $\eta_b: C(b, c) \to X(b)$ must take an element $f \in C(b, c)$ (i.e., a morphism $f: b \to c$) to $X(f)(\xi)$, according to the commutativity of this diagram.

The crucial point is that the naturality condition on any [[natural transformation]] $\eta : C(-,c) \Rightarrow X$ is sufficient to ensure that $\eta$ is already entirely fixed by the value $\eta_c(Id_c) \in X(c)$ of its component $\eta_c : C(c,c) \to X(c)$ on the [[identity morphism]] $Id_c$.
And every such value extends to a natural transformation $\eta$.

More in detail, the bijection is established by the map

$$
  [C^{op}, Set](C(-,c),X)
  \stackrel{|_{c}}{\to}
  Set(C(c,c), X(c))
  \stackrel{ev_{Id_c}}{\to}
  X(c)
$$

where the first step is taking the component of a [[natural transformation]] at $c \in C$ and the second step is [[evaluation]] at $Id_c \in C(c,c)$.

The inverse of this map takes $f \in X(c)$ to the natural transformation $\eta^f$ with components
$$
  \eta^f_d := X(-)(f) : C(d,c) \to X(d)
  \,.
$$

=--

### In homotopy type theory

Discussion in [[homotopy type theory]].

Note: the [[HoTT book]] calls a [[internal category in HoTT]] a "precategory" and a [[univalent category]] a "category", but here we shall refer to the standard terminology of "category" and "univalent category" respectively.

By Lemma 9.5.3 in the HoTT book (see [[product category]]), we have an induced functor $\mathbf{y} : A \to \mathit{Set}^{A^{op}}$ which we call the **yoneda embedding**.

**Theorem 9.5.4 (The Yoneda Lemma)**
For any [[category]] $A$, any $a:A$, and any functor $F: \mathit{Set}^{A^{op}}$, we have an [[isomorphism]]

$$hom_{\mathit{Set}^{A^{op}}}(\mathbf{y}a,F) \cong F a \qquad \qquad(9.5.5)$$

Moreover this is natural in both $a$ and $F$.

**Proof.** Given a [[natural transformation]] $\alpha : \mathbf{y}a \to F$, we can consider the component $\alpha_a : \mathbf{y}a(a) \to F a$. Since $\mathbf{y} a(a)\equiv hom_A(a,a)$, we have $1_a:\mathbf{y}a(a)$, so that $\alpha_a(1_a):F a$. This gives a function $\alpha \mapsto \alpha_a(1_a)$ from left to right in (9.5.5).

In the other direction, given $x: F a$, we define $\alpha : \mathbf{y} a \to F$ by

$$\alpha_{a'}(f) \equiv F_{a,a'}(f)(x)$$

Naturality is easy to check, so this gives a function from right to left in (9.5.5).

To show that these are inverses, first suppose given $x: F a$. Then with $\alpha$ defined as above, we have $\alpha : \mathbf{y}a \to F$ and define $x$ as above, then for any $f:hom_A(a',a)$ we have

$$
\begin{aligned}
  \alpha_{a'}(f) &= \alpha_{a'}(\mathbf{y} a_{a,a'}(f)(1_a))\\
  &= (\alpha_{a'} \circ \mathbf{y}a_{a,a'}(f))(1_a)\\
  &= (F_{a,a'}(f) \circ \alpha_a)(1_a)\\
  &= F_{a,a'}(f_(\alpha_a(1_a))\\
  &= F_{a,a'}(f)(x).
\end{aligned}
$$

Thus, both composites are equal to identities. The proof of naturality follows from this. $\square$

**Corollary 9.5.6**
The Yoneda embedding $\mathbf{y} : A \to \mathit{Set}^{A^{op}}$ is [[fully faithful]].

**Proof.** By the Yoneda lemma, we have

$$hom_{\mathit{Set}^{A^{op}}}(\mathbf{y}a,\mathbf{y}b) \cong \mathbf{y} b(a) \equiv hom_A(a,b)$$

It is easy to check that this isomorphism is in fact the action of $\mathbf{y}$ on hom-sets. $\square$

## Corollaries {#YonedaCorollaries}

The Yoneda lemma has the following direct consequences. As the Yoneda lemma itself, these are as easily established as they are useful and important.

### corollary I: Yoneda embedding

The Yoneda lemma implies that the [[Yoneda embedding]] functor $y \colon C \to [C^op,Set]$ really is an _embedding_ in that it is a [[full and faithful functor]], because for $c,d \in C$ it naturally induces the isomorphism of Hom-sets.

$$
  [C^{op},Set](C(-,c),C(-,d)) \simeq (C(-,d))(c) = C(c,d)
$$

### corollary II: uniqueness of representing objects

Since the [[Yoneda embedding]] is a [[full and faithful functor]], an [[isomorphism]] of [[representable functor|representable presheaves]] $y(c) \simeq y(d)$ must come from an [[isomorphism]] of the representing objects $c \simeq d$:

$$
  y(c) \simeq y(d) \;\; \Leftrightarrow \;\;
  c \simeq d
$$

### corollary III: universality of representing objects

A [[presheaf|presheaf]] $X \colon C^{op} \to Set$ is [[representable functor|representable]] precisely if the [[comma category|comma category]] $(y,const_X)$ has a [[terminal object]]. If a [[terminal object]] is $(d, g : y(d) \to X) \simeq (d, g \in X(d))$ then $X \simeq y(d)$.

This follows from unwrapping the definition of [[morphisms]] in the [[comma category]] $(y,const_X)$ and applying the Yoneda lemma to find

$$
  (y,const_X)((c,f \in X(c)), (d, g \in X(d)))
  \simeq
  \{
    u \in C(c,d) : X(u)(g) = f
  \}
  \,.
$$

Hence $(y,const_X)((c,f \in X(c)), (d, g \in X(d))) \simeq pt$ says precisely that $X(-)(f) \colon C(c,d) \to X(c)$ is a bijection.

### Interpretation

For emphasis, here is the interpretation of these three corollaries in words:

* **corollary I** says that the interpretation of presheaves on $C$ as generalized objects probeable by objects $c$ of $C$ is consistent: the probes of $X$ by $c$ are indeed the maps of generalized objects from $c$ into $X$;

* **corollary II** says that probes by objects of $C$ are sufficient to distinguish objects of $C$: two objects of $C$ are the same if they have the same probes by other objects of $C$.

* **corollary III** characterizes [[representable functor|representable functors]] by a [[universal property]] and is hence the bridge between the notion of [[representable functor]] and [[universal construction|universal constructions]].

## Generalizations

The Yoneda lemma tends to carry over to all important generalizations of the context of [[locally small category|categories]]:

* There is an analog of the Yoneda lemma in [[enriched category theory]]. See [[enriched Yoneda lemma]].

* In the context of [[module|modules]] (see also [[Day convolution]]) the Yoneda lemma becomes the important statement of [[Yoneda reduction]], which identifies the bimodule $\hom_C(-, -)$ as a unit bimodule.

* There is a [[Yoneda lemma for bicategories]].

* There is a [[Yoneda lemma for tricategories]].

* There is a [[Yoneda lemma for (∞,1)-categories]].

* In [[functional programming]], the Yoneda embedding is related to the [[continuation passing style]] transform.

* Formulation of the lemma in [[dependent type theory]]: [A type theoretical Yoneda lemma](http://homotopytypetheory.org/2012/05/02/a-type-theoretical-yoneda-lemma/) at [homotopytypetheory.org](http://homotopytypetheory.org)

## Necessity of naturality

The assumption of naturality is necessary for the Yoneda lemma to hold. A simple counter-example is given by a category with two objects $A$ and $B$, in which $Hom(A,A) = Hom(A,B) = Hom(B,B) = \mathbb{Z}_{\geq 0}$, the set of integers greater than or equal to $0$, in which $Hom(B,A) = \mathbb{Z}_{\geq 1}$, the set of integers greater than or equal to $1$, and in which composition is addition. Here it is certainly the case that $Hom(A,-)$ is isomorphic to $Hom(B,-)$ for any choice of $-$, but $A$ and $B$ are not isomorphic (composition with any arrow $B \rightarrow A$ is greater than or equal to $1$, so cannot have an inverse, since $0$ is the identity on $A$ and $B$).

A finite counter-example is given by the category with two objects $A$ and $B$, in which $Hom(A,A) = Hom(A,B) = Hom(B,B) = \{0, 1\}$, in which $Hom(B,A) = \{0, 2\}$, and composition is multiplication modulo 2. Here, again, it is certainly the case that $Hom(A,-)$ is isomorphic to $Hom(B,-)$ for any choice of $-$, but $A$ and $B$ are not isomorphic (composition with any arrow $B \rightarrow A$ is $0$, so cannot have an inverse, since $1$ is the identity on $A$ and $B$).

On the other hand, there have been examples of locally finite categories where naturality is not necessary. For example, ([Lovász, Theorem 3.6 (iv)](#Lovasz)) states precisely that finite relational structures $A$ and $B$ are isomorphic if, and only if, $Hom(C,A) \cong Hom(C,B)$ for every finite relational structure $C$. Later ([Pultr, Theorem 2.2](#Pultr)) generalised the result to finitely well-powered, locally finite categories with (extremal epi, mono) [[factorization system]].

## The Yoneda lemma in semicategories

An interesting phenomenon arises in the case of [[semicategory|semicategories]] i.e. "categories" (possibly) lacking [[identity morphisms]]:

the Yoneda lemma fails in general, since its validity in a semicategory $\mathcal{G}$ implies that $\mathcal{G}$ is in fact already a category because the Yoneda lemma permits to embed $\mathcal{G}$ into $PrSh(\mathcal{G})$ and the latter is always a category, the embedding then implying that $\mathcal{G}$ is itself a category!

But for [[regular semicategories]] $\mathcal{R}$ there is a [[unity of opposites]] in the category of all [[semipresheaves]] on $\mathcal{R}$ between the so called regular presheaves that are [[colimits]] of [[representable presheaf|representables]] and presheaves satisfying the Yoneda lemma, whence _the Yoneda lemma holds dialectically for regular presheaves!_

For some of the details see at _[[regular semicategory]]_ and the references therein.

## Applications

* The Yoneda lemma is the or a central ingredient in various [[reconstruction theorem|reconstruction theorems]], such as those of [[Tannaka duality]]. See there for a detailed account.

* In its incarnations as [[Yoneda reduction]] the Yoneda lemma governs the algebra of [[end|ends]] and [[coend|coends]] and hence that of [[bimodule|bimodules]] and [[profunctor|profunctors]].

* The Yoneda lemma is effectively the reason that [[Isbell conjugation]] exists. This is a fundamental duality that relates [[geometry]] and [[algebra]] in large part of mathematics.

## Related entries

* [[Yoneda reduction]]

* [[co-Yoneda lemma]]

* [[Yoneda structure]]

* [[Brown representability theorem]]

* [[continuation-passing style]]

* [[regular semicategory]]

* [[Yoneda lemma for (∞,1)-categories]]

\linebreak

## References
 {#References}

For general references see any text on [[category theory]], as listed in the references [there](category+theory#References).

The term _Yoneda lemma_ originated in an interview of [[Nobuo Yoneda]] by [[Saunders Mac Lane]] at Paris Gare du Nord:

* Yoshiki Kinoshita, *Nobuo Yoneda (1930-1996)*

  &

  [[Saunders MacLane]], *The Yoneda Lemma*,

  Math. Japonica, **47**, No. 1 (1998) 155-156

  ([pdf](https://dmitripavlov.org/scans/yoneda.pdf) [[MathJaponica_YonedaObituary.pdf:file]], also: [posting to catlist in 1996](http://www.mta.ca/~cat-dist/catlist/1999/yoneda))

In _[[Categories for the Working Mathematician]]_ MacLane writes that this happened in 1954.

<img src="http://ncatlab.org/nlab/files/YonedaObituary.jpg">

Review and exposition:

* [Using Yoneda rather than J to present the identity type](https://www.cs.bham.ac.uk/~mhe/yoneda/yoneda.html)

* [[Alexander Grothendieck]], Section A.1 of: *Technique de descente et théorèmes d'existence en géométrie algébriques. II. Le théorème d'existence en théorie formelle des modules*, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195, 22 p.  ([numdam:SB_1958-1960__5__369_0](http://www.numdam.org/item/?id=SB_1958-1960__5__369_0))

* [[Saunders MacLane]], §III.2 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Tom Leinster]], _[[LeinsterYoneda.ps:file]]_

* [[Emily Riehl]], _Category Theory in Context. Chapter 2. Universal Properties, Representability, and the Yoneda Lemma_ [pdf](http://www.math.jhu.edu/~eriehl/context.pdf)

* {#RRZ2004} Marie La Palme Reyes, Gonzalo E. Reyes, and Houman Zolfaghari, _Generic figures and their glueings: A constructive approach to functor categories_, Polimetrica sas, 2004 ([author page](https://reyes-reyes.com/2004/06/01/generic-figures-and-their-glueings-a-constructive-approach-to-functor-categories/),[pdf](https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf)).

* [[Paolo Perrone]], _Notes on Category Theory with examples from basic mathematics_, Chapter 2. ([arXiv](http://arxiv.org/abs/1912.10642))

A discussion of the Yoneda lemma from the point of view of [[universal algebra]] is in

* [[Vaughan Pratt]], _The Yoneda lemma without category theory: algebra and applications_ ([pdf](http://boole.stanford.edu/pub/yon.pdf)).

A treatment of the Yoneda lemma for [[internal category in an (infinity,1)-category|categories internal to an (∞,1)-topos]] is in

* Louis Martini, _Yoneda's lemma for internal higher categories_, ([arXiv:2103.17141](https://arxiv.org/abs/2103.17141))

Early Lovász-Type results include

* {#Lovasz} László Lovász, _Operations with structures_, Acta Mathematica Academiae Scientiarum Hungarica 18.3-4 (1967): 321-328.
* {#Pultr} Aleš Pultr. _Isomorphism types of objects in categories determined by numbers of morphisms_, Acta Scientiarum Mathematicarum, 35:155–160, 1973.

[[!redirects yoneda lemma]]
[[!redirects Yoneda Lemma]]
[[!redirects Yoneda embedding lemma]]
[[!redirects Yoneda imbedding lemma]]


Yoneda lemma - contents

**[[Yoneda lemma]]**

## Ingredients

* [[category]]

* [[functor]]

* [[natural transformation]]

* [[presheaf]]

* [[category of presheaves]]

* [[representable presheaf]]

* [[Yoneda embedding]]

## Incarnations

* [[Yoneda lemma]]

* [[enriched Yoneda lemma]]

* [[co-Yoneda lemma]]

* [[Yoneda reduction]]

## Properties

* [[free cocompletion]]

* [[Yoneda extension]]

## Universal aspects

* [[representable functor]]

* [[universal construction]]

* [[universal element]]

## Classification

* [[classifying space]], [[classifying stack]]

* [[moduli space]], [[moduli stack]], [[derived moduli space]]

* [[classifying topos]]

* [[subobject classifier]]

* [[universal principal bundle]], [[universal principal ∞-bundle]]

* [[classifying morphism]]

## Induced theorems

* [[Tannaka duality]]

...

## In higher category theory

[[Yoneda lemma for higher categories]]:

* [[Yoneda lemma for (∞,1)-categories]]

* [[Yoneda lemma for bicategories]]

* [[Yoneda lemma for tricategories]]

[Edit this sidebar](/nlab/edit/Yoneda+lemma+-+contents)


Yoneda lemma for (infinity,1)-categories

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

# Contents#
* table of contents
{: toc}

## Idea

The statement of the [[Yoneda lemma]] generalizes from [[categories]] to [[(∞,1)-categories]].

## Yoneda embedding
 {#YonedaEmbedding}

\begin{definition}
For $C$ an [[(∞,1)-category]] and $PSh(C)$ its [[(∞,1)-category of (∞,1)-presheaves]], the **$(\infty,1)$-Yoneda embedding** is the [[(∞,1)-functor]]

$$
  y \colon C \to PSh(C)
$$

given by $y(X) \colon U \mapsto C(U,X)$.
\end{definition}

Seen under the [[(infinity,1)-Grothendieck construction|$\infty$-Grothendieck construction]] this is [Riehl & Verity 2018, Def. 6.2.3](#RiehlVerity18).

## Properties

### Yoneda lemma
 {#YonedaLemma}

\begin{prop}
**$(\infty,1)$-Yoneda embedding**

Let $C$ be an [[(∞,1)-category]] and $PSh(C) \coloneqq Func(C^\op, \infty Grpd)$ be the corresponding [[(∞,1)-category of (∞,1)-presheaves]]. Then the canonical [[(∞,1)-functor]]

$$
  Y \colon C \to PSh(C)
$$

is a [[full and faithful (∞,1)-functor]].
\end{prop}

For small $\infty$-categories this is [[Higher Topos Theory|HTT, prop. 5.1.3.1]]. For possibly large $\infty$-categories see [Riehl & Verity 2018, Thm. 7.2.22](#RiehlVerity18) (which considers $\infty$-presheaves regarded under the [[(infinity,1)-Grothendieck construction|$\infty$-Grothendieck construction]]) and [[Kerodon]], [Thm. 8.2.5.4](https://kerodon.net/tag/03NJ).

+-- {: .num_prop }
###### Proposition
**$(\infty,1)$-Yoneda theorem**

For $C$ a small $(\infty,1)$-category and $F \colon C^{op} \to \infty Grpd$ an $(\infty,1)$-functor, the composite

$$
  C^{op}
   \to PSh_{(\infty,1)}(C)^{op} \stackrel{Hom(-,F)}{\to}
  \infty Grpd
$$

is equivalent to $F$.

=--

For small $\infty$-sites this is [[Higher Topos Theory|HTT, Lemma 5.5.2.1]]. For possibly large $\infty$-sites see [[Kerodon]], [Prop. 8.2.1.3](https://kerodon.net/tag/03M5).

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

For small $\infty$-sites, the statement may be obtained as a consequence of the [[sSet]]-[[enriched category theory|enriched]] [[Yoneda lemma]] by using the fact that the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C)$ is modeled by the [[enriched functor category]] $[C^{op}, sSet]_{proj}$ with $C$ regarded as a [[simplicially enriched category]] and using the global [[model structure on simplicial presheaves]].

=--

### Naturality
 {#Naturality}

+-- {: .num_prop }

###### Proposition

$PSh$ can be extended to a functor $PSh \colon (\infty,1)Cat \to (\infty,1)\widehat{Cat}$ so that the yoneda embedding $C \to PSh(C)$ is a natural transformation.

Here, $(\infty,1)\widehat{Cat}$ is the (∞,1)-category of large (∞,1)-categories.

=--

This follows from ([[Higher Topos Theory|HTT, prop. 5.3.6.10]]), together with the identification of $PSh(C)$ with the category obtained by freely adjoining small colimits to $C$. This functor is locally left adjoint to the contravariant functor $C \mapsto Func(C^\op, \infty Grpd)$.

### Preservation of limits

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

The $(\infty,1)$-Yoneda embedding $y : C \to PSh(C)$ preserves all [[(∞,1)-limit]]s that exist in $C$.

=--

([[Higher Topos Theory|HTT, prop. 5.1.3.2]])

### Local Yoneda embedding
  {#LocalYonedaEmbedding}

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

For $C$ an [[(∞,1)-site]] and $\mathcal{X}$ an [[(∞,1)-topos]], [[(∞,1)-geometric morphism]]s $(f^* \dashv f_*) Sh(C) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{X}$ from the [[(∞,1)-sheaf (∞,1)-topos]] $Sh(C)$ to $\mathcal{X}$ correspond to the **local** [[(∞,1)-functor]]s $f^* : C \to \mathcal{X}$, those that

* are left [[exact (∞,1)-functor]]s;

* send [[covering]] families $\{U_i \to X\}$ in $\mathcal{G}$ to [[effective epimorphism]]

  $$
    \coprod_i f^*(U_i) \to f^*(X)
    \,.
  $$

More preseicely, the [[(∞,1)-functor]]

$$
  Topos(\mathcal{X}, Sh_{(\infty,1)}(\mathcal{G}))
  \stackrel{L}{\to}
  Topos(\mathcal{X}, PSh_{(\infty,1)}(\mathcal{G}))
  \stackrel{y}{\to}
  Func(\mathcal{X}, \mathcal{G})
$$

given by precomposition of [[inverse image]] functors by [[∞-stackification]] and by the [(∞,1)-Yoneda embedding](#YonedaEmbedding)
is a [[full and faithful (∞,1)-functor]] and its essential image is spanned by these local morphisms.

=--

([[Higher Topos Theory|HTT, prop. 6.2.3.20]])

## Related concepts

* [[Brown representability theorem]]

* [[Yoneda Lemma for higher categories]]

## References

* {#LurieHTT} [[Jacob Lurie]], Prop. 5.1.3.1 and Lemma 5.5.2.1 in: _[[Higher Topos Theory]]_ (2009)

* {#RiehlVerity18} [[Emily Riehl]], [[Dominic Verity]], Def. 6.2.3 and Thm. 7.2.22 in: *The comprehension construction*, Higher Structures **2** 1 (2018)  ([arXiv:1706.10023](https://arxiv.org/abs/1706.10023), [hs:39](http://137.111.162.45/index.php/higher_structures/article/view/39))

* MathOverflow, _The Yoneda Lemma for $(\infty,1)$-categories?_ ([MO:9737/381](https://mathoverflow.net/q/9737/381))

* {#Kerodon} [[Kerodon]], Part 2, Chapter 8: *The Yoneda Embedding* $[$[kerodon:03JA](https://kerodon.net/tag/03JA), esp. [Prop. 8.2.1.3](https://kerodon.net/tag/03M5) and [Thm. 8.2.5.4](https://kerodon.net/tag/03NJ)$]$

Discussion in the context of an [[∞-cosmos]]:

* {#RiehlVerity17} [[Emily Riehl]], [[Dominic Verity]], Section 6 of: _Fibrations and Yoneda's lemma in an $\infty$-cosmos_, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 ([arXiv:1506.05500](https://arxiv.org/abs/1506.05500), [doi:10.1016/j.jpaa.2016.07.003](https://doi.org/10.1016/j.jpaa.2016.07.003))

Discussion [[category internal to an (infinity,1)-topos|internal to]] any [[(∞,1)-topos]]:

* [[Louis Martini]], *Yoneda's lemma for internal higher categories*, [[arXiv:2103.17141](https://arxiv.org/abs/2103.17141)]

Formalization of the $(\infty,1)$-Yoneda lemma via [[simplicial homotopy type theory]] (in [[Rzk]]):

* [[Nikolai Kudasov]], [[Emily Riehl]], [[Jonathan Weinberger]]. *Formalizing the $\infty$-categorical Yoneda lemma* (2023) [[arXiv:2309.08340](https://arxiv.org/abs/2309.08340)]

[[!redirects Yoneda lemma for (∞,1)-categories]]
[[!redirects (infinity,1)-Yoneda lemma]]
[[!redirects (∞,1)-Yoneda lemma]]

[[!redirects Yoneda embedding for (∞,1)-categories]]
[[!redirects (∞,1)-Yoneda embedding]]
[[!redirects (infinity,1)-Yoneda embedding]]

[[!redirects Yoneda lemma for infinity-categories]]

[[!redirects Yoneda lemma for infinity1-categories]]


Yoneda lemma for bicategories

+--{: .rightHandSide}
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### Context
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

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

## Statement

The **Yoneda lemma for bicategories** is a version of the [[Yoneda lemma]] that applies to [[bicategories]], the most common algebraic sort of weak [[2-category]].  It says that for any bicategory $C$, any object $x\in C$, and any [[pseudofunctor]] $F\colon C^{op}\to Cat$, there is an equivalence of categories

$$ [C^{op},Cat](C(-,x), F) \simeq F(x) $$

which is [[pseudonatural transformation|pseudonatural]] in $x$ and $F$, and which is given by evaluation at $1_x$, i.e. $\alpha\colon C(-,x)\to F$ maps to $\alpha_x(1_x)$.

For bicategories $A$ and $B$, $[A,B]$ denotes the bicategory of [[pseudofunctors]], [[pseudonatural transformations]], and [[modifications]] from $A$ to $B$.  Note that it is a strict 2-category as soon as $B$ is.

## Implications

In particular, the Yoneda lemma for bicategories implies that there is a [[Yoneda embedding]] for bicategories $C\to [C^{op},Cat]$ which is [[2-fully-faithful functor|2-fully-faithful]], i.e. an equivalence on hom-categories.  Therefore, $C$ is [[equivalence|equivalent]] to a sub-bicategory of $[C^{op},Cat]$.  Since [[Cat]] is a [[strict 2-category]], it follows that $C$ is equivalent to a strict 2-category, which is one form of the [[coherence theorem for bicategories]].  (Conversely, another form of the coherence theorem can be used to prove the Yoneda lemma; see below.)

## Proof

A detailed proof of the bicategorical Yoneda lemma is given in ([Johnson & Yau 20, Chap. 8](#JohnsonYau)).

### Explicit proof

An explicit proof, involving many diagrams, has been written up by [[Igor Baković]] and can be found [here](http://www.irb.hr/korisnici/ibakovic/yoneda.pdf).

### High-technology proof

We will take it for granted that $[C^{op},Cat]$ is a well-defined bicategory; this is a basic fact having nothing to do with the Yoneda lemma.  We also take it as given that "evaluation at $1_x$" functor
$$ [C^{op},Cat](C(-,x), F) \to F(x) $$
is well-defined and pseudonatural in $F$ and $x$; our goal is to prove that it is an equivalence.  (Granted, these basic facts require a fair amount of verification as well.)

We will use part of the [[coherence theorem for pseudoalgebras]], which says that for a suitably well-behaved strict [[2-monad]] $T$, the inclusion $T$-$Alg_{strict} \hookrightarrow Ps$-$T$-$Alg$ of the 2-category of strict $T$-algebras and strict $T$-morphisms into the 2-category of pseudo $T$-algebras and pseudo $T$-morphisms has a left adjoint, usually written as $(-)'$.  Moreover, for any pseudo $T$-algebra $A$, the unit $A\to A'$ is an equivalence in $Ps$-$T$-$Alg$.

First, there is a 2-monad $T$ such that strict $T$-algebras are strict 2-categories, strict $T$-morphisms are strict 2-functors, pseudo $T$-algebras are [[biased|unbiased bicategories]], and pseudo $T$-morphisms are [[pseudofunctors]].  By Mac Lane's coherence theorem for bicategories, any ordinary bicategory can equally well be considered as an unbiased one.  Thus, since $Cat$ is a strict 2-category, for any bicategory $C$ there is a strict 2-category $C'$ such that pseudofunctors $C\to Cat$ are in bijection with strict 2-functors $C'\to Cat$.

Now note that a pseudonatural transformation between two pseudofunctors (resp. strict 2-functors) $C\to D$ is the same as a single pseudofunctor (resp. strict 2-functor) $C\to Cyl(D)$, where $Cyl(D)$ is the bicategory whose objects are the 1-cells of $D$, whose 1-cells are squares in $D$ commuting up to isomorphism, and whose 2-cells are "cylinders" in $D$.  Likewise, a modification between two such transformations is the same as a single functor (of whichever sort) $C\to 2Cyl(D)$, where the objects of $2Cyl(D)$ are the 2-cells of $D$, and so on.  Therefore, $C'$ classifies not only pseudofunctors out of $C$, but transformations and modifications between them; thus we have an isomorphism
$$[C^{op},Cat] \cong [(C')^{op},Cat]_{strict,pseudo}$$
where $[A,B]_{strict,pseudo}$ denotes the 2-category of strict 2-functors, pseudonatural transformations, and modifications between two strict 2-categories.  Thus we can equally well analyze the functor
$$ [(C')^{op},Cat]_{strict,pseudo}(\overline{C(-,x)}, \overline{F}) \to \overline{F}(x) = F(x) $$
given by evaluation at $1_x$.  Here $\overline{C(-,x)}$ and $\overline{F}$ denote the strict 2-functors $(C')^{op}\to Cat$ corresponding to the pseudofunctors $C(-,x)$ and $F$ under the $(-)'$ adjunction.  However, we also have a strict 2-functor $C'(-,x)$, and the equivalence $C\simeq C'$ induces an equivalence $C'(-,x)\simeq \overline{C(-,x)}$.  Therefore, it suffices to analyze the functor
$$ [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F}) \to \overline{F}(x). $$
Now for any $A$ and $B$, we have an inclusion functor $[A,B]_{strict,strict} \to [A,B]_{strict,pseudo}$ where $[A,B]_{strict,strict}$ denotes the 2-category of strict 2-functors, strict 2-natural transformations, and modifications.  This functor is [[bijective on objects functor|bijective on objects]] and [[locally fully faithful 2-functor|locally fully faithful]].  Moreover, the composite
$$ [(C')^{op},Cat]_{strict,strict}(C'(-,x), \overline{F}) \to [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F}) \to \overline{F}(x). $$
is an *isomorphism*, by the [[enriched Yoneda lemma]], in the special case of $Cat$-enrichment.  Since
$$[(C')^{op},Cat]_{strict,strict}(C'(-,x), \overline{F}) \to [(C')^{op},Cat]_{strict,pseudo}(C'(-,x), \overline{F})$$
is fully faithful, if we can show that it is essentially surjective, then the [[2-out-of-3 property]] for equivalences of categories will imply that the desired functor is an equivalence.

Here we at last descend to something concrete.  Given $\alpha\colon C'(-,x)\to \overline{F}$, we have an obvious choice for a strict transformation for it to be equivalent to, namely $\beta$ whose components $\beta_y\colon C'(y,x)\to \overline{F}(y)$ is given by $f \mapsto \overline{F}(f)(a)$ where $a = \alpha_x(1_x)\in \overline{F}(x)$.  Since $\alpha$ is pseudonatural, for any $f\colon y\to x$ in $C'$ we have an isomorphism
$$\alpha_y(f) = \alpha_y(f\circ 1_x) \cong \overline{F}(f)(\alpha_x(1_x)) = \overline{F}(f)(a) = \beta_y(f).$$
We then simply verify that these isomorphisms are the components of an (invertible) modification $\alpha\cong \beta$.  This completes the proof.

##Related entries

* [[Yoneda lemma for tricategories]]

## References

Review:

* [[Angelo Vistoli]], §3.6.2 in: *Grothendieck topologies, fibered categories and descent theory*, in: *[[Fundamental algebraic geometry -- Grothendieck's FGA explained]]*, Mathematical Surveys and Monographs **123**, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [math.AG/0412512](http://arxiv.org/abs/math/0412512)]

See also:

* {#JohnsonYau} [[Niles Johnson]], [[Donald Yau]], Chap. 8: _2-Dimensional Categories_, ([arXiv:2002.06055](https://arxiv.org/abs/2002.06055))

An account of [[Morita equivalence]] as a corollary of the Yoneda lemma for bicategories is in

* [[Niles Johnson]], _Morita Theory For Derived Categories: A Bicategorical Perspective_ ([arXiv:0805.3673](http://arxiv.org/abs/0805.3673))

The stricter case of 2-categories is detailed in

* Jonas Hedman, _2-Categories and Yoneda lemma_ ([pdf](http://www.diva-portal.org/smash/get/diva2:1064822/FULLTEXT01.pdf))
* Max Kelly, _Basic Concepts of Enriched Category Theory_ ([TAC](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf))

[[!redirects bicategorical Yoneda lemma]]
[[!redirects 2-Yoneda lemma]]

[[!redirects Yoneda embedding for bicategories]]
[[!redirects bicategorical Yoneda embedding]]
[[!redirects 2-Yoneda embedding]]


Yoneda lemma for higher categories

[[!redirects Yoneda Lemma for higher categories]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

# Contents#
* table of contents
{: toc}

## Idea

One expects the [[Yoneda lemma]] to generalize to essentially every flavor of [[higher category theory]]. Various special cases have been (defined and) proven, such as the:

* [[Yoneda lemma for (∞,1)-categories]]

* [[Yoneda lemma for bicategories]]

* [[Yoneda lemma for tricategories]]

## Yoneda embedding
 {#YonedaEmbedding}

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

For $C$ an [[(∞,n)-category]] and $PSh(C)\coloneqq Func(C^\op, (\infty,(n-1)) Cat)$ its [[(∞,n)-category of (∞,n)-presheaves]], the **$(\infty,n)$-Yoneda embedding** is the [[(∞,n)-functor]]

$$
  y : C \to PSh(C)
$$

given by $y(X) : U \mapsto C(U,X)$.

=--

## Properties

### Yoneda lemma

+-- {: .num_prop }
###### Proposition
**$(\infty,n)$-Yoneda embedding**

Let $C$ be an [[(∞,n)-category]] and $PSh(C)$ be the corresponding [[(∞,n)-category of (∞,n)-presheaves]]. Then the canonical [[(∞,n)-functor]]

$$
  Y : C \to PSh(C)
$$

is a [[full and faithful (∞,n)-functor]].

=--

+-- {: .num_prop }
###### Proposition
**$(\infty,n)$-Yoneda theorem**

For $C$ a small $(\infty,n)$-category and $F : C^{op} \to (\infty,(n-1)) Cat$ an $(\infty,n)$-functor, the composite

$$
  C^{op} \to PSh_{(\infty,1)}(C)^{op} \stackrel{Hom(-,F)}{\to}
  (\infty,(n-1)) Cat$$

is equivalent to $F$.

=--

### Preservation of limits

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

The $(\infty,n)$-Yoneda embedding $y : C \to PSh(C)$ preserves all [[(∞,n)-limit]]s that exist in $C$.

=--

## Related concepts

* [[higher topos theory]]


Yoneda lemma for tricategories

+--{: .rightHandSide}
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### Context
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

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

## Idea

The **Yoneda lemma for tricategories** is a version of the [[Yoneda lemma]] that applies to [[tricategories]], the most common algebraic sort of weak [[3-category]] ([Buhné, Thrm 2.12](#Buhné)).

## References

* {#Buhné} Lukas Buhné, _Topics in three-dimensional descent theory_, ([pdf](https://d-nb.info/1072553694/34))

[[!redirects Yoneda lemma for 3-categories]]


Yoneda reduction

+-- {: .rightHandSide}
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###Context###
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
=--
=--

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

## Idea

The term _Yoneda reduction_ was coined by [[Todd Trimble]] in his (unpublished) thesis. It refers to a technique based on the [[Yoneda lemma]] for performing a number of [[end|end and coend]] calculations which arise in [[coherence]] theory and [[enriched category theory]].

## The module perspective on the Yoneda lemma

There are various formulations of the Yoneda lemma. One says that given a [[presheaf]] $F: C^{op} \to Set$, there is a canonical isomorphism

$$F(c) \cong Nat(\hom_C(-, c), F)$$

where "Nat" refers to the set of natural transformations between presheaves $C^{op} \to Set$; in other words, the hom

$$Set^{C^{op}}(\hom_C(-, c), F)$$

appropriate to the presheaf category.

There is an $V$-[[enriched category]] version, whenever $C$ is a category enriched in a [[complete category|complete]], [[cocomplete category|cocomplete]], [[symmetric monoidal category|symmetric]] [[closed monoidal category|monoidal closed category]] $V$. Here "Nat" is constructed as an enriched [[end]] (an example of a [[weighted limit]]):

$$V^{C^{op}}(C(-, c), F) = \int_d F(d)^{C(d, c)}$$

and therefore the enriched Yoneda lemma gives an [[isomorphism]]

$$F(c) \cong \int_d F(d)^{C(d, c)} \qquad (1)$$

which is ($V$-)natural in $c$; we may therefore write

$$F(-) \cong \int_d F(d)^{C(d, -)} \qquad (2)$$

and this isomorphism is $V$-natural in $F$.

We pause to give an instance of the Yoneda lemma which is both familiar and which serves to inform much of the module-theoretic terminology in the discussion below. Let $V = Ab$; let $R$ be a ring (conceived as an $Ab$-enriched category with exactly one object $\bullet$). Then $Ab^{R^{op}}$ is the ($Ab$-enriched) category of right $R$-modules, or equivalently, left $R^{op}$-modules). The presheaf $\hom_R(-, \bullet)$ is just the underlying abelian group of $R$ seen as a right [[module]] over the ring $R$, also known as the _[[regular representation]]_.

The first formulation (1) of the Yoneda lemma would simply say that at the level of _abelian groups_, we have for any right $R$-module $M$

$$M(\bullet) \cong RightMod_R(R, M)$$

Further taking into account the "naturality" in the argument bullet, the formulation (2) says that actually we have an isomorphism at the level of _right $R$-modules_

$$M \cong RightMod_R(R, M)$$

where the module structure on the right side arises by considering the argument $R$ now as a _[[bimodule]]_ over the (ring) $R$.

The (enriched) Yoneda lemma is nothing but a far-reaching extrapolation of this basic isomorphism: it says

$$F \cong RightMod_C(\hom_C, F)$$

where the $C$-presheaf or right $C$-module hom on the right is appropriately constructed as an enriched end, and $\hom_C: C^{op} \otimes C \to V$ is a treated as a $V$-enriched "bimodule" over $C$, and plays the role of the "[[regular representation]]" of $C$.

## Calculus of bimodules

The analogy between [[presheaves]] and [[module]]s can be pursued considerably further. Again, we start with the perhaps more familiar context of rings and modules.

In the first place, given a ring $R$, there is a familiar monoidal category of $R$-bimodules (and bimodule morphisms). If $M, N$ are bimodules over $R$, with left $R$-actions denoted by $\lambda$'s and the right actions by $\rho$'s, their tensor product $M \otimes_R N$, defined by the coequalizer

$$M \otimes R \otimes N \stackrel{\to}{\to} M \otimes N \to M \otimes_R N$$

(where the two parallel arrows are $M \otimes \lambda$, $\rho \times N$) carries an evident $R$-bimodule structure. Each of the functors $M \otimes_R -$ and $- \otimes_R N$ admits a [[adjunction|right adjoint]] expressed by natural isomorphisms of abelian groups

$$Bimod(N, Left_R(M, Q)) \cong Bimod(M \otimes_R N, Q) \cong Bimod(M, Right_R(N, Q))$$

where $Left_R(M, Q)$ denotes the abelian group of left $R$-module maps $M \to Q$, equipped with its natural $R$-bimodule structure; $Right(N, Q)$ is similar. Thus the monoidal category of $R$-bimodules is biclosed.

More generally, there is a bicategory whose objects or 0-cells are rings $R, S, \ldots$, and whose morphisms or 1-cells $R \to S$ are left $R$-, right $S$-bimodules. 2-cells are homomorphisms of bimodules. If $M: R \to S$ and $N: S \to T$ are bimodules, then their bimodule composite is $M \otimes_S N: R \to T$. This too is a biclosed bicategory, meaning that

This generalized module theory can be pursued much further.

(Lost a bunch of work, due to vagaries of computers. Sigh. Will return later.)

## Examples

+-- {: .un_theorem}
###### Relative coherence theorem for symmetric monoidal categories

If $V$ is symmetric monoidal, then the monoid of endomorphisms on the $n$-fold tensor functor
$$\bigotimes^n: V^{\otimes n} \to V$$
is in bijection with the monoid of endomorphisms on the unit object $I$.
=--

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

By fully and faithfully embedding $V$ (as a symmetric monoidal category) into $Set^{V^{op}}$, we may without loss of generality suppose $V$ is complete, cocomplete, symmetric monoidal closed.

The result is by induction on $n$: observe that a map

$$x_1 \otimes \x_2 \otimes \ldots \otimes x_n \to x_1 \otimes x_2 \otimes \ldots \otimes x_n$$

natural in all the arguments $x_i$, in particular in $x_n$, corresponds to a map dinatural in $x_n$:

$$x_1 \otimes \ldots \otimes x_{n-1} \to x_1 \otimes \ldots \otimes x_{n-1} \otimes x_n)^{x_n}$$

and hence to a map to the end

$$x_1 \otimes \ldots \otimes x_{n-1} \to \int_{x_n} (x_1 \otimes \ldots \otimes x_{n-1} \otimes x_n)^{x_{n}^{I}}$$

where the end exists and is isomorphic to

$$x_1 \otimes \ldots \otimes x_{n-1} \otimes I \cong x_1 \otimes \ldots \otimes x_{n-1}$$

by Yoneda reduction. This completes the induction.
=--

(It's been ages since I've thought about this. I need to think through the argument carefully again.)

## Blog resources

[[Todd Trimble]] talks about Yoneda reduction on the $n$Café [here](http://golem.ph.utexas.edu/category/2008/01/the_concept_of_a_space_of_stat.html#c014365).


Yoneda structure

+-- {: .rightHandSide}
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### Context
#### 2-Category theory
+-- {: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

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

## Idea

The concept of a **Yoneda structure** provides in a general 2-categorical setting the axiomatic description of the formal properties of the usual [[category of presheaves|presheaf construction]] and [[Yoneda embedding]] of ([[locally small category|locally]]) [[small category|small categories]]. The size issues arising in that context are absorbed directly into the structure via a class of (locally) "small" maps.

## Preliminaries

The axioms of a Yoneda structure are out to capture the properties of the presheaf construction with [[CAT]] replaced by general 2-category $\mathcal{K}$. In order to handle size issues a class of "legitimate" or "admissible" 0-cells is singled out in $|\mathcal{K}|$ as well as a class of 1-cells that behave well with respect to this class and the presheaf construction. In fact, it suffices to describe the admissible 1-cells since one can then identify the admissible 0-cells with the admissible identity 1-cells.

In $CAT$ relative to the usual presheaf construction one should think of the [[locally small category|locally small categories]] as the admissible 0-cells i.e. those categories $\mathcal{C}$ with all Hom-sets $\mathcal{C}(x,y)$ contained in a category $Set$ of "small" sets itself contained as object in a larger [[Grothendieck universe]] $U_0$. In this setting admissible functors $f:\mathcal{A}\to\mathcal{B}$ are those with all relative Hom-sets $\mathcal{B}(f(a),b)\in Set$. Furthermore, one can show ([Freyd-Street 1995](FS95)) that _a category_ $\mathcal{C}\in CAT$ _is small_ i.e. $|\mathcal{C}|\in Set$ precisely if $\mathcal{C}$ and $Set^{\mathcal{C}^{op}}$ _are locally small_.

Admissible functors $f$ in this sense in $CAT$ are closed under precomposition not only among themselves but with respect to arbitrary (composable) $g$ since the relative Hom-sets $\mathcal{B}(f{}g(x),b)$ are simply a subclass of $\mathcal{B}(f(a),b)$ namely those for which $a\in im(g)$. Whence these "relatively small" functors form a [[right ideal]]. Given the close connection between [[KZ doctrines]] and Yoneda structures it will nevertheless be useful to consider the more general case of closure in itself under composition as well, a situation which we acknowledge terminologically with the prefix "proto".

+-- {: .num_defn #admissible_1cell}
###### Definition
Let $\mathcal{K}$ be a [[2-category]] and $\mathbb{A}$ be a class of 1-cells. The 1-cells $f\in \mathbb{A}$ (and by abuse, the class $\mathbb{A}$ as well) are called _admissible_ if for all $f\in \mathbb{A}$ and composable 1-cells $g\in \mathcal{K}$,  $f\circ g\in \mathbb{A}$. $f\in \mathbb{A}$ and $\mathbb{A}$ are called _proto-admissible_ if this closure property holds for $g\in \mathbb{A}$.
=--

+-- {: .num_defn #admissible_0cell}
###### Definition
Let $\mathcal{K}$ be a [[2-category]] and $\mathbb{A}$ be an admissible (resp. proto-admissible) class of 1-cells. A 0-cell $C\in |\mathcal{K}|$ is called _admissible_ (resp. _proto-admissible_) if $id_C$ is admissible (resp. proto-admissible). We denote the corresponding class of 0-cells by $|\mathbb{A}|$.
=--

For admissible $\mathbb{A}$, $C\in|\mathbb{A}|$ iff all 1-cells with codomain $C$ are admissible. This formulation has the advantage that it makes sense for [[semi-category|semi-categories]] as well.

Having now "taken care" of the size issues we recall/introduce some terminology concerning [[Kan extensions]] and [[relative adjoint functor|relative adjoint functors]] that will prove effective in yielding a surprisingly concise axiomatic description of the presheaf construction.

+-- {: .num_defn #left_extension}
###### Definition
Let $\mathcal{K}$ be a [[2-category]] and $\eta:f\Rightarrow e\circ g$ be a 2-cell:
$$
  \array{
     A& & \overset{f}{\longrightarrow} & &B
     \\
     &{}_g\searrow & \Downarrow _\eta& \nearrow _e&
     \\
      & & C & &
  }
$$
We say that $\eta$ (or, by abuse, the diagram) exhibits $e:C\to B$ as a _left extension_ of $f:A\to B$ _along_ $g:A\to C$ if for all parallel maps $k:C\to B$ pasting with $\eta$ induces a bijection between 2-cells $\sigma:e\Rightarrow k$ and 2-cells $f\Rightarrow k\circ g$.

We say that a 1-cell $h:B\to D$ _preserves_ this left extension if the following diagram exhibits $h\circ e$ as a left extension of $h\circ f$ along $g$ :

$$
  \array{ &     &D    & &
     \\

      &{}^{h{}f}{\nearrow}&\seArrow^{id} &{\nwarrow}^h&
     \\
     A& & \overset{f}{\longrightarrow} & &B
     \\
     &{}_g\searrow & \Downarrow _\eta& \nearrow _e&
     \\
      & & C & &
  }
$$
The left extension is called _absolute_ if it is preserved by all 1-cells with domain $B$.
=--

+-- {: .num_defn #left_lifting}
###### Definition
Let $\phi:f\Rightarrow g\circ l$ be a 2-cell:

$$
  \array{

     A& & \overset{l}{\longrightarrow} & &C
     \\
     &{}_f\searrow & \neArrow _\phi& \swarrow _g&
     \\
      & & B & &
  }
$$

We say that $\phi$ (or, by abuse, the diagram) exhibits $l:A\to C$ as a _left lifting_ of $f:A\to B$ _through_ $g:C\to B$ iff for all parallel maps $k:A\to C$ pasting with $\phi$ induces a bijection between 2-cells $\sigma:l\Rightarrow k$ and 2-cells $f\Rightarrow g\circ k$.

We say that a 1-cell $j:D\to A$ preserves this left lifting if the following diagram exhibits $l\circ j$ as a left lifting of $f\circ j$ through $g$ :

$$
  \array{
     D &\overset{j}{\longrightarrow} & A& \overset{l}{\longrightarrow}  &C
     \\
     &{}_{f{}j}\searrow{}^{id}{\neArrow} &{}_f\downarrow & \overset{\phi}{\Rightarrow} \swarrow_g &
     \\
      & & B & &
  }
$$

We say that the left lifting is _absolute_ if it is preserved by all 1-cells with codomain $A$.

=--

+-- {: .num_example #fully_faithfull}
###### Example
The following diagram

$$
  \array{

     A& & \overset{id_A}{\longrightarrow} & &A
     \\
     &{}_f\searrow & \neArrow _{id}& \swarrow _f&
     \\
      & & B & &
  }
$$
exhibits $id_A$ as an absolute left lifting of $f:A\to B$ through itself iff $f:A\to B$ is _representably fully-faithful_ i.e. the functor "postcomposition with $f$" $\mathcal{K}(X,f):\mathcal{K}(X,A)\to\mathcal{K}(X,B)$ is fully-faithful for all $X\in|\mathcal{K}|$. This holds since the Hom-set $Hom_{\mathcal{K}(X,A)}(k,g)$ is precisely the set of 2-cells $k\Rightarrow g$ and $\mathcal{K}(X,f)$ acts on them by pasting with $id_f$.

=--

+-- {: .num_example #adjunction}
###### Example
The following diagram

$$
  \array{

     B& & \overset{g}{\longrightarrow} & &A
     \\
     &{}_{id_B}\searrow & \neArrow _\eta& \swarrow _f&
     \\
      & & B & &
  }
$$
exhibits $g$ as an absolute left lifting of $id_B$ through $f$ iff there exists a 2-cell
$$
  \array{

     A& & \overset{id_A}{\longrightarrow} & &A
     \\
&{}_{f}\searrow & \Uparrow _\epsilon& \nearrow _g&
     \\
      & & B & &
  }
$$
such that the pasting of $\epsilon$ on $\eta$ at $g$
$$
  \array{
    & &A & &
     \\
&{}_{f}\swarrow & \neArrow _\epsilon& \searrow _{id_A}&
     \\
     B& & \overset{g}{\longrightarrow} & &A
     \\
     &{}_{id_B}\searrow & \neArrow _\eta& \swarrow _f&
     \\
      & & B & &
  }
$$
and the pasting of $\epsilon$ on $\eta$ at $f$
$$
  \array{
     B &\overset{g}{\longrightarrow} & A& \overset{id_A}{\longrightarrow}  &A
     \\
     &{}_{id_B}\searrow{}^{\eta}{\neArrow} &{}_f\downarrow & \overset{\epsilon}{\neArrow} \nearrow_g &
     \\
      & & B & &
  }
$$

yield identity 2-cells.

Of course, this situation expresses an **adjunction** $g\dashv f$ with _unit_ $\eta$ and _counit_ $\epsilon$.  Here the absolute left lifting property of $\eta$ is furthermore equivalent to the left lifting property of $\eta$ plus preservation by $f \colon A\to B$ (cf. [Street-Walters 1978](#SW78), prop.2).

=--

## Definition

We are now ready to give the definition of a Yoneda structure:

+-- {: .num_defn #yoneda_structure}
###### Definition
Let $\mathcal{K}$ be a [[2-category]] and $\mathbb{A}$ be an admissible class of 1-cells.

A _presheaf construction_ $\mathcal{P}$ for $\mathbb{A}$ assigns to every admissible object $A\in |\mathbb{A}|$ an object $\mathcal{P}A\in |\mathcal{K}|$ called its _object of presheaves_ and an admissible 1-cell $y_A:A\to\mathcal{P}A$ called its _Yoneda morphism_ subject to the following conditions:

* (**YS1**) For each admissible 1-cell $f:A\to B$ with admissible domain $A\in |\mathbb{A}|$ there is given a 2-cell $\chi_f:y_A\Rightarrow e_f\circ f$ such that the following diagram

$$
  \array{
     A& & \overset{f}{\longrightarrow} & &B
     \\
     &{}_{y_A}\searrow & {}^{\chi_f}\neArrow& \swarrow _{e_f}&
     \\
      & & \mathcal{P}A & &
  }
$$
exhibits $f$ as an absolute left lifting of $y_A$ through $e_f$ and $e_f$ as a left extension of $y_A$ along $f$.

* (**YS2**) For all $A\in|\mathbb{A}|$, $id_{\mathcal{P}A}$ is the left extension of $y_A$ along itself as exhibited in

$$
  \array{
     A& & \overset{y_A}{\longrightarrow} & &\mathcal{P}A
     \\
     &{}_{y_A}\searrow & \Downarrow _{id}& \nearrow _{id_{\mathcal{P}A}}&
     \\
      & & \mathcal{P}A & &
  }
$$

* (**YS3**) For all $i:A\to B$, $j:B\to C$ such that $A,B\in |\mathbb{A}|$ and $i,j\in\mathbb{A}$ the following diagram

$$
  \array{
   A &\overset{i}{\rightarrow}& B& \overset{j}{\rightarrow} & C
  \\
  {}_{y_A}\downarrow &\overset{\chi_{{y_B}\circ i}}{\Rightarrow}&\downarrow_{y_B}&\overset{\chi_j}{\Rightarrow}\swarrow &{}_{e_j}
  \\
  \mathcal{P}A&\underset{e_{y_B\circ i}}{\leftarrow} &\mathcal{P}B& &
  }
$$

exhibits $e_{{y_B}i}\circ e_j$ as the left extension of $y_A$ along $j\circ i$.

The pair $(\mathbb{A},\mathcal{P})$ is called a **Yoneda structure** on the 2-category $\mathcal{K}$.
=--

We desisted from tracking the prefix 'proto' through the foregoing but it should clear that a _proto-Yoneda structure_ results from replacing 'admissible' by 'proto-admissible' throughout the definition. Indeed, in (YS3) the assumption that $i\in\mathbb{A}$ was made in proviso for the case of proto-admissible 1-cells (cf. [Walker 2017](#Walker17)) since in presence of the right ideal property this already follows from the admissibility of $id_B$.

In cases where we need to keep track of from which (proto-)Yoneda structure the various structural 1- and 2-cells come from we will use the presheaf construction as a superscript for disambiguation: for (proto-)Yoneda structure $(\mathbb{A},\mathcal{P})$ we write $y_A^\mathcal{P}$ and $\chi_f^{\mathcal{P}}$ etc.

In Street-Walters ([1978](#SW78)) a stronger axiom (YS2') is also considered:

* (**YS2'**) If the following diagram

$$
  \array{
     A& & \overset{f}{\longrightarrow} & B&
     \\
     &{}_{y_A}\searrow & \overset{\chi_f}{\Rightarrow}& \swarrow \overset{\sigma}{\Rightarrow}\swarrow_g
     \\
      & & \mathcal{P}A & &
  }
$$
exhibits $f$ as an absolute left lifting of $y_A$ through $g$ then $\sigma:e_f\Rightarrow g$ is an isomorphism.

It is shown there (prop.11) that (YS1) and (YS2') imply (YS2) and (YS3).

Together with a finite-completeness assumption on $\mathcal{K}$ and pointwiseness of the left extensions the resulting Yoneda structures are called **good** in Weber ([2007](#Weber07)) where it is also shown that Yoneda structures arising from presheaf constructions of the form $\mathcal{P}({}_-) = [({}_-)^{op},\Omega]$ have this property, $\Omega$ being an abstract "object of small sets" in a cartesian closed 2-category $\mathcal{K}$ with an involution $({}_-)^{op}$.

## Properties

....

## Examples

* The primordial example is $CAT$ with $\mathcal{P}:C\mapsto Set^{C^{op}}$ with $y_C:a\mapsto Hom_C({}_-,a)$ , the Yoneda embedding. A functor $f:C\to D$ is admissible if the relative $Hom_D(f{}_-,{}_-):D\to SET^{C^{op}}$ factors through $Set^{C^{op}}$. Admissible categories are precisely the locally small categories.

## The relation to KZ doctrines{#extension_KZ-doctrine}

Whereas it was shown already in the 1970s that ordinary monads on a 1-category can be brought into an [[extension system|extension form]] that avoids the iteration of the endofunctor similar presentations for 2-dimensional monad theory evolved more recently. For the comparison of [[lax-idempotent 2-monad|lax-idempotent 2-monads]] aka _KZ doctrines_ to Yoneda structures such presentations provided in the work of Marmolejo and Wood ([2012](#MW12)) come in handy. The link to Yoneda structures has been made in Walker ([2017](#Walker17)).

+-- {: .num_defn #KZ_doctrine}
###### Definition
Let $\mathcal{K}$ be a [[2-category]]. A _KZ-doctrine (in extension form)_ on $\mathcal{K}$ is a pair $(P,y)$ where $P$ assigns to every 0-cell $A\in\mathcal{K}$ a 0-cell $P(A)\in\mathcal{K}$ and $y$ is a family of 1-cells $y_A:A\to P(A)$ indexed by the 0-cells $A\in\mathcal{K}$ such that

* For every pair of 0-cells $A$, $B$ and 1-cells $f:A\to P(B)$ there exists an invertible 2-cell $\epsilon_f$

$$
  \array{
     A& & \overset{f}{\longrightarrow} & &P(B)
     \\
     &{}_{y_A}\searrow & \Downarrow _{\epsilon_f}& \nearrow _\overline{f}&
     \\
      & & P(A) & &
  }
$$

that exhibits $\overline{f}$ as left extension of $f$ along $y_A$. Furthermore, in case $B=A$ and $f=y_A$ then $\epsilon_{y_A}$ is given by the identity 2-cell $id_{y_A}:y_A\Rightarrow id_{P(A)}\circ y_A$.

* For any 1-cell $g:B\to P(C)$, the 1-cell $\overline{g}:P(B)\to P(C)$ given itself by the left extension along $y_B$ preserves the left extension of $f:A\to B$ along $y_A$ exhibited by $\epsilon_f$.
=--

Clearly, with $(P,y)$ the presheaf construction of a Yoneda structure comes into sight though we still need to define a suitable class of admissible maps from $(P,y)$.

But before we do this we will introduce the concept that corresponds to the familiar notion of a pseudoalgebra for a lax-idempotent 2-monad thereby hopefully making it plausible that $(P,y)$ indeed is equivalent to the usual algebraic concept.

The main idea of the following definition is that the "pseudoalgebras" $X\in|\mathcal{K}|$ mimic the extension properties of the $P(A)$, in particular, all $P(A)$ satisfy the condition trivially and should be thought of as free algebras.

+-- {: .num_defn #P-complete}
###### Definition
Given a KZ-doctrine $(P,y)$ on $\mathcal{K}$. A 0-cell $X\in|\mathcal{K}|$ is called _P-cocomplete_ if for every $g:B\to X$ there exists an invertible 2-cell $\epsilon_g$

$$
  \array{
     B& & \overset{g}{\longrightarrow} & &X
     \\
     &{}_{y_B}\searrow & \Downarrow _{\epsilon_g}& \nearrow _\overline{g}&
     \\
      & & P(B) & &
  }
$$
that exhibits $\overline{g}$ as left extension of $g$ along $y_B$. Moreover, this left extension $\overline{g}:P(B)\to X$ preserves all the left extensions $\overline{f}:P(A)\to P(B)$ along $y_A$ of arbitrary $f:A\to P(B)$.
=--
+-- {: .num_defn #P-cell}
###### Definition
A 1-cell $h:X\to Y$ between two P-cocomplete objects $X$, $Y$ is called a _P-homomorphism_ (, or a _P-cell_) if $h$ preserves the left extension $\overline{f}:P(A)\to X$ along $y_A:A\to P(A)$ for every $f:A\to X$.
=--

+-- {: .num_prop #P-algebra}
###### Proposition
Given a KZ-doctrine $(P,y)$ on $\mathcal{K}$. The following are equivalent:

* $A\in|\mathcal{K}|$ is P-cocomplete;

* $y_A:A\to P(A)$ has a left adjoint with invertible counit;

* $A$ is the underlying object of a pseudoalgebra.

=--

**Proof**.  A combination of results from Bunge-Funk ([1999](#BF99)) and Marmolejo-Wood ([2012](#MW12)). $\qed$

We now attend the problem of defining a class $\mathbb{A}$ of admissible maps for $(P,y)$.

+-- {: .num_defn #P-admissible}
###### Definition
Given a KZ-doctrine $(P,y)$ on $\mathcal{K}$. A 1-cell $a:B\to C$ is called _P-admissible_ if the left extension of $y_B:B\to P(B)$ along $a$ exists and moreover is preserved by all left extensions $\overline{h}:P(B)\to X$ along $y_B$ of 1-cells $h:B\to X$ into a P-cocomplete 0-cell $X$.
=--

A crucial property of the Yoneda embedding is of course that is in fact an _embedding_ whence we must demand the same for the units of a KZ-doctrine:

+-- {: .num_defn #locally_fully_faithful}
###### Definition
A KZ-doctrine $(P,y)$ on a 2-category $\mathcal{K}$ is called _locally fully-faithful_ if all $y_A: A\to P(A)$ are representably fully-faithful (cf. [ex.](#fully_faithful)).
=--

Now we are ready to state the main result of this section.

+-- {: .num_prop #Yoneda_monad}
###### Theorem (Walker)
Let $(P,y)$ be a locally fully-faithful KZ-doctrine on 2-category $\mathcal{K}$ with $\mathbb{A}_P$ the class of P-admissible 1-cells. The pair $(\mathbb{A}_P,P)$ defines a proto-Yoneda structure on $\mathcal{K}$.

=--

**Proof**. cf. Walker ([2017](#Walker17), p.9). $\qed$

Jokingly, we may say that _a general KZ-doctrine is nothing but unfaithful Yoneda-structure without size problems!_ Conversely, the main difference between a locally fully-faithful KZ-doctrine and a Yoneda structure concerns size: For the KZ-doctrine every identity morphism is admissible and, accordingly, its presheaf construction is total whereas this need not be the case for general Yoneda structures.

## In retrospective

The following is a quote from [[Ross Street|Ross Street's]] "Australian conspectus of higher categories" ([Street 2010](#Street10), p. 241):

>In 1971 Bob Walters and I began work on Yoneda structures on 2-categories $[$[KS1](#KS73)$]$, $[$[StW](#SW78)$]$. The idea was to axiomatize the deeper aspects of categories beyond their merely being algebraic structures. This work centred on the Yoneda embedding $A\to \mathcal{P}A$ of a category $A$ into its presheaf category $\mathcal{P}A =[ A^{op} ,Set]$ . We covered the more general example of categories enriched in a base $\mathcal{V}$ where $\mathcal{P}A = [A^{op} ,\mathcal {V}]$ . Clearly size considerations needed to be taken seriously although a motivating size-free example was preordered sets with $\mathcal{P}A$ the inclusion-ordered set of right order ideals in $A$. Size was just an extra part of the structure. With the advent of elementary topos theory and the stimulation of the work of Anders Kock and Christian Mikkelsen, we showed that the preordered objects in a topos provided a good example. We were happy to realize $[$[KS1](#KS73)$]$ that an elementary topos was precisely a finitely complete category with a _power object_ (that is, a relations classifier). This meant that my work with Walters could be viewed as a higher-dimensional version of topos theory. As usual when raising dimension, what we might mean by a 2-dimensional topos could be many things, several of which could be useful. I looked $[$[St6](#Street74)$]$, $[$[St8](#Street80)$]$ at those special Yoneda structures where $\mathcal{P}A$ classified two-sided discrete fibrations.

## Related entries

* [[Yoneda embedding]]

* [[Yoneda lemma]]

* [[pro-arrow equipment]]

* [[2-topos]]

* [[cosmos]]

* [[relative adjoint functor]]

* [[Kan extension]]

* [[lax-idempotent 2-monad]]

* [[exact square]]

## Link

* n-café seminar blog on the Street-Walters paper: ([link](http://golem.ph.utexas.edu/category/2014/03/an_exegesis_of_yoneda_structur.html))

## References

The original sources are

* {#W71}[[Bob Walters]], _Yoneda 2Categories_ , talk at the University of New South Wales December 1971. ([manuscript](https://web.archive.org/web/20170211062709/https://dl.dropboxusercontent.com/u/92056191/Archive/temporary_new_material/yoneda.pdf))

* {#KS73}[[Max Kelly]], [[Ross Street]] (eds.), _Abstracts of the Sydney Category Theory Seminar 1972/73_ , Macquarie University.

* {#SW78} [[Ross Street]], [[Bob Walters]], _Yoneda structures on 2-categories_,  JPAA **50** (1978) 350-379 [doi:10.1016/0021-8693(78)90160-6]

Early variations on the theme are in

* {#Street74}[[Ross Street]], _Elementary cosmoi I_ , pp.104-133 in Springer LNM **420** 1974.

* {#Street80}[[Ross Street]], _Cosmoi of internal categories_ , Transactions AMS **258** (1980) pp.271-318.

The Street quote stems from

* {#Street10}[[Ross Street]], _An Australian conspectus of higher categories_ , pp.237-264 in Baez, May (eds.), _Towards Higher Categories_ , Springer Heidelberg 2010. ([draft](http://maths.mq.edu.au/~street/Minneapolis.pdf))

The result on locally small categories suggesting the definition of a small object is reported in

* {#FS95}[[Peter Freyd]], [[Ross Street]], _On the size of categories_ , TAC **1** (1995) pp.174-185. ([abstract](http://www.tac.mta.ca/tac/volumes/1995/n9/1-09abs.html))

Exact squares in Yoneda structures are studied in

* [[René Guitart]], _Relations et carrés exacts_ , Ann. Sc. Math. Québec **IV** no.2 (1980) pp.103-125. ([draft](http://rene.guitart.pagesperso-orange.fr/textespublications/rg42.pdf))

* L. Van den Bril, _Exactitude dans les Yoneda-structures_ , Cah. Top. Géo. Diff. Cat. **XXIII** no.2 (1982) pp.215-224.

The following explores Yoneda structures arising from 2-categories with a discrete-opfibration classifier (such as [[2-toposes]]):

* {#Weber07} [[Mark Weber]], *Yoneda structures from 2-toposes*, Appl Categor Struct **15** (2007) 259–323 [[doi:10.1007/s10485-007-9079-2](https://doi.org/10.1007/s10485-007-9079-2), [pdf](https://sites.google.com/site/markwebersmaths/home/yoneda-structures-from-2-toposes)]

The following two investigate the connections with KZ doctrines:

* {#Walker17} C. Walker, _Yoneda Structures and KZ Doctrines_ , arXiv:1703.08693 (2017). ([abstract](http://arxiv.org/pdf/1703.08693v3))

* {#Walker18} C. Walker, _Distributive Laws via Admissibility_ , arXiv:1706.09575 (2018). ([abstract](https://arxiv.org/abs/1706.09575))

The interplay of Yoneda structures and KZ doctrines is employed to some effect in

* {#DL18} Ivan Di Liberti, [[Fosco Loregian]], _Accessibility and Presentability in 2-Categories_ , arXiv:1804.08710 (2018). ([abstract](https://arxiv.org/abs/1804.08710))

The pertaining technical ingredients on KZ doctrines are due to the following two papers

* {#BF99} [[Marta Bunge]], [[Jonathon Funk]], _On a bicomma object condition for KZ-doctrines_ , JPAA **143** (1999) pp.69-105.

* {#MW12} [[Francisco Marmolejo]], [[Richard J. Wood]], *Kan extensions and lax idempotent pseudomonads*, [[TAC]] **26** 1 (2012) 1-29 [[26-01](http://www.tac.mta.ca/tac/volumes/26/1/26-01abs.html)]

The relation to pro-arrow equipments, the presheaf construction and Isbell duality is discused in

* {#DL19} [[Ivan Di Liberti]], [[Fosco Loregian]], _On the Unicity of Formal Category Theories_ , arXiv:1901.01594 (2019). ([abstract](https://arxiv.org/abs/1901.01594))

[[!redirects yoneda structure]]
[[!redirects Yoneda Structure]]
[[!redirects Yoneda-structure]]
[[!redirects Yoneda structures]]
[[!redirects admissible 1-cell]]


Yong-Baek Kim

* [research group page](https://sites.google.com/view/ybkimgroup/yong-baek-kim)

* [institute page](https://www.physics.utoronto.ca/members/kim-yong-baek/)

## Selected writings

On ([[non-perturbative effect|strongly]]) [[interaction|interacting]] [[topological phases of matter]] (such as [[topological order]]):

* [[Jason Alicea]], [[Matthew Fisher]], [[Marcel Franz]], [[Yong-Baek Kim]],  *Strongly Interacting Topological Phases*, report on Banff workshop [15w5051](http://www.birs.ca/events/2015/5-day-workshops/15w5051) (2015) [[pdf](https://www.birs.ca/workshops/2015/15w5051/report15w5051.pdf), [[AliceaEtAl-InteractingTopPhases.pdf:file]]]

category: people


Yong-Geun Oh

* [website](http://www.math.wisc.edu/~oh/)

category: people


Yong-Shi Wu

* [institute page](https://faculty.utah.edu/u0028335-YONG-SHI_WU/research/index.hml)

* [institute page](https://web.physics.utah.edu/~wu/)

* [GoogleScholar page](https://scholar.google.com/citations?user=9NM7cUkAAAAJ&hl=en)

## Selected writings

On [[anyon]]-[[wavefunctions]] as [[multi-valued functions]] on a [[configuration space of points]]:

* {#Wu84} [[Yong-Shi Wu]], *Multiparticle Quantum Mechanics Obeying Fractional Statistics*, Phys. Rev. Lett. **53** (1984) 111 $[$[doi:10.1103/PhysRevLett.53.111](https://doi.org/10.1103/PhysRevLett.53.111), [pdf](https://core.ac.uk/download/pdf/276286925.pdf)$]$

category: people


Yongbin Ruan

* [webpage](http://www-personal.umich.edu/~ruan/)

## Selected writings

Introducing [[quantum cohomology]]:

* [[Yongbin Ruan]], [[Gang Tian]], _A mathematical theory of quantum cohomology_, Mathematical Research Letters **1** 2 (1994) 269-278 [[doi:10.4310/MRL.1994.v1.n2.a15](https://doi.org/10.4310/MRL.1994.v1.n2.a15), [pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1994/0001/0002/MRL-1994-0001-0002-a015.pdf)]

* [[Yongbin Ruan]], [[Gang Tian]], _A mathematical theory of quantum cohomology_, J. Diff. Geometry __42__ 2 (1995) 259-367 [[doi:10.4310/jdg/1214457234](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-42/issue-2/A-mathematical-theory-of-quantum-cohomology/10.4310/jdg/1214457234.full)]

Introducing [[Chen-Ruan cohomology]] for [[orbifolds]]:

* {#ChenRuan00} [[Weimin Chen]], [[Yongbin Ruan]], _A New Cohomology Theory for Orbifold_, Commun. Math. Phys. 248 (2004) 1-31 ([arXiv:math/0004129](https://arxiv.org/abs/math/0004129))

On [[orbifolds]], [[orbifold cohomology]] and specifically on [[Chen-Ruan cohomology]] and [[equivariant K-theory]]:

* {#ALR07} [[Alejandro Adem]], [[Johann Leida]], [[Yongbin Ruan]], _Orbifolds and Stringy Topology_, Cambridge Tracts in Mathematics **171** (2007) ([doi:10.1017/CBO9780511543081](https://doi.org/10.1017/CBO9780511543081), [pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))

On [[discrete torsion]] in relation to twisted [[Chen-Ruan cohomology|Chen-ruan]] [[orbifold cohomology]]:

* [[Yongbin Ruan]], *Discrete torsion and twisted orbifold cohomology*, J. Sympl. Geom. 2 (2003), 1–24 ([arXiv:math/0005299](https://arxiv.org/abs/math/0005299))

On [[twisted equivariant K-theory|twisted]] [[orbifold K-theory]]:

* {#AdemRuan01} [[Alejandro Adem]], [[Yongbin Ruan]], _Twisted Orbifold K-Theory_, Commun. Math. Phys. 237 (2003) 533-556 ([arXiv:math/0107168](https://arxiv.org/abs/math/0107168))

* [[Alejandro Adem]], [[Yongbin Ruan]], [[Bin Zhang]], _A Stringy Product on Twisted Orbifold K-theory_, Morfismos (10th Anniversary Issue), Vol. 11, No 2 (2007), 33-64.  ([arXiv:math/0605534](https://arxiv.org/abs/math/0605534), [Morfismos pdf](www.morfismos.cinvestav.mx/Portals/morfismos/SiteDocs/Articulos/Volumen11/No2/Zhang/arz.pdf))

On [[orbifolds]] in [[mathematical physics]] and in particular in [[string theory]]:

* [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]], _[[Orbifolds in Mathematics and Physics]]_, Contemporary Mathematics 310, American Mathematical Society, 2002

On [[gauged linear sigma-models]]:

* Huijun Fan, Tyler Jarvis, [[Yongbin Ruan]], _A Mathematical Theory of the Gauged Linear Sigma Model_ ([arXiv:1506.02109](https://arxiv.org/abs/1506.02109))

category: people


Yoram Moses

* [Wikipedia entry](https://en.wikipedia.org/wiki/Yoram_Moses)

* [GoogleScholar page](https://scholar.google.com/citations?user=X90c-SYAAAAJ&hl=en)

## Selected writings

On [[S5 modal logic]] as [[epistemic logic]]:

* [[Joseph Y. Halpern]], [[Yoram Moses]], *A guide to completeness and complexity for modal logics of knowledge and belief*, Artificial Intelligence **54** 3 (1992) 319-379

* {#FaginHalpernMosesVardi95} Ronald Fagin, [[Joseph Y. Halpern]], [[Yoram Moses]], Moshe Y. Vardi, *Reasoning About Knowledge*, The MIT Press (1995) $[$[ISBN:9780262562003](https://mitpress.mit.edu/9780262562003/reasoning-about-knowledge/)$]$

category: people


Yoseph Ayoub > history

* [[Joseph Ayoub]]


Yoshinori Matsuo

* [Institute page](http://www.dma.jim.osaka-u.ac.jp/view?l=en&u=10010757&a2=0000005&a3=0000134&o=affiliation&sm=affiliation&sl=en&sp=1)

## Selected writings

On the [[nuclear matrix model]] for [[baryons]]/[[nucleons]] the [[Witten-Sakai-Sugimoto model]]:

* {#HashimotoMatsuoMorita19} [[Koji Hashimoto]], [[Yoshinori Matsuo]], [[Takeshi Morita]], _Nuclear states and spectra in holographic QCD_, JHEP12 (2019) 001 ([arXiv:1902.07444](https://arxiv.org/abs/1902.07444))

Discussion of [[nucleon]] binding energies with the [[nuclear matrix model]]:

* [[Koji Hashimoto]], [[Yoshinori Matsuo]], _Nuclear binding energy in holographic QCD_ ([arXiv:2103.03563](https://arxiv.org/abs/2103.03563))

category: people


Yosida duality

## Idea

An analogue of the [[Gelfand duality]] for [[Riesz spaces]].

## Statement

The category of [[compact Hausdorff topological spaces]]
and [[continuous maps]] is contravariantly [[equivalent]]
to the category of uniformly complete Archimedean unital [[Riesz spaces]].

## Related concepts

* [[Gelfand duality]]

* [[Riesz space]]

## References

* [[Bas Westerbaan]], _Yosida Duality_, [arXiv:1612.03327](https://arxiv.org/abs/1612.03327).

* [[Kôsaku Yosida]], _On the representation of the vector lattice_, Proceedings of the Imperial Academy, Tokyo 18 (1942), 339–342.  [ProjectEuclid](http://projecteuclid.org/euclid.pja/1195573861).


You Qi

* [personal page](https://you-qi2121.github.io/mypage/index.html)

## Selected writings

Introduction to [[categorification]] with focus on [[Khovanov homology]]:

* [[Mikhail Khovanov]] (notes by [[You Qi]]), *Introduction to categorification*, lecture notes, Columbia University (2010, 2020) [[web](https://www.math.columbia.edu/~khovanov/cat2020/), [web](https://you-qi2121.github.io/mypage/categorificationnotes.html), full:[pdf](https://www.dropbox.com/scl/fi/wdesax1c8il6tgwbi20t4/KhovanovYouQi-Categorification.pdf?rlkey=l5cm3khnzu604ijdnl06o89od&dl=0)]

[[!redirects Qi You]]

category: people


Younesse Kaddar

A theoretical computer science student at the École Normale Supérieure Paris-Saclay

* [Web page](http://younesse.net)

## Related pages

* [[homotopy type theory]]

category: people


Young diagram

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

# Contents
* table of contents
{: toc}

## The idea

Young diagrams are used to describe many objects in algebra and combinatorics, including:

* integer [[partitions]].  For example, the integer partition

  $$ 17 = 5 + 4 + 4 + 2 + 1 + 1 $$

  is drawn as the Young diagram

+-- {: #Young style="text-align:center"}
<svg width="120" height="140" xmlns="http://www.w3.org/2000/svg" se:nonce="39384" xmlns:se="http://svg-edit.googlecode.com" xmlns:xlink="http://www.w3.org/1999/xlink">
 <!-- Created with SVG-edit - http://svg-edit.googlecode.com/ -->
 <desc>Young diagram (5,4,4,2,1,1)</desc>
 <g>
  <title>Layer 1</title>
  <rect x="10" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_1"/>
  <rect x="30" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_2"/>
  <rect x="50" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_3"/>
  <rect x="70" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_4"/>
  <rect x="90" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_5"/>
  <rect x="10" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_6"/>
  <rect x="30" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_7"/>
  <rect x="50" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_8"/>
  <rect x="70" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_9"/>
  <rect x="10" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_10"/>
  <rect x="30" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_11"/>
  <rect x="50" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_12"/>
  <rect x="70" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_13"/>
  <rect x="10" y="70" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_14"/>
  <rect x="30" y="70" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_15"/>
  <rect x="10" y="90" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_16"/>
  <rect x="10" y="110" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_17"/>
 </g>
</svg>
=--

* [[conjugacy classes]] in $S_n$.

* [[irreducible representations]]:

  * [[irreducible representations]] of the [[symmetric groups]] $S_n$ over any field of characteristic zero

  * [[irreducible representations|irreducible]] (algebraic) representation of the [[special linear groups]] $SL(N,\mathbb{C})$

  * [[irreducible representation|irreducible]] [[unitary representations]] of the [[special unitary group|special unitary groups]] $SU(N)$

* [[elementary symmetric functions]]

* [[Schur functors]]

* [[linear basis|basis]] vectors for the free [[lambda-ring]] on one generator, $\Lambda$

* [[flag varieties]] for the [[special linear groups]] $SL(N,k)$, where $k$ is any field

* [[characteristic classes]] for complex [[vector bundles]]: that is, [[cohomology]] classes on the [[classifying spaces]] of the [[general linear groups]] $GL(N,\mathbb{C})$

* [[characteristic classes]] for hemitian vector bundles: that is, [[cohomology]] classes on the [[classifying spaces]] of the [[unitary groups]] $U(N)$

* finite-dimensional [[C*-algebras]]: any such algebra is of the form $M_{n_1}(\mathbb{C}) \oplus \cdots \oplus M_{n_k}(\mathbb{C})$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.

* finite [[abelian group|abelian]] [[p-groups]]: any such group is of the form $\mathbb{Z}/p^{n_1} \oplus \cdots \oplus \mathbb{Z}/p^{n_k}$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.

* finite [[commutative algebra|commutative]] [[semisimple algebras]] over a [[prime field]] $\mathbb{F}_p$: any such algebra is of the form $\mathbb{F}_{p^{n_1}} \oplus \cdots \oplus \mathbb{F}_{p^{n_k}}$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.

* the [[trace of a category|trace of the category of finite sets]] has isomorphism classes of objects corresponding to Young diagrams.

## Young diagram

A **Young diagram** $F^\lambda$, also called **Ferrers diagram**, is a graphical representation of an unordered integer partition $\lambda = (\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_l$). If $\lambda\vdash n$ is a partition of $n$ then the Young diagram has $n$ boxes. A partition can be addressed as a multiset over $\mathbb{N}$.

There are two widely used such representations. The _English_ one uses matrix-like indices, and the _French_ one uses Cartesian coordinate-like indices for the boxes $x_{i,j}$ in the diagram $F^\lambda$.

In the English representation the boxes are adjusted to the north-west in the 4th quadrant of a 2-dimensional Cartesian coordinate system, with the 'y'-axis being downward oriented. For instance the diagram $F^{(5,4,4,2,1,1)}$ representing the partition $(5,4,4,2,1,1)$ of $17$ is given in the English representation as:

+-- {: #Young style="text-align:center"}
<svg width="120" height="140" xmlns="http://www.w3.org/2000/svg" se:nonce="39384" xmlns:se="http://svg-edit.googlecode.com" xmlns:xlink="http://www.w3.org/1999/xlink">
 <!-- Created with SVG-edit - http://svg-edit.googlecode.com/ -->
 <desc>Young diagram (5,4,4,2,1,1)</desc>
 <g>
  <title>Layer 1</title>
  <rect x="10" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_1"/>
  <rect x="30" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_2"/>
  <rect x="50" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_3"/>
  <rect x="70" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_4"/>
  <rect x="90" y="10" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_5"/>
  <rect x="10" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_6"/>
  <rect x="30" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_7"/>
  <rect x="50" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_8"/>
  <rect x="70" y="30" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_9"/>
  <rect x="10" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_10"/>
  <rect x="30" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_11"/>
  <rect x="50" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_12"/>
  <rect x="70" y="50" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_13"/>
  <rect x="10" y="70" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_14"/>
  <rect x="30" y="70" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_15"/>
  <rect x="10" y="90" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_16"/>
  <rect x="10" y="110" width="20" height="20" fill="#ffdddd" stroke="#000000" stroke-width="2" id="svg_39384_17"/>
 </g>
</svg>
=--

Let $\mathbb{Y}$ be the set of Young diagrams. Important functions on Young
diagrams include:

* {#Conjugation} **conjugation**: denoted by a prime $\prime : \mathbb{Y} \rightarrow \mathbb{Y}$
  reflects the Young diagram along its main diagonal (north-west to south-east).
  In the above example the conjugated partition would be $\lambda^\prime=(6,4,3,3,1)$.
* **weight**: $wt \colon \mathbb{Y} \rightarrow \mathbb{N}$ provides the number
  of boxes.
* **length**: $\ell \colon \mathbb{Y} \rightarrow \mathbb{N}$ provides the number
  of rows or equivalently the number of positive parts of the partition $\lambda$.
  The length of the conjugated diagram gives the number of columns.
* **plus**: $+ \colon \mathbb{Y}\times \mathbb{Y} \rightarrow \mathbb{Y} ::
  (\mu,\nu) \mapsto \mu + \nu = (\mu_1+\nu_1,\ldots,\mu_l+\nu_l)$
* **times**: $\times \colon \mathbb{Y}\times \mathbb{Y} \rightarrow \mathbb{Y} ::
 (\mu,\nu) \mapsto (\mu \cup \nu)_{\ge}$ the unordered union of the multisets. It
  follows that $\mu\times \nu =(\mu^\prime + \nu^\prime)^\prime$.

A **filling** of a Young diagram with elements from a set $S$ is called a [[Young tableau]].

## Skew Young diagram

A generalization of a Young diagram is a skew Young diagram. Let $\mu,\nu$ be two partitions, and let $\nu \le \mu$ be defined as $\forall i : \nu_i\le \mu_i$ (possibly adding trailing zeros). The **skew Young diagram** $F^{\mu/\nu}$ is given by the Young diagram $F^\mu$ with all boxes belonging to $F^\nu$ when superimposed removed. If $\mu=(5,4,4,2,1,1)$ and $\nu=(3,3,2,1)$ then $F^{\mu/\nu}$ looks like:

+-- {: #SkewYoung style="text-align:center"}
<svg width="120" height="140" xmlns="http://www.w3.org/2000/svg" se:nonce="39424" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:se="http://svg-edit.googlecode.com">
 <!-- Created with SVG-edit - http://svg-edit.googlecode.com/ -->
 <desc>Skew Young diagram (5,4,4,2,1,1)/(3,3,2,1)</desc>
 <g>
  <title>Layer 1</title>
  <rect id="svg_39424_1" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="10" x="0"/>
  <rect id="svg_39424_2" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="10" x="20"/>
  <rect id="svg_39424_3" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="10" x="40"/>
  <rect id="svg_39424_4" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="10" x="60"/>
  <rect id="svg_39424_5" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="10" x="80"/>
  <rect id="svg_39424_6" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="30" x="0"/>
  <rect id="svg_39424_7" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="30" x="20"/>
  <rect id="svg_39424_8" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="30" x="40"/>
  <rect id="svg_39424_9" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="30" x="60"/>
  <rect id="svg_39424_10" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="50" x="0"/>
  <rect id="svg_39424_11" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="50" x="20"/>
  <rect id="svg_39424_12" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="50" x="40"/>
  <rect id="svg_39424_13" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="50" x="60"/>
  <rect id="svg_39424_14" stroke-width="2" stroke="#cccccc" fill="none" height="20" width="20" y="70" x="0"/>
  <rect id="svg_39424_15" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="70" x="20"/>
  <rect id="svg_39424_16" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="90" x="0"/>
  <rect id="svg_39424_17" stroke-width="2" stroke="#000000" fill="#ffdddd" height="20" width="20" y="110" x="0"/>
 </g>
</svg>
=--

* A skew diagram is called **connected** if all boxes share an edge.
* A skew diagram is called a **horizontal strip** if every column contains at most one box.
* A skew diagram is called a **vertical strip** if every row contains at most one box.
* conjugation, weight, length extend to skew diagrams accordingly.

## Young tableau

...

## Related concepts

* [[semistandard Young tableau]],

* [[hook length formula]], [[hook-content formula]]

* [[Schur polynomial]]

* [[representation theory of the symmetric group]]

* [[representation theory of the general linear group]]

* [[Schur-Weyl duality]]

* [[Schur-Weyl measure]]

## References ##

Quick introduction

* Alexander Yong, What is ... a Young Tableau, _Notices of the American Mathematical Society_ *54* (February 2007), 240–241.   ([pdf](http://www.ams.org/notices/200702/whatis-yong.pdf))

Textbook accounts are in any book on [[representation theory]] in general and on the [[representation theory of the symmetric group]] in particular; such as:

* {#FultonHarris91} [[William Fulton]], [[Joe Harris]], _Representation Theory: a First Course_, Springer, Berlin, 1991 ([pdf](http://isites.harvard.edu/fs/docs/icb.topic1381051.files/fulton-harris-representation-theory.pdf))

* {#Sagan2001} [[Bruce E. Sagan]], _The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions_, Springer, 2001

More details:

* Kazuhiko Koike, Itaru Terada, _Young-diagrammatic methods for the representation theory of the classical groups of type $B_n$, $C_n$, $D_n$_,  Journal of Algebra, Volume 107, Issue 2, May 1987, Pages 466-511

* [[William Fulton]], _Young Tableaux, with Applications to Representation Theory and Geometry_, Cambridge U. Press, 1997 ([doi:10.1017/CBO9780511626241](https://doi.org/10.1017/CBO9780511626241))

* [[Ron M. Adin]], [[Yuval Roichman]], *Enumeration of Standard Young Tableaux*, Chapter 14 in: Miklós Bóna, *Handbook of Enumerative Combinatorics*, CRC Press 2015  ([arXiv:1408.4497](https://arxiv.org/abs/1408.4497), [ISBN:9781482220858](https://www.routledge.com/Handbook-of-Enumerative-Combinatorics/Bona/p/book/9781482220858))

Connection to [[algebraic geometry]]:

* C. de Concini, [[D. Eisenbud]], C. Procesi, _Young diagrams and determinantal varieties_, Invent. Math. __56__ (1980), 129-165.

With an eye towards application to (the [[standard model of particle physics|standard model]] of) [[particle physics]]:

* [[Howard Georgi]], §12  in: *Lie Algebras In Particle Physics*, Westview Press (1999), CRC Press (2019) [[doi:10.1201/9780429499210](https://doi.org/10.1201/9780429499210)]

category: combinatorics

[[!redirects Young diagram]]
[[!redirects Young diagrams]]
[[!redirects Ferrers diagram]]
[[!redirects Ferrers diagrams]]
[[!redirects Young tableau]]
[[!redirects Young tableaus]]
[[!redirects Young tableaux]]


Young measure

Related entries: [[calculus of variations]], [[microlocal defect functional]], [[measure theory]]

* L. C. Young, _Generalized curves and the existence of an attained absolute minimum in the calculus of variations_,  Comptes Rendus de la Soc. des Sciences et des Lettres de Varsovie, classe III, 30, 212-234 (1937); _Generalized surfaces in the calculus of variations_, Ann. of Math. (2) 43 (1942), 84–103 [MR0006023]
(http://www.ams.org/mathscinet-getitem?mr=0006023); _Generalized surfaces in the calculus of variations. II_, Ann. of Math. (2) 43 (1942), 530–544 [MR0006832](http://www.ams.org/mathscinet-getitem?mr=0006832); _Lectures on the calculus of variations and optimal control theory_, Foreword by Wendell H. Fleming, W. B. Saunders Co., Philadelphia-London-Toronto 1969 [MR0259704](http://www.ams.org/mathscinet-getitem?mr=0259704)
* Pablo Pedregal, _Optimization, relaxation and Young measures_, Bull. Amer. Math. Soc. 36 (1999), 27-58 [link](http://www.ams.org/journals/bull/1999-36-01/S0273-0979-99-00774-0)
* Luc Tartar, _On mathematical tools for studying partial differential equations of continuum physics: $H$-measures and Young measures_, Developments in partial differential equations and applications to mathematical physics (Ferrara, 1991) Plenum, New York, 1992, pp. 201–217 [MR1213932 (94c:35031)](http://www.ams.org/mathscinet-getitem?mr=1213932)

category: analysis


Young subgroup

## Idea

Every composition of an integer $n$ induces a [[subgroup]] of $\mathfrak{S}_{n}$.

## Definition

Let $n \ge 1$. Given a [[composition]] $(\lambda_{1},...,\lambda_{k})$ of the integer $n$ (where $\lambda_{i} \ge 1$), we have the Young subgroup:

$\mathfrak{S}_{\lambda_{1}} \times ... \times \mathfrak{S}_{\lambda_{k}} \hookrightarrow \mathfrak{S}_{n}$

associated to this composition.

## Example

If $(k,1,l)$ is a composition of $n$ as above, then:

$\mathfrak{S}_{k} \times \mathfrak{S}_{1} \times \mathfrak{S}_{l} \hookrightarrow \mathfrak{S}_{n}$

is a subgroup of permutations $\sigma \in \mathfrak{S}_{n}$ which verify that $k+1$ is a [[fixed point]] of $\sigma$. However some permutations fix $k+1$ without being an element of this subgroup.

## Related notions

[[symmetric group]]


Young's inequality

\tableofcontents

## Young's inequality for products

\begin{proposition}

Given

* $a, \,b \,\in\, \mathbb{R}_{\gt 0}$, non-[[negative number|negative]] [[real numbers]];

* $p, \, q \,\in\, \mathbb{R}_{\gt 1}$ such that

  $$
    \frac{1}{p}
    +
    \frac{1}{q}
    =
    1
  $$

then the following [[inequality]] holds:

$$
  a b
  \;\leq\;
  \frac{a^p}{p}
  +
  \frac{b^q}{q}
  \,,
$$

which is an [[equality]] if and only if $a^p = b^q$.

\end{proposition}

One proof is by [[convex function|convexity]] of the [[exponential function]]: choosing $x, y, t$ such that $\exp(x) = a^p$, $\exp = b^q$ and $t = \frac1{p}$, Young's inequality is identical to the convexity constraint

$$\exp(tx + (1-t)y) \leq t\exp(x) + (1-t)\exp(y).$$

## Related concepts

* from Young's inequality follows [[Hölder's inequality]]

## References

See also:

* Wikipedia, *[Young's inequality for products](https://en.m.wikipedia.org/wiki/Young%27s_inequality_for_products)*

[[!redirects Young's inequality for products]]


Young–Fibonacci lattice

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

#Contents#
* table of contents
{:toc}

## Idea

A kind of [[modular lattice]].

## References

* Wikipedia, _[Young–Fibonacci lattice](http://en.wikipedia.org/wiki/Young%E2%80%93Fibonacci_lattice)_


Yu Qiu

* [webpage](https://www.math.cuhk.edu.hk/people/academic-staff/yqiu)

## related $n$Lab entries

* [[Bridgeland stability condition]]

* [[Dynkin quiver]]

category: people


Yu-tin Huang

* [Institute page](https://www.phys.ntu.edu.tw/enphysics/yutin.html)

* [InSpire page](https://inspirehep.net/authors/1038150)

## Selected writings

On [[scattering amplitudes]] in [[perturbative quantum field theory]]:

* [[Henriette Elvang]], [[Yu-tin Huang]], *Scattering Amplitudes*, Cambridge University Press (2015) [[arXiv:1308.1697](http://arxiv.org/abs/1308.1697), [doi:10.1017/CBO9781107706620]( https://doi.org/10.1017/CBO9781107706620)]

category: people


Yuan Feng

* [Institute page](https://profiles.uts.edu.au/Yuan.Feng)

* [Research group page](https://quantum-lab.org/yuan/)

## Selected writings

Introducing the [[QPMC]] quantum protocol model checker:

* [[Yuan Feng]], Ernst Moritz Hahn, Andrea Turrini, Lijun Zhang, *`QPMC`: A Model Checker for Quantum Programs and Protocols*, in *Formal Methods. FM 2015*, Lecture Notes in Computer Science **9109**, Springer (2015)  [[doi:10.1007/978-3-319-19249-9_17](https://doi.org/10.1007/978-3-319-19249-9_17)]

On [[software verification]] in [[quantum computing]], hence with/of [[quantum programming languages]]:

* {#YingFeng18} [[Mingsheng Ying]], [[Yuan Feng]], *Model Checking Quantum Systems --- A Survey* [[arXiv:1807.09466](https://arxiv.org/abs/1807.09466)]

* Ji Guan, [[Yuan Feng]], Andrea Turrini, [[Mingsheng Ying]], *Model Checking Applied to Quantum Physics* [[arXiv:1902.03218](https://arxiv.org/abs/1902.03218)]

* {#YingFeng21} [[Mingsheng Ying]], [[Yuan Feng]], *Model Checking Quantum Systems -- Principles and Algorithms*, Cambridge University Press (2021) [[ISBN:9781108484305](https://www.cambridge.org/ae/academic/subjects/computer-science/programming-languages-and-applied-logic/model-checking-quantum-systems-principles-and-algorithms?format=HB)]

On [[quantum computing|quantum]] [[software verification]] with the [[Coq]] [[proof assistant]]:

* Wenjun Shi, Qinxiang Cao, Yuxin Deng, Hanru Jiang, [[Yuan Feng]], *Symbolic Reasoning about Quantum Circuits in Coq*, Journal of Computer Science and Technology (JCST), **36** 6 (2021) 1291-1306 $[$[arXiv:2005.11023](https://arxiv.org/abs/2005.11023), [doi:10.1007/s11390-021-1637-9](https://doi.org/10.1007/s11390-021-1637-9)$]$

category: people


Yufei Zhao

* [personal page](https://yufeizhao.com/)

## Selected writings

On [[Young tableaux]] and the [[representation theory of the symmetric group]]:

* [[Yufei Zhao]], _Young Tableaux and the Representations of the Symmetric Group_ ([pdf](https://yufeizhao.com/research/youngtab-hcmr.pdf), [[ZhaoYoungTableaux.pdf:file]])

category: people


Yuhma Asano

* [SPIRE page](https://inspirehep.net/authors/1274238)

## Selected writings

On [[M2-M5-brane bound states]] in the [[BMN matrix model]]:

* {#AIST17b} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _Spherical transverse M5-branes from the plane wave matrix model_, JHEP 02 (2018) 076 $[$[arXiv:1711.07681](https://arxiv.org/abs/1711.07681), doi:10.1007/JHEP02(2018)076$]$

* {#AIST17a} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _On the transverse M5-branes in matrix theory_, Phys. Rev. D **96** 126003 (2017) $[$[arXiv:1701.07140](https://arxiv.org/abs/1701.07140), [doi:10.1103/PhysRevD.96.126003](https://doi.org/10.1103/PhysRevD.96.126003)$]$

On the emergence of [[higher spin gauge theory|higher spin]] [[gravity]] from the [[IKKT matrix model]]:

* [[Yuhma Asano]], [[Harold Steinacker]], *Spherically symmetric solutions of higher-spin gravity in the IKKT matrix model* ([arXiv:2112.08204](https://arxiv.org/abs/2112.08204))


Yuho Sakatani

* [inspire page](https://inspirehep.net/authors/1077874)

## Selected writings

On [[M-brane]] [[sigma-models]] on [[exceptional geometry|exceptional]] [[exceptional generalized geometry|generalized geometric]] [[target spacetimes]]:

* [[Yuho Sakatani]], [[Shozo Uehara]], _Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry_, Phys. Rev. Lett. **117** 191601 (2016) [[arXiv:1607.04265](https://arxiv.org/abs/1607.04265), [talk slides](http://www2.yukawa.kyoto-u.ac.jp/~qft.web/2016/slides/sakatani.pdf)]

* [[Yuho Sakatani]], [[Shozo Uehara]], _Exceptional M-brane sigma models and $\eta$-symbols_ [[arXiv:1712.10316](https://arxiv.org/abs/1712.10316)]

On [[exotic branes]] in [[string theory]]/[[M-theory]]:

* [[Jose J. Fernandez-Melgarejo]], [[Tetsuji Kimura]], [[Yuho Sakatani]], *Weaving the Exotic Web*, JHEP **2018** 72 (2018) [doi:10.1007/JHEP09(2018)072]

category: people