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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 . In order to handle size issues a class of βlegitimateβ or βadmissibleβ 0-cells is singled out in 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 relative to the usual presheaf construction one should think of the locally small categories as the admissible 0-cells i.e. those categories with all Hom-sets contained in a category of βsmallβ sets itself contained as object in a larger Grothendieck universe . In this setting admissible functors are those with all relative Hom-sets . Furthermore, one can show (Freyd-Street 1995) that a category is small i.e. precisely if and are locally small.
Admissible functors in this sense in are closed under precomposition not only among themselves but with respect to arbitrary (composable) since the relative Hom-sets are simply a subclass of namely those for which . 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 be a 2-category and be a class of 1-cells. The 1-cells (and by abuse, the class as well) are called admissible if for all and composable 1-cells , . and are called proto-admissible if this closure property holds for .
Let be a 2-category and be an admissible (resp. proto-admissible) class of 1-cells. A 0-cell is called admissible (resp. proto-admissible) if is admissible (resp. proto-admissible). We denote the corresponding class of 0-cells by .
For admissible , iff all 1-cells with codomain 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 be a 2-category and be a 2-cell:
We say that (or, by abuse, the diagram) exhibits as a left extension of along if for all parallel maps pasting with induces a bijection between 2-cells and 2-cells .
We say that a 1-cell preserves this left extension if the following diagram exhibits as a left extension of along :
The left extension is called absolute if it is preserved by all 1-cells with domain .
Let be a 2-cell:
We say that (or, by abuse, the diagram) exhibits as a left lifting of through iff for all parallel maps pasting with induces a bijection between 2-cells and 2-cells .
We say that a 1-cell preserves this left lifting if the following diagram exhibits as a left lifting of through :
We say that the left lifting is absolute if it is preserved by all 1-cells with codomain .
The following diagram
exhibits as an absolute left lifting of through itself iff is representably fully-faithful i.e. the functor βpostcomposition with β is fully-faithful for all . This holds since the Hom-set is precisely the set of 2-cells and acts on them by pasting with .
The following diagram
exhibits as an absolute left lifting of through iff there exists a 2-cell
such that the pasting of on at
and the pasting of on at
yield identity 2-cells.
Of course, this situation expresses an adjunction with unit and counit . Here the absolute left lifting property of is furthermore equivalent to the left lifting property of plus preservation by (cf. Street-Walters 1978, prop.2).
We are now ready to give the definition of a Yoneda structure:
Let be a 2-category and be an admissible class of 1-cells.
A presheaf construction for assigns to every admissible object an object called its object of presheaves and an admissible 1-cell called its Yoneda morphism subject to the following conditions:
exhibits as an absolute left lifting of through and as a left extension of along .
exhibits as the left extension of along .
The pair is called a Yoneda structure on the 2-category .
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 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 .
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 we write and etc.
In Street-Walters (1978) a stronger axiom (YS2β) is also considered:
exhibits as an absolute left lifting of through then is an isomorphism.
It is shown there (prop.11) that (YS1) and (YS2β) imply (YS2) and (YS3).
Together with a finite-completeness assumption on 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 have this property, being an abstract βobject of small setsβ in a cartesian closed 2-category with an involution .
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 be a 2-category. A KZ-doctrine (in extension form) on is a pair where assigns to every 0-cell a 0-cell and is a family of 1-cells indexed by the 0-cells such that
that exhibits as left extension of along . Furthermore, in case and then is given by the identity 2-cell .
Clearly, with the presheaf construction of a Yoneda structure comes into sight though we still need to define a suitable class of admissible maps from .
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 indeed is equivalent to the usual algebraic concept.
The main idea of the following definition is that the βpseudoalgebrasβ mimic the extension properties of the , in particular, all satisfy the condition trivially and should be thought of as free algebras.
Given a KZ-doctrine on . A 0-cell is called P-cocomplete if for every there exists an invertible 2-cell
that exhibits as left extension of along . Moreover, this left extension preserves all the left extensions along of arbitrary .
A 1-cell between two P-cocomplete objects , is called a P-homomorphism (, or a P-cell) if preserves the left extension along for every .
Given a KZ-doctrine on . The following are equivalent:
is P-cocomplete;
has a left adjoint with invertible counit;
is the underlying object of a pseudoalgebra.
Proof. A combination of results from Bunge-Funk (1999) and Marmolejo-Wood (2012).
We now attend the problem of defining a class of admissible maps for .
Given a KZ-doctrine on . A 1-cell is called P-admissible if the left extension of along exists and moreover is preserved by all left extensions along of 1-cells into a P-cocomplete 0-cell .
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 on a 2-category is called locally fully-faithful if all are representably fully-faithful (cf. ex.).
Now we are ready to state the main result of this section.
Let be a locally fully-faithful KZ-doctrine on 2-category with the class of P-admissible 1-cells. The pair defines a proto-Yoneda structure on .
Proof. cf. Walker (2017, p.9).
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 of a category into its presheaf category . We covered the more general example of categories enriched in a base where . Clearly size considerations needed to be taken seriously although a motivating size-free example was preordered sets with the inclusion-ordered set of right order ideals in . 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 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) 350-379 [doi:10.1016/0021-8693(78)90160-6]
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 (such as 2-toposes):
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 1 (2012) 1-29 [26-01]
The relation to pro-arrow equipments, the presheaf construction and Isbell duality is discused in
Last revised on October 12, 2023 at 10:09:58. See the history of this page for a list of all contributions to it.