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) βsmallβ maps.
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 βlegitimateβ or βadmissibleβ 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 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 βsmallβ 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) 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 βrelatively smallβ 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 βprotoβ.
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 $g\in \mathbb{A}$.
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-categories as well.
Having now βtaken careβ 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.
Let $\mathcal{K}$ be a 2-category and $\eta:f\Rightarrow e\circ g$ be a 2-cell:
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 $g$ :
The left extension is called absolute if it is preserved by all 1-cells with domain $B$.
Let $\phi:f\Rightarrow g\circ l$ be a 2-cell:
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 $g$ :
We say that the left lifting is absolute if it is preserved by all 1-cells with codomain $A$.
The following diagram
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 βpostcomposition with $f$β $\mathcal{K}(X,f):\mathcal{K}(X,A)\to\mathcal{K}(X,B)$ 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 $id_f$.
The following diagram
exhibits $g$ as an absolute left lifting of $id_B$ through $f$ iff there exists a 2-cell
such that the pasting of $\epsilon$ on $\eta$ at $g$
and the pasting of $\epsilon$ on $\eta$ at $f$
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 $f:A\to B$ (cf. Street-Walters 1978, prop.2).
We are now ready to give the definition of a Yoneda structure:
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:
exhibits $f$ as an absolute left lifting of $y_A$ through $e_f$ and $e_f$ as a left extension of $y_A$ along $f$.
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 βprotoβ through the foregoing but it should clear that a proto-Yoneda structure results from replacing βadmissibleβ by βproto-admissibleβ 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) 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) a stronger axiom (YS2β) is also considered:
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 (YS2β) 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) 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 βobject of small setsβ in a cartesian closed 2-category $\mathcal{K}$ with an involution $({}_-)^{op}$.
β¦.
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).
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
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 $id_{y_A}:y_A\Rightarrow id_{P(A)}\circ y_A$.
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 βpseudoalgebrasβ $X\in|\mathcal{K}|$ 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.
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 $\epsilon_g$
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 $f:A\to P(B)$.
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 $f:A\to X$.
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) and Marmolejo-Wood (2012). $\qed$
We now attend the problem of defining a class $\mathbb{A}$ of admissible maps for $(P,y)$.
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:
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.).
Now we are ready to state the main result of this section.
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, 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.
The following is a quote from Ross Street's βAustralian conspectus of higher categoriesβ (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 $A\to \mathcal{P}A$ 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$]$ 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 $\mathcal{P}A$ classified two-sided discrete fibrations.
The original sources are
Bob Walters, Yoneda 2Categories , talk at the University of New South Wales December 1971. (manuscript)
Max Kelly, Ross Street (eds.), Abstracts of the Sydney Category Theory Seminar 1972/73 , Macquarie University.
Ross Street, Bob Walters, Yoneda structures on 2-categories , JPAA 50 (1978) pp.350-379.
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
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 structures arising from 2-categories with a discrete-opfibration classifier:
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
The pertaining technical ingredients on KZ doctrines are due to the following two papers
Marta Bunge, Jonathon Funk, On a bicomma object condition for KZ-doctrines , JPAA 143 (1999) pp.69-105.
Francisco Marmolejo, Richard J. Wood, Kan extensions and lax idempotent pseudomonads , TAC 26 no.1 (2012) pp.1-19. (abstract)
The relation to pro-arrow equipments, the presheaf construction and Isbell duality is discused in
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