nLab
Yoneda structure

Contents

Idea

The concept of a Yoneda structure provides in a general 2-categorical setting the axiomatic description of the formal properties of the usual presheaf construction and Yoneda embedding of (locally) small categories. The size issues arising in that context are absorbed directly into the structure via a class of (locally) β€œsmall” 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 𝒦\mathcal{K}. In order to handle size issues a class of β€œlegitimate” or β€œamissible” 0-cells is singled out in |𝒦||\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 CATCAT relative to the usual presheaf construction one should think of the locally small categories as the admissible 0-cells i.e. those categories π’ž\mathcal{C} with all Hom-sets π’ž(x,y)\mathcal{C}(x,y) contained in a category SetSet of β€œsmall” sets itself contained as object in a larger Grothendieck universe U 0U_0. In this setting admissible functors f:π’œβ†’β„¬f:\mathcal{A}\to\mathcal{B} are those with all relative Hom-sets ℬ(f(a),b)∈Set\mathcal{B}(f(a),b)\in Set. Furthermore, one can show (Freyd-Street 1995) that a category π’žβˆˆCAT\mathcal{C}\in CAT is small i.e. |π’ž|∈Set|\mathcal{C}|\in Set precisely if π’ž\mathcal{C} and Set π’ž opSet^{\mathcal{C}^{op}} are locally small.

Admissible functors ff in this sense in CATCAT are closed under precomposition not only among themselves but with respect to arbitrary (composable) gg since the relative Hom-sets ℬ(fg(x),b)\mathcal{B}(f{}g(x),b) are simply a subclass of ℬ(f(a),b)\mathcal{B}(f(a),b) namely those for which a∈im(g)a\in im(g). Whence these β€œrelatively 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 β€œproto”.

Definition

Let 𝒦\mathcal{K} be a 2-category and 𝔸\mathbb{A} be a class of 1-cells. The 1-cells fβˆˆπ”Έf\in \mathbb{A} (and by abuse, the class 𝔸\mathbb{A} as well) are called admissible if for all fβˆˆπ”Έf\in \mathbb{A} and composable 1-cells gβˆˆπ’¦g\in \mathcal{K}, f∘gβˆˆπ”Έf\circ g\in \mathbb{A}. fβˆˆπ”Έf\in \mathbb{A} and 𝔸\mathbb{A} are called proto-admissible if this closure property holds for gβˆˆπ”Έg\in \mathbb{A}.

Definition

Let 𝒦\mathcal{K} be a 2-category and 𝔸\mathbb{A} be an admissible (resp. proto-admissible) class of 1-cells. A 0-cell C∈|𝒦|C\in |\mathcal{K}| is called admissible (resp. proto-admissible) if id Cid_C is admissible (resp. proto-admissible). We denote the corresponding class of 0-cells by |𝔸||\mathbb{A}|.

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

Having now β€œtaken care” of the size issues we recall/introduce some terminology concerning Kan extensions and relative adjoint functors that will prove effective in yielding a surprisingly concise axiomatic description of the presheaf construction.

Definition

Let 𝒦\mathcal{K} be a 2-category and Ξ·:fβ‡’e∘g\eta:f\Rightarrow e\circ g be a 2-cell:

A ⟢f B gβ†˜ ⇓ Ξ· β†— e C \array{ A& & \overset{f}{\longrightarrow} & &B \\ &{}_g\searrow & \Downarrow _\eta& \nearrow _e& \\ & & C & & }

We say that Ξ·\eta (or, by abuse, the diagram) exhibits e:Cβ†’Be:C\to B as a left extension of f:Aβ†’Bf:A\to B along g:Aβ†’Cg:A\to C if for all parallel maps k:Cβ†’Bk:C\to B pasting with Ξ·\eta induces a bijection between 2-cells Οƒ:eβ‡’k\sigma:e\Rightarrow k and 2-cells fβ‡’k∘gf\Rightarrow k\circ g.

We say that a 1-cell h:Bβ†’Dh:B\to D preserves this left extension if the following diagram exhibits h∘eh\circ e as a left extension of h∘fh\circ f along gg :

D hfβ†— β‡˜ id β†– h A ⟢f B gβ†˜ ⇓ Ξ· β†— e C \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 BB.

Definition

Let Ο•:fβ‡’g∘l\phi:f\Rightarrow g\circ l be a 2-cell:

A ⟢l C fβ†˜ β‡— Ο• ↙ g B \array{ A& & \overset{l}{\longrightarrow} & &C \\ &{}_f\searrow & \neArrow _\phi& \swarrow _g& \\ & & B & & }

We say that Ο•\phi (or, by abuse, the diagram) exhibits l:Aβ†’Cl:A\to C as a left lifting of f:Aβ†’Bf:A\to B through g:Cβ†’Bg:C\to B iff for all parallel maps k:Aβ†’Ck:A\to C pasting with Ο•\phi induces a bijection between 2-cells Οƒ:lβ‡’k\sigma:l\Rightarrow k and 2-cells fβ‡’g∘kf\Rightarrow g\circ k.

We say that a 1-cell j:Dβ†’Aj:D\to A preserves this left lifting if the following diagram exhibits l∘jl\circ j as a left lifting of f∘jf\circ j through gg :

D ⟢j A ⟢l C fjβ†˜ idβ‡— f↓ ⇒ϕ↙ g B \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 AA.

Example

The following diagram

A ⟢id A A fβ†˜ β‡— id ↙ f B \array{ A& & \overset{id_A}{\longrightarrow} & &A \\ &{}_f\searrow & \neArrow _{id}& \swarrow _f& \\ & & B & & }

exhibits id Aid_A as an absolute left lifting of f:Aβ†’Bf:A\to B through itself iff f:Aβ†’Bf:A\to B is representably fully-faithful i.e. the functor β€œpostcomposition with ff” 𝒦(X,f):𝒦(X,A)→𝒦(X,B)\mathcal{K}(X,f):\mathcal{K}(X,A)\to\mathcal{K}(X,B) is fully-faithful for all X∈|𝒦|X\in|\mathcal{K}|. This holds since the Hom-set Hom 𝒦(X,A)(k,g)Hom_{\mathcal{K}(X,A)}(k,g) is precisely the set of 2-cells kβ‡’gk\Rightarrow g and 𝒦(X,f)\mathcal{K}(X,f) acts on them by pasting with id fid_f.

Example

The following diagram

B ⟢g A id Bβ†˜ β‡— Ξ· ↙ f B \array{ B& & \overset{g}{\longrightarrow} & &A \\ &{}_{id_B}\searrow & \neArrow _\eta& \swarrow _f& \\ & & B & & }

exhibits gg as an absolute left lifting of id Bid_B through ff iff there exists a 2-cell

A ⟢id A A fβ†˜ ⇑ Ο΅ β†— g B \array{ A& & \overset{id_A}{\longrightarrow} & &A \\ &{}_{f}\searrow & \Uparrow _\epsilon& \nearrow _g& \\ & & B & & }

such that the pasting of Ο΅\epsilon on Ξ·\eta at gg

A f↙ β‡— Ο΅ β†˜ id A B ⟢g A id Bβ†˜ β‡— Ξ· ↙ f B \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 Ο΅\epsilon on Ξ·\eta at ff

B ⟢g A ⟢id A A id Bβ†˜ Ξ·β‡— f↓ β‡—Ο΅β†— g B \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⊣fg\dashv f with unit Ξ·\eta and counit Ο΅\epsilon. Here the absolute left lifting property of Ξ·\eta is furthermore equivalent to the left lefting property of Ξ·\eta plus preservation by f:Aβ†’Bf:A\to B (cf. Street-Walters 1978, prop.2).

Definition

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

Definition

Let 𝒦\mathcal{K} be a 2-category and 𝔸\mathbb{A} be an admissible class of 1-cells.

A presheaf construction 𝒫\mathcal{P} for 𝔸\mathbb{A} assigns to every admissible object A∈|𝔸|A\in |\mathbb{A}| an object 𝒫A∈|𝒦|\mathcal{P}A\in |\mathcal{K}| called its object of presheaves and an admissible 1-cell y A:A→𝒫Ay_A:A\to\mathcal{P}A called its Yoneda morphism subject to the following conditions:

  • (YS1) For each admissible 1-cell f:Aβ†’Bf:A\to B with admissible domain A∈|𝔸|A\in |\mathbb{A}| there is given a 2-cell Ο‡ f:y Aβ‡’e f∘f\chi_f:y_A\Rightarrow e_f\circ f such that the following diagram
A ⟢f B y Aβ†˜ Ο‡ fβ‡— ↙ e f 𝒫A \array{ A& & \overset{f}{\longrightarrow} & &B \\ &{}_{y_A}\searrow & {}^{\chi_f}\neArrow& \swarrow _{e_f}& \\ & & \mathcal{P}A & & }

exhibits ff as an absolute left lifting of y Ay_A through e fe_f and e fe_f as a left extension of y Ay_A along ff.

  • (YS2) For all A∈|𝔸|A\in|\mathbb{A}|, id 𝒫Aid_{\mathcal{P}A} is the left extension of y Ay_A along itself as exhibited in
A ⟢y A 𝒫A y Aβ†˜ ⇓ id β†— id 𝒫A 𝒫A \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:Aβ†’Bi:A\to B, j:Bβ†’Cj:B\to C such that A,B∈|𝔸|A,B\in |\mathbb{A}| and i,jβˆˆπ”Έi,j\in\mathbb{A} the following diagram
A β†’i B β†’j C y A↓ β‡’Ο‡ y B∘i ↓ y B β‡’Ο‡ j↙ e j 𝒫A ←e y B∘i 𝒫B \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 Bi∘e je_{{y_B}i}\circ e_j as the left extension of y Ay_A along j∘ij\circ i.

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

We desisted from tracking the prefix β€˜proto’ through the foregoing but it should clear that a proto-Yoneda structure results from replacing β€˜admissible’ by β€˜proto-admissible’ throughout the definition. Indeed, in (YS3) the assumption that iβˆˆπ”Έi\in\mathbb{A} was made in proviso for the case of proto-admissible 1-cells (cf. Walker 2017) since in presence of the right ideal property this already follows from the admissibility of id Bid_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 (𝔸,𝒫)(\mathbb{A},\mathcal{P}) we write y A 𝒫y_A^\mathcal{P} and Ο‡ f 𝒫\chi_f^{\mathcal{P}} etc.

In Street-Walters (1978) a stronger axiom (YS2’) is also considered:

  • (YS2’) If the following diagram
A ⟢f B y Aβ†˜ β‡’Ο‡ f ↙⇒σ↙ g 𝒫A \array{ A& & \overset{f}{\longrightarrow} & B& \\ &{}_{y_A}\searrow & \overset{\chi_f}{\Rightarrow}& \swarrow \overset{\sigma}{\Rightarrow}\swarrow_g \\ & & \mathcal{P}A & & }

exhibits ff as an absolute left lifting of y Ay_A through gg then σ:e f⇒g\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 𝒦\mathcal{K} and pointwiseness of the left extensions the resulting Yoneda structures are called good in Weber (2007) where it is also shown that Yoneda structures arising from presheaf constructions of the form 𝒫( βˆ’)=[( βˆ’) op,Ξ©]\mathcal{P}({}_-) = [({}_-)^{op},\Omega] have this property, Ξ©\Omega being an abstract β€œobject of small sets” in a cartesian closed 2-category 𝒦\mathcal{K} with an involution ( βˆ’) op({}_-)^{op}.

Properties

….

Examples

  • The primordial example is CATCAT with 𝒫:C↦Set C op\mathcal{P}:C\mapsto Set^{C^{op}} with y C:a↦Hom C( βˆ’,a)y_C:a\mapsto Hom_C({}_-,a) , the Yoneda embedding. A functor f:Cβ†’Df:C\to D is admissible if the relative Hom D(f βˆ’, βˆ’):Dβ†’SET C opHom_D(f{}_-,{}_-):D\to SET^{C^{op}} factors through Set C opSet^{C^{op}}. Admissible categories are precisely the locally small categories.

The relation to KZ doctrines

Whereas it was shown already in the 1970s that ordinary monads on a 1-category can be brought into an 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-monads aka KZ doctrines to Yoneda structures such presentations provided in the work of Marmolejo and Wood (2012) come in handy. The link to Yoneda structures has been made in Walker (2017).

Definition

Let 𝒦\mathcal{K} be a 2-category. A KZ-doctrine (in extension form) on 𝒦\mathcal{K} is a pair (P,y)(P,y) where PP assigns to every 0-cell Aβˆˆπ’¦A\in\mathcal{K} a 0-cell P(A)βˆˆπ’¦P(A)\in\mathcal{K} and yy is a family of 1-cells y A:Aβ†’P(A)y_A:A\to P(A) indexed by the 0-cells Aβˆˆπ’¦A\in\mathcal{K} such that

  • For every pair of 0-cells AA, BB and 1-cells f:Aβ†’P(B)f:A\to P(B) there exists an invertible 2-cell Ο΅ f\epsilon_f
A ⟢f P(B) y Aβ†˜ ⇓ Ο΅ f β†— fΒ― P(A) \array{ A& & \overset{f}{\longrightarrow} & &P(B) \\ &{}_{y_A}\searrow & \Downarrow _{\epsilon_f}& \nearrow _\overline{f}& \\ & & P(A) & & }

that exhibits fΒ―\overline{f} as left extension of ff along y Ay_A. Furthermore, in case B=AB=A and f=y Af=y_A then Ο΅ y A\epsilon_{y_A} is given by the identity 2-cell id y A:y Aβ‡’id P(A)∘y Aid_{y_A}:y_A\Rightarrow id_{P(A)}\circ y_A.

  • For any 1-cell g:Bβ†’P(C)g:B\to P(C), the 1-cell gΒ―:P(B)β†’P(C)\overline{g}:P(B)\to P(C) given itself by the left extension along y By_B preserves the left extension of f:Aβ†’Bf:A\to B along y Ay_A exhibited by Ο΅ f\epsilon_f.

Clearly, with (P,y)(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)(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)(P,y) indeed is equivalent to the usual algebraic concept.

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

Definition

Given a KZ-doctrine (P,y)(P,y) on 𝒦\mathcal{K}. A 0-cell X∈|𝒦|X\in|\mathcal{K}| is called P-cocomplete if for every g:Bβ†’Xg:B\to X there exists an invertible 2-cell Ο΅ g\epsilon_g

B ⟢g X y Bβ†˜ ⇓ Ο΅ g β†— gΒ― P(B) \array{ B& & \overset{g}{\longrightarrow} & &X \\ &{}_{y_B}\searrow & \Downarrow _{\epsilon_g}& \nearrow _\overline{g}& \\ & & P(B) & & }

that exhibits g¯\overline{g} as left extension of gg along y By_B. Moreover, this left extension g¯:P(B)→X\overline{g}:P(B)\to X preserves all the left extensions f¯:P(A)→P(B)\overline{f}:P(A)\to P(B) along y Ay_A of arbitrary f:A→P(B)f:A\to P(B).

Definition

A 1-cell h:X→Yh:X\to Y between two P-cocomplete objects XX, YY is called a P-homomorphism (, or a P-cell) if hh preserves the left extension f¯:P(A)→X\overline{f}:P(A)\to X along y A:A→P(A)y_A:A\to P(A) for every f:A→Xf:A\to X.

Proposition

Given a KZ-doctrine (P,y)(P,y) on 𝒦\mathcal{K}. The following are equivalent:

  • A∈|𝒦|A\in|\mathcal{K}| is P-cocomplete;

  • y A:Aβ†’P(A)y_A:A\to P(A) has a left adjoint with invertible counit;

  • AA is the underlying object of a pseudoalgebra.

Proof. A combination of results from Bunge-Funk (1999) and Marmolejo-Wood (2012). β–ͺ\qed

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

Definition

Given a KZ-doctrine (P,y)(P,y) on 𝒦\mathcal{K}. A 1-cell a:Bβ†’Ca:B\to C is called P-admissible if the left extension of y B:Bβ†’P(B)y_B:B\to P(B) along aa exists and moreover is preserved by all left extensions hΒ―:P(B)β†’X\overline{h}:P(B)\to X along y By_B of 1-cells h:Bβ†’Xh:B\to X into a P-cocomplete 0-cell XX.

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:

Definition

A KZ-doctrine (P,y)(P,y) on a 2-category 𝒦\mathcal{K} is called locally fully-faithful if all y A:Aβ†’P(A)y_A: A\to P(A) are representably fully-faithful (cf. ex.).

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

Theorem (Walker)

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

Proof. cf. Walker (2017, p.9). β–ͺ\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's β€œAustralian conspectus of higher categories” (Street 2010, p. 241):

In 1971 Bob Walters and I began work on Yoneda structures on 2-categories [[KS1]], [[StW]]. The idea was to axiomatize the deeper aspects of categories beyond their merely being algebraic structures. This work centred on the Yoneda embedding A→𝒫AA\to \mathcal{P}A of a category AA into its presheaf category 𝒫A=[A op,Set]\mathcal{P}A =[ A^{op} ,Set] . We covered the more general example of categories enriched in a base 𝒱\mathcal{V} where 𝒫A=[A op,𝒱]\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 𝒫A\mathcal{P}A the inclusion-ordered set of right order ideals in AA. 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]] 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]], [[St8]] at those special Yoneda structures where 𝒫A\mathcal{P}A classified two-sided discrete fibrations.

  • n-cafΓ© seminar blog on the Street-Walters paper: (link)

References

The original sources are

Early variations on the theme are in

  • Ross Street, Elementary cosmoi I , pp.104-133 in Springer LNM 420 1974.

  • Ross Street, Cosmoi of internal categories , Transactions AMS 258 (1980) pp.271-318.

The Street quote stems from

  • Ross Street, An Australian conspectus of higher categories , pp.237-264 in Baez, May (eds.), Towards Higher Categories , Springer Heidelberg 2010. (draft)

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

Exact squares in Yoneda structures are studied in

  • RenΓ© Guitart, Relations et carrΓ©s exacts , Ann. Sc. Math. QuΓ©bec IV no.2 (1980) pp.103-125. (draft)

  • L. Van den Bril, Exactitude dans les Yoneda-structures , Cah. Top. GΓ©o. Diff. Cat. XXIII no.2 (1982) pp.215-224.

The following explores Yoneda structure arising from 2-categories with a discrete-opfibration classifier:

  • Mark Weber, Yoneda structures from 2-toposes , Appl. Cat. Struc. 15 (2007) pp.259-323. (preprint)

The following two investigate the connections with KZ doctrines:

  • C. Walker, Yoneda Structures and KZ Doctrines , arXiv:1703.08693 (2017). (abstract)

  • C. Walker, Distributive Laws via Admissibility , arXiv:1706.09575 (2018). (abstract)

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

  • Ivan Di Liberti, Fosco Loregian, Accessibility and Presentability in 2-Categories , arXiv:1804.08710 (2018). (abstract)

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

Last revised on June 18, 2018 at 16:59:18. See the history of this page for a list of all contributions to it.