C-Lenses Explained:
Bx Foundations for the Rest of Us
Anthony Anjorin1 , Hsiang-Shang Ko2 and Erhan Leblebici
1
IAV GmbH, Germany
2
Institute of Information Science, Academia Sinica, Taiwan
Abstract
Bidirectional transformations (bx) are mechanisms for maintaining the consistency of multiple artefacts.
Some of the challenges bx research aims to address include answering fundamental questions such as
how best to precisely characterise consistency, “good” consistency maintainers, and the required input
and assumptions to guarantee this good behaviour.
While substantial progress has been made towards unifying the different (variants of) formal frame-
works for bx, many of these formal results are not yet easily accessible to more practically-oriented bx
researchers, often due to a missing background in (advanced) category theory. This is an unfortunate
situation, as many bx tools are developed and maintained by practically-oriented bx researchers, who
would but currently cannot fully profit from formal results and insights. In particular, we are not aware
of any practical implementation of C-lenses, not even preliminary discussions about its relevance to
practical bx. We believe this is because of the inaccessibility of the categorical language, and write this
paper to decipher C-lenses for bx researchers without any substantial background in category theory.
Our goal is to spark broader interest in C-lenses and in discussions on their usefulness in practice and
potential for improving existing, and inspiring new bx tools.
We start by reviewing the well-known state-based bx framework, already accessible to a broad au-
dience due to its simplicity. We then explain how this framework can be elegantly generalised to the
richer, delta-based setting of C-lenses using multiple examples and illustrative diagrams.
Keywords
Bidirectional Transformations, Formal Foundations, C-Lenses
1. Introduction
Bidirectional transformations (bx) are mechanisms for maintaining the consistency of multiple
artefacts [1]. To provide support for consistency maintenance, a formal understanding of what
constitutes a “good” consistency maintainer, as well as tooling support for automation are
equally important. It is also essential that formal and practical bx go hand-in-hand: bx tools
should faithfully instantiate a bx formal framework, and bx formal frameworks should cover
aspects of practical relevance.
There has been considerable progress made regarding bx foundations. Johnson et al. [2, 3, 4]
and Diskin et al. [5] have both suggested generalisations of the state-based lens bx framework [6]
to a richer delta-based setting, addressing certain limitations inherent to a state-based setting.
Bx 2021: 9th International Workshop on Bidirectional Transformations, part of STAF, June 21, 2021
" anthony.anjorin@iav.de (A. Anjorin); joshko@iis.sinica.edu.tw (H. Ko); erhan.leblebici@outlook.com
(E. Leblebici)
© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
While there are relatively direct implementations of the state-based characterisation of bx
such as Boomerang [7], GRoundTram [8], BiGUL [9] and HOBiT [10] (to name just a few),
we are not aware of any bx tool that faithfully implements C-lenses according to Johnson
et al. [3]. This is certainly not because practical bx tools are inherently state-based, or that
working in a rich delta-based setting is often infeasible in practice. Indeed, bx tools from the
Model-Driven Engineering (MDE) community are often able to record and work with deltas (see
a recent comparison of bx tools for details [11]). Diskin and Hermann [12] have collaborated to
discuss how Triple Graph Grammars (TGGs) [13] can be aligned with D-lenses according to
Diskin et al. [5]. This is a helpful step in the right direction, as there are multiple TGG-based
bx tools available. We claim that most results of Johnson et al. [3], concerning C-lenses as
an elegant generalisation of state-based to delta-based bx, remain largely inaccessible to most
practically-oriented bx researchers mainly due to a missing background in category theory.
Our goal in this paper is to contribute to the ongoing endeavour of bridging formal and
practical bx. We aim to spark interest in C-lenses from a practical point of view, and enable
discussions on their potential to improve existing bx tools and perhaps inspire a new generation
of bx tools in the different bx sub-communities. To serve the primary purpose of the bx commu-
nity — to enable communication and exchange of ideas between the various sub-communities —
we target practical bx researchers without any substantial background in category theory, and
aim to be as self-contained and understandable as possible. As the paper is intentionally kept
lightweight in terms of technicality, we refer to and rely on Johnson et al. [2, 3, 4] for all formal
details and proofs.
The paper starts with lenses in the state-based setting (Sect. 2) as a warm-up for the generali-
sation to C-lenses in our main, delta-based setting (Get and Put operations in Sect. 3 and their
laws in Sect. 4). The main advantage of C-lenses is universality, which we explain and discuss
in Sect. 5. Related work is in Sect. 6 and our conclusion in Sect. 7.
2. State-based bx
We start with the basic notions of consistency maintenance in a simple state-based setting, as a
warm-up for our main delta-based setting in Sect. 3. Consider a simple consistency maintenance
scenario (known as the composers example [14]) where we wish to synchronise multisets
(sources) of composers with their name, date of birth, and nationality, with multisets (views) of
composers with only their name and nationality. This minimal example exhibits asymmetry,
a property of consistency maintenance scenarios that we will assume1 throughout the paper:
when a pair of source and view are consistent, the view is completely determined by the source.
Assuming asymmetry, a lens [6] keeps a pair of source and view consistent using two functions:
Get to obtain the view of a source, and Put to update an existing source given a new view. This
can be formalised in a straightforward manner as follows:
Get
Get : 𝑆 → 𝑉, 𝑠 ↦−→ 𝑣 (1)
Put
Put : 𝑉 × 𝑆 → 𝑆, (𝑣, 𝑠) ↦−→ 𝑠′ (2)
1
Johnson et al. [15] argue that this is not a fundamental restriction as symmetric lenses can be constructed as
spans of asymmetric lenses.
𝑆 and 𝑉 are sets of objects, Get and Put are functions, and 𝑉 × 𝑆 is the product set consisting
of pairs (𝑣, 𝑠) of objects with 𝑣 ∈ 𝑉 and 𝑠 ∈ 𝑆. Independent of the exact behaviour of a lens
— a pair (Get, Put) — the challenge is to characterise “good” synchronisation behaviour in a
general manner. The lens framework does this by requiring the three laws depicted in Fig. 1
for very well-behaved lenses. We have chosen a diagrammatic notation for the laws as this
generalises well to the delta-based setting in Sect. 3. We now explain the meaning of these laws.
PutGet GetPut PutPut
P ut (Get, S) V ⇥ P ut
V ⇥S V ⇥V ⇥S V ⇥S
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S AAAB6nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8eK9gPaUDbbTbt0swm7E6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z9wD49aJsk0402WyER3Qmq4FIo3UaDknVRzGoeSt8Px7cxvP3FtRKIecZLyIKZDJSLBKFrpoZFh3616NW8Oskr8glShQKPvfvUGCctirpBJakzX91IMcqpRMMmnlV5meErZmA5511JFY26CfH7qlJxZZUCiRNtSSObq74mcxsZM4tB2xhRHZtmbif953Qyj6yAXKs2QK7ZYFGWSYEJmf5OB0JyhnFhCmRb2VsJGVFOGNp2KDcFffnmVtC5qvlfz7y+r9ZsijjKcwCmcgw9XUIc7aEATGAzhGV7hzZHOi/PufCxaS04xcwx/4Hz+AFW6jdE=
V S V ⇥S S
P ut
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Figure 1: State-based lens laws for very well-behaved lenses
Starting with PutGet, the diagram to the far left: 𝜋0 is the projection to the first (0th) di-
mension, i.e., to 𝑉 in this case — meaning that 𝜋0 maps pairs (𝑣, 𝑠) to 𝑣. The equals sign “=”
inside the diagram means that it must commute for the law to hold, i.e., Put ; Get = 𝜋0 where
the operator “;” — read as “then” — composes functions as required. The intuition here is that
if you put a view 𝑣 into an old source 𝑠 to obtain a new source 𝑠′ , the view you get from 𝑠′
immediately after that should be exactly 𝑣. This ensures that Put synchronises 𝑣 fully with 𝑠.
Example 1. To make things more concrete, consider our composers example: Let 𝑆 be a set
of multisets of triples (name, date, nat) representing composers, and 𝑉 be a set of multisets
of pairs (name, nat) representing composers without their date of birth. This means that
source and view objects here are multisets of triples and pairs, respectively. Let Get be the
function mapping sources to views by mapping the triples (name, date, nat) in a source to
pairs (name, nat) in the resulting view. Now let us define Put(𝑣, 𝑠) = 𝑠′ as follows:
1. Set 𝑠′ = 𝑠.
2. For every pair (name, nat) in 𝑣, if there exists no triple of the form (name, _, nat)2 in 𝑠′
then add a new triple (name, ???, nat) to 𝑠′ .
3. For every triple (name, date, nat) in 𝑠′ , if there exists no pair (name, nat) in 𝑣 then
remove the triple (name, date, nat) from 𝑠′ .
While this choice of Get and Put might seem reasonable, the PutGet law tells us that it is
actually a bad idea: Consider 𝑣 = { (J.S. Bach, German), (J.S. Bach, German) } and 𝑠 = { (J.S.
Bach, 31/03/1685, German) }. This is a valid input pair for Put — in particular, 𝑣 is a valid view,
i.e., 𝑣 ∈ 𝑉 . As Put does not care about duplicates, however, it will yield an unchanged 𝑠 as the
result 𝑠′ . As Get(𝑠) = { (J.S. Bach, German) } =
̸ 𝑣, the PutGet law is violated.
Intuitively, Put as we have defined it does not take 𝑣 fully into account. Specifically, it
ignores the fact that some pairs can be present multiple times in the view. As this information
2
“_” means any value.
is, however, not ignored by Get, something is wrong. To avoid violating PutGet, 𝐺𝑒𝑡 can be
adjusted to avoid duplicate pairs in the view.
For the second law GetPut in the diagram of Fig. 1: 1𝑆 denotes the identity function 𝑠 ↦→ 𝑠,
(Get, 1𝑆 ) denotes a pair of functions that maps 𝑠 to the pair (Get(𝑠), 1𝑆 (𝑠)) = (𝑣, 𝑠). As the
diagram is to be commutative, the law demands that (Get, 1𝑆 ) ; Put = 1𝑆 . Intuitively, this
means getting a view 𝑣 from a source 𝑠 and directly putting the same view 𝑣 back in 𝑠 must do
nothing to 𝑠. This ensures that Put does not fabricate random information that does not come
from the view.
Example 2. Revisiting our composers example, let Put be defined as for Ex. 1. Let Get project
triples to pairs as in Ex. 1 and additionally avoid duplicates so PutGet is fulfilled. To explore
a practical violation of GetPut, let us assume that Get is further adjusted to keep only last
names, e.g., J.S. Bach becomes Bach in the view to provide a more compact list of composers.
Without changing Put, the PutGet law is still satisfied for this new Get. Consider the source
containing J.S. Bach as a composer. Applying Get results in an abbreviated name in the view.
Put as defined for Ex. 1 will not, however, recognise this and instead delete the composer J.S.
Bach in the source and add the new composer Bach with a default date of birth. As this changes
the source, we see that (Get, 1𝑆 ) ; Put ̸= 1𝑆 , violating GetPut.
The final law PutPut is depicted to the right of Fig. 1. 𝑉 × 𝑉 × 𝑆 is a product set consisting
of triples (𝑣, 𝑣 ′ , 𝑠). 𝜋0,2 is the projection to the first (0th) and third (2nd) components, i.e., maps
triples (𝑣, 𝑣 ′ , 𝑠) to pairs (𝑣, 𝑠). The function 1𝑉 ×Put applies the identity to the first component
of a triple, and Put to the second and third components, i.e., maps triples (𝑣, 𝑣 ′ , 𝑠) to pairs
(𝑣, Put(𝑣 ′ , 𝑠)) = (𝑣, 𝑠′ ). As the diagram is to be commutative, this means that 1𝑉 × Put ; Put,
i.e., applying Put to the first view 𝑣 ′ , then to the second 𝑣, is to be the same as 𝜋0,2 ; Put, i.e.,
skipping the first view 𝑣 ′ and just applying Put to the second view 𝑣. Intuitively, this means
that it should make no difference if intermediate views such as 𝑣 ′ are put into the source or not.
The last and final view 𝑣 fully determines the resulting source.
Example 3. Consider our composers example with Put as defined in Ex. 1 and Get as the
projection to names and nationalities avoiding duplicates. PutPut can be shown to be violated
as follows: Starting with a source 𝑠 = { (J.S. Bach, 31/03/1685, German) }, let 𝑣 ′ = {} and 𝑣 =
{ (J.S. Bach, German) }. As Put(𝑣, 𝑃 𝑢𝑡(𝑣 ′ , 𝑠)) = Put(𝑣, {}) = { (J.S. Bach, ???, German) } ̸=
Put(𝑣, 𝑠) = 𝑠, PutPut is violated. Intuitively, the first view 𝑣 ′ deletes the single entry in the
source, while the second view 𝑣 can be interpreted as attempting to recreate it. This does not
work, however, as there is no way to reconstruct the date of birth of the composer from 𝑣.
Numerous authors including Johnson et al. [2] and Diskin et al. [5] use the PutPut law to
motivate a generalisation of the lens state-based framework to a delta-based setting. In some
state-based scenarios, the PutPut law is indeed unreasonable: it basically demands that putting
arbitrary intermediate views make no difference to the final result of putting the final view.
Considering Ex. 3, the practical consequence of PutPut would be to require tracking a com-
plete history of all changes so that all previous values can be restored. This can be undesirable
or even infeasible in practice due to limited memory and other practical constraints, and is also
potentially confusing as there is no formal way to distinguish between (i) deleting and restoring
a composer and (ii) deleting a composer and adding a new composer that just happens to have
the same name and nationality. In a richer, delta-based setting this distinction becomes possible,
making the PutPut law just as fundamental as PutGet and GetPut.
3. Delta-based get and put
In this section and the next, we generalise the Get and Put functions and the (first two) well-
behavedness laws to a delta-based setting, in which we can express not only what sources
and views become but also how they are modified. In this setting, 𝑆 is no longer simply a
collection of all possible sources but is now a category, consisting not only of all sources as its
objects obj (𝑆), but also of all possible arrows arr (𝑆) between all sources. As it helps (in a bx
context) to think of these arrows as “deltas” that represent a means of obtaining (computing,
transforming) one source from another, we shall often use the term “delta” instead of the more
generic “arrow” in the rest of this paper. Analogously, 𝑉 is also a category consisting of all
possible views obj (𝑉 ) and view deltas arr (𝑉 ). While we shall use the notation 𝛿𝑠𝑠′ : 𝑠 → 𝑠′
to represent deltas, it is crucial to bear in mind that deltas are in general not fully determined
by their endpoints. This means that unless explicitly stated otherwise, there can in general be
* : 𝑠 → 𝑠′ between the same objects 𝑠 and 𝑠′ .
multiple, different deltas 𝛿𝑠𝑠′ ̸= 𝛿𝑠𝑠 ′
Example 4. Some objects and deltas in the source and view categories 𝑆 and 𝑉 for the composers
example are shown in Fig. 2. Within the two categories (depicted as rectangles), objects are
multisets/states (depicted as ellipses), and deltas are modifications from one multiset to another
(depicted as bold arrows). Only three sources and three deltas are shown in 𝑆. The source 𝑠
contains just one composer J.S. Bach (represented as a triple). The source delta 𝛿𝑠𝑠′ adds a new
composer with the same name and nationality but a different date of birth 08/03/1714 to result
in 𝑠′ , while 𝛿𝑠′ 𝑠′′ then edits this new composer by changing the name to C.P.E. Bach.
S (J.S. Bach, 31/03/1685, German)
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s
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(J.S. Bach, German)
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ss0
Get!
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id(v)
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0
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ss0 ; s0 s00 (J.S. Bach, 31/03/1685, German)
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s
AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0mkUI8FLx6r2A9oQ9lsJ+3SzSbsboQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ7dzvPKHSPJaPZpqgH9GR5CFn1FjpQV8OyhW36i5A1omXkwrkaA7KX/1hzNIIpWGCat3z3MT4GVWGM4GzUj/VmFA2oSPsWSpphNrPFpfOyIVVhiSMlS1pyEL9PZHRSOtpFNjOiJqxXvXm4n9eLzXhjZ9xmaQGJVsuClNBTEzmb5MhV8iMmFpCmeL2VsLGVFFmbDglG4K3+vI6aV9XPbfq3dcqjVoeRxHO4ByuwIM6NOAOmtACBiE8wyu8ORPnxXl3PpatBSefOYU/cD5/ADrxjRo=
Get
AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeCBz1WsB/YxrLZbtqlm03YnQil9F948aCIV/+NN/+NmzYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsYX2d++4lrI2J1j5OE+xEdKhEKRtFKDzccH3tMaFbqlytu1Z2DrBIvJxXI0eiXv3qDmKURV8gkNabruQn6U6pRMMlnpV5qeELZmA5511JFI2786fziGTmzyoCEsbalkMzV3xNTGhkziQLbGVEcmWUvE//zuimGV/5UqCRFrthiUZhKgjHJ3icDoTlDObGEMi3srYSNqKYMbUhZCN7yy6ukdVH13Kp3V6vUa3kcRTiBUzgHDy6hDrfQgCYwUPAMr/DmGOfFeXc+Fq0FJ585hj9wPn8A3z6QVQ==
AAAB9HicbVBNS8NAEN3Ur1q/qh69BIu0p5KIoMeCF48VbCu0oWw2k3bpZhN3J4ES+ju8eFDEqz/Gm//GbZuDtj4YeLw3w8w8PxFco+N8W6WNza3tnfJuZW//4PCoenzS1XGqGHRYLGL16FMNgkvoIEcBj4kCGvkCev7kdu73MlCax/IBpwl4ER1JHnJG0UjeIACBdJhnWb0+G1ZrTtNZwF4nbkFqpEB7WP0aBDFLI5DIBNW67zoJejlVyJmAWWWQakgom9AR9A2VNALt5YujZ/aFUQI7jJUpifZC/T2R00jraeSbzojiWK96c/E/r59ieOPlXCYpgmTLRWEqbIzteQJ2wBUwFFNDKFPc3GqzMVWUocmpYkJwV19eJ93Lpus03furWqtRxFEmZ+ScNIhLrkmL3JE26RBGnsgzeSVvVma9WO/Wx7K1ZBUzp+QPrM8fiY+R2Q==
vv 00
(J.S. Bach, 08/03/1714, German)
Get!
AAACBnicbVDLSsNAFJ34rPUVdSOIMFiECiVNtNAuiy4UVxXtA9pQJtNpO3QyCTMToYS6ceOvuHGhiFu/wZ1/47TNQlsPXDiccy/33uOFjEpl29/GwuLS8spqai29vrG5tW3u7NZkEAlMqjhggWh4SBJGOakqqhhphIIg32Ok7g0uxn79nghJA36nhiFxfdTjtEsxUlpqm4fZa+vWejhHuJ+zS3n7LO8UnULukggf8ZO2mbEtewI4T5yEZECCStv8anUCHPmEK8yQlE3HDpUbI6EoZmSUbkWShAgPUI80NeXIJ9KNJ2+M4LFWOrAbCF1cwYn6eyJGvpRD39OdPlJ9OeuNxf+8ZqS6JTemPIwU4Xi6qBsxqAI4zgR2qCBYsaEmCAuqb4W4jwTCSieX1iE4sy/Pk9qp5diWc1PIlPeTOFLgAByBLHBAEZTBFaiAKsDgETyDV/BmPBkvxrvxMW1dMJKZPfAHxucPAMmVgQ==
s0 s00
AAAB+HicbVBNSwMxEM3Wr1o/uurRS7AInsquFPRY8KDHCvYD2rVk02wbmk2WZFapS3+JFw+KePWnePPfmLZ70NYHA4/3ZpiZFyaCG/C8b6ewtr6xuVXcLu3s7u2X3YPDllGppqxJlVC6ExLDBJesCRwE6ySakTgUrB2Or2Z++4Fpw5W8g0nCgpgMJY84JWClvlu+ZnDf03w4AqK1euy7Fa/qzYFXiZ+TCsrR6LtfvYGiacwkUEGM6fpeAkFGNHAq2LTUSw1LCB2TIetaKknMTJDND5/iU6sMcKS0LQl4rv6eyEhszCQObWdMYGSWvZn4n9dNIboMMi6TFJiki0VRKjAoPEsBD7hmFMTEEkI1t7diOiKaULBZlWwI/vLLq6R1XvW9qn9bq9RreRxFdIxO0Bny0QWqoxvUQE1EUYqe0St6c56cF+fd+Vi0Fpx85gj9gfP5Ax/Xk1Y=
Get!
AAAB9XicbVBNS8NAEJ34WetX1aOXxSLtqSQi6LHgxWMF+wFtLJvNtF262YTdjVJC/4cXD4p49b9489+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8M/Pbj6g0j+W9mSToR3Qo+YAzaqz00AtRGNrPdEVXKtN+qezW3DnIKvFyUoYcjX7pqxfGLI1QGiao1l3PTYyfUWU4Ezgt9lKNCWVjOsSupZJGqP1sfvWUnFslJINY2ZKGzNXfExmNtJ5Ege2MqBnpZW8m/ud1UzO49jMuk9SgZItFg1QQE5NZBCTkCpkRE0soU9zeStiIKsqMDapoQ/CWX14lrYua59a8u8tyvZrHUYBTOIMqeHAFdbiFBjSBgYJneIU358l5cd6dj0XrmpPPnMAfOJ8/5MmSBA==
s 00 AAAB+HicbVBNSwMxEM3Wr1o/uurRS7AInsquFPRY8KDHCvYD2rVk02wbmk2WZFapS3+JFw+KePWnePPfmLZ70NYHA4/3ZpiZFyaCG/C8b6ewtr6xuVXcLu3s7u2X3YPDllGppqxJlVC6ExLDBJesCRwE6ySakTgUrB2Or2Z++4Fpw5W8g0nCgpgMJY84JWClvlu+ZnDf03w4AqK1euy7Fa/qzYFXiZ+TCsrR6LtfvYGiacwkUEGM6fpeAkFGNHAq2LTUSw1LCB2TIetaKknMTJDND5/iU6sMcKS0LQl4rv6eyEhszCQObWdMYGSWvZn4n9dNIboMMi6TFJiki0VRKjAoPEsBD7hmFMTEEkI1t7diOiKaULBZlWwI/vLLq6R1XvW9qn9bq9RreRxFdIxO0Bny0QWqoxvUQE1EUYqe0St6c56cF+fd+Vi0Fpx85gj9gfP5Ax/Xk1Y=
v 00
Get
AAAB6nicbVBNS8NAEJ34WetX1aOXxSL1VBIp6LHgxWNF+wFtKJvtpF262YTdTaGE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvSATXxnW/nY3Nre2d3cJecf/g8Oi4dHLa0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4bu63J6g0j+WTmSboR3QoecgZNVZ6nFQq/VLZrboLkHXi5aQMORr90ldvELM0QmmYoFp3PTcxfkaV4UzgrNhLNSaUjekQu5ZKGqH2s8WpM3JplQEJY2VLGrJQf09kNNJ6GgW2M6JmpFe9ufif101NeOtnXCapQcmWi8JUEBOT+d9kwBUyI6aWUKa4vZWwEVWUGZtO0Ybgrb68TlrXVc+teg+1cr2Wx1GAc7iAK/DgBupwDw1oAoMhPMMrvDnCeXHenY9l64aTz5zBHzifP6AfjU4=
AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBapp5JIQY8FLx4r2g9oQ9lsN+3SzSbsToQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IJHCoOt+O4WNza3tneJuaW//4PCofHzSNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNMbud+54lrI2L1iNOE+xEdKREKRtFKD6ZaHZQrbs1dgKwTLycVyNEclL/6w5ilEVfIJDWm57kJ+hnVKJjks1I/NTyhbEJHvGepohE3frY4dUYurDIkYaxtKSQL9fdERiNjplFgOyOKY7PqzcX/vF6K4Y2fCZWkyBVbLgpTSTAm87/JUGjOUE4toUwLeythY6opQ5tOyYbgrb68TtpXNc+teff1SqOex1GEMziHS/DgGhpwB01oAYMRPMMrvDnSeXHenY9la8HJZ07hD5zPH5uNjUs=
(J.S. Bach, 31/03/1685, German) AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeCBz1WsB/YxrLZbtqlm03YnQil9F948aCIV/+NN/+NmzYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsYX2d++4lrI2J1j5OE+xEdKhEKRtFKDzccH3tMaFbqlytu1Z2DrBIvJxXI0eiXv3qDmKURV8gkNabruQn6U6pRMMlnpV5qeELZmA5511JFI2786fziGTmzyoCEsbalkMzV3xNTGhkziQLbGVEcmWUvE//zuimGV/5UqCRFrthiUZhKgjHJ3icDoTlDObGEMi3srYSNqKYMbUhZCN7yy6ukdVH13Kp3V6vUa3kcRTiBUzgHDy6hDrfQgCYwUPAMr/DmGOfFeXc+Fq0FJ585hj9wPn8A3z6QVQ==
(J.S. Bach, German)
AAAB+3icbVDLSsNAFL2pr1pfsW4EN8EiVJCQiKDLogvFVUX7gDaUyXTSDp2ZhJmJWEL9FDcuFHHrj7jzb5w+Flo9cOFwzr3ce0+YMKq0531ZuYXFpeWV/GphbX1jc8veLtZVnEpMajhmsWyGSBFGBalpqhlpJpIgHjLSCAcXY79xT6SisbjTw4QEHPUEjShG2kgdu1i+dm/dx3OE+0eXRHIkDjt2yXO9CZy/xJ+REsxQ7dif7W6MU06Exgwp1fK9RAcZkppiRkaFdqpIgvAA9UjLUIE4UUE2uX3kHBil60SxNCW0M1F/TmSIKzXkoenkSPfVvDcW//NaqY7OgoyKJNVE4OmiKGWOjp1xEE6XSoI1GxqCsKTmVgf3kURYm7gKJgR//uW/pH7s+p7r35
sha1_base64="GxTkCr6VfunAo1wqdLC5n4PvkUU=">AAAB+3icbVDLSsNAFL2pr1pfsW4EN8EiVJCQiKDLogvFVUX7gDaUyXTSDp2ZhJmJWEL9FDcuFHHrj7jzb5w+Flo9cOFwzr3ce0+YMKq0531ZuYXFpeWV/GphbX1jc8veLtZVnEpMajhmsWyGSBFGBalpqhlpJpIgHjLSCAcXY79xT6SisbjTw4QEHPUEjShG2kgdu1i+dm/dx3OE+0eXRHIkDjt2yXO9CZy/xJ+REsxQ7dif7W6MU06Exgwp1fK9RAcZkppiRkaFdqpIgvAA9UjLUIE4UUE2uX3kHBil60SxNCW0M1F/TmSIKzXkoenkSPfVvDcW//NaqY7OgoyKJNVE4OmiKGWOjp1xEE6XSoI1GxqCsKTmVgf3kURYm7gKJgR//uW/pH7s+p7r35yUKruzOPKwB/tQBh9OoQJXUIUaYHiAJ3iBV2tkPVtv1vu0NWfNZnbgF6yPb+7FkvE=
AAACBnicbVDLSsNAFJ3UV62vqBtBhGARKpQ0sT66LLpQXFW0D2hDmUwn7dDJJMxMhBLqxo2/4saFIm79Bnf+jdM2C209cOFwzr3ce48bUiKkZX1rqbn5hcWl9HJmZXVtfUPf3KqJIOIIV1FAA95wocCUMFyVRFLcCDmGvktx3e1fjPz6PeaCBOxODkLs+LDLiEcQlEpq63u5a/PWfDiHqJcv2gWrWLBPSyf5S8x9yA7betYyrTGMWWInJAsSVNr6V6sToMjHTCIKhWjaViidGHJJEMXDTCsSOISoD7u4qSiDPhZOPH5jaBwopWN4AVfFpDFWf0/E0Bdi4Luq04eyJ6a9kfif14ykV3JiwsJIYoYmi7yIGjIwRpkYHcIxknSgCEScqFsN1IMcIqmSy6gQ7OmXZ0ntyLQt0745zpZ3kjjSYBfsgxywwRkogytQAVWAwCN4Bq/gTXvSXrR37WPSmtKSmW3wB9rnDwVVlYQ=
(C.P.E. Bach, 08/03/1714, German) (C.P.E. Bach, German)
AAAB/XicbVDLSgMxFM3UV62v8bEQ3ASLUEGGGRF0WSyiywr2Ae1QMmmmDU0yQ5IR6lD9FTcuFHHrf7jzb0zbWWjrgQuHc+7l3nuCmFGlXffbyi0sLi2v5FcLa+sbm1v29k5dRYnEpIYjFslmgBRhVJCappqRZiwJ4gEjjWBQGfuNeyIVjcSdHsbE56gnaEgx0kbq2HulilN1rpzHS4T7J9dEciSOO3bRddwJ4DzxMlIEGaod+6vdjXDCidCYIaVanhtrP0VSU8zIqNBOFIkRHqAeaRkqECfKTyfXj+CRUbowjKQpoeFE/T2RIq7UkAemkyPdV7PeWPzPayU6vPBTKuJEE4Gni8KEQR3BcRSwSyXBmg0NQVhScyvEfSQR1iawggnBm315ntRPHc91vNuzYnk/iyMPDsAhKAEPnIMyuAFVUAMYPIBn8ArerCfrxXq3PqatOSub2QV/YH3+AN/0k24=
AAACCHicbVDLSsNAFJ34rPUVdSO4MFiECiWdaKFdFovosoJ9QBvKZDpth04mYWYilFB3bvwVNy4UcesnuPNvnLZZaOuBC4dz7uXee7yQUakg/DaWlldW19ZTG+nNre2dXXNvvy6DSGBSwwELRNNDkjDKSU1RxUgzFAT5HiMNb1iZ+I17IiQN+J0ahcT1UZ/THsVIaaljHmcrdtW+sh8uER7kYCkPL/JO0SnkronwET/rmBlowymsReIkJAMSVDvmV7sb4MgnXGGGpGw5MFRujISimJFxuh1JEiI8RH3S0pQjn0g3nj4ytk610rV6gdDFlTVVf0/EyJdy5Hu600dqIOe9ifif14pUr+TGlIeRIhzPFvUiZqnAmqRidakgWLGRJggLqm+18AAJhJXOLq1DcOZfXiT1c9uBtnNbyJQPkzhS4AicgCxwQBGUwQ2oghrA4BE8g1fwZjwZL8a78TFrXTKSmQPwB8bnD/aYlf4=
Figure 2 Excerpt of de ta-based Get for the composers examp e
Categories must have a composition operator “;” with which deltas can be composed in an
associative manner (without caring about the order in which a chain of deltas is composed),
and an identity operator id that maps every object 𝑠 to an identity delta id (𝑠) : 𝑠 → 𝑠 that
represents doing nothing to 𝑠. Identity is expected to be compatible with composition: doing
nothing before or after doing something should make no difference, that is, id (𝑠) ; 𝛿𝑠𝑠′ = 𝛿𝑠𝑠′
and 𝛿𝑠′ 𝑠 ; id (𝑠) = 𝛿𝑠′ 𝑠 for all incoming deltas 𝛿𝑠𝑠′ : 𝑠 → 𝑠′ and outgoing deltas 𝛿𝑠′ 𝑠 : 𝑠′ → 𝑠.
Example 5. Back to Fig. 2, the composition of both deltas, i.e., adding C.P.E. Bach directly, is
depicted as the delta 𝛿𝑠𝑠′ ; 𝛿𝑠′ 𝑠′′ leading directly from 𝑠 to 𝑠′′ . Get ∘ maps 𝑠 to a view 𝑣 in 𝑉
containing J.S. Bach represented as a pair. The identity deltas are not shown in the figure, but
there is one from each state back to itself, representing a no-op.
As 𝑆 and 𝑉 are now categories, Get : 𝑆 → 𝑉 has to operate on both objects and deltas,
mapping source objects to view objects, and source deltas to view deltas. Get is now a so called
functor with two parts: Get ∘ , which maps sources to views, and Get → , which maps source
deltas 𝛿𝑠𝑠′ : 𝑠 → 𝑠′ to corresponding view deltas 𝛿𝑣𝑣′ : Get ∘ (𝑠) → Get ∘ (𝑠′ ). Such functors
must respect the composition and identity operators in the source and target categories, i.e.,
Get → (𝛿𝑠𝑠′ ; 𝛿𝑠′ 𝑠′′ ) = Get → (𝛿𝑠𝑠′ ); Get → (𝛿𝑠′ 𝑠′′ ) and Get → (id (𝑠)) = id (Get ∘ (𝑠)).
Example 6. Figure 2 contains a small excerpt of a delta-based Get : 𝑆 → 𝑉 for the composers
example. Mappings for Get ∘ are depicted as dashed “↦→” arrows between sets, mappings for
Get → as dotted “↦→” arrows between deltas. The functor Get is given by the entirety of all
dashed and dotted mappings. As we are using the variant of Get that avoids duplicates, Get ∘
also maps 𝑠′ to 𝑣. As the new composer becomes distinct in the view after his name is changed
to C.P.E. Bach, 𝑠′′ is mapped to a new set 𝑣 ′′ now with both J.S. Bach and C.P.E. Bach. As Get is
a functor, we also have to specify what happens to deltas: 𝛿𝑠𝑠′ is mapped to id (𝑣) as it does
not correspond to any change in the view. 𝛿𝑠′ 𝑠′′ is mapped to 𝛿𝑣𝑣′′ , i.e., in this case editing a
composer in the source corresponds to adding a composer in the view if the corresponding
composer becomes distinct. As functors must respect composition, 𝛿𝑠𝑠′ ; 𝛿𝑠′ 𝑠′′ must be mapped
to Get → (𝛿𝑠𝑠′ ; 𝛿𝑠′ 𝑠′′ ) = Get → (𝛿𝑠𝑠′ ); Get → (𝛿𝑠′ 𝑠′′ ) = id (𝑣); 𝛿𝑣𝑣′′ = 𝛿𝑣𝑣′′ .
To generalise Put to a functor, let
s0
us start by discussing the input for
Put ∘ , the object part of Put. From s P ut s AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipqQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/epYz3
AAAB8HicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Bj04jGCeUiyhtnJbDJkZnaZ6RVCyFd48aCIVz/Hm3/jJNmDJhY0FFXddHdFqRQWff/bW1ldW9/YLGwVt3d29/ZLB4cNm2SG8TpLZGJaEbVcCs3rKFDyVmo4VZHkzWh4M/WbT9xYkeh7HKU8VLSvRSwYRSc91DJ87DBhWLdU9iv+DGSZBDkpQ45at/TV6SUsU1wjk9TaduCnGI6pQcEknxQ7meUpZUPa521HNVXchuPZwRNy6pQeiRPjSiOZqb8nxlRZO1KR61QUB3bRm4r/ee0M46twLHSaIddsvijOJMGETL8nPWE4QzlyhDIj3K2EDaihDF1GRRdCsPjyMmmcVwK/EtxdlKvXeRwFOIYTOIMALqEKt1CDOjBQ8Ayv8OYZ78V79z7mrStePnMEf+B9/gDSeZBo
AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqMeiF48t2A9oQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ3dzvPKHSPJYPZpqgH9GR5CFn1FipqQflilt1FyDrxMtJBXI0BuWv/jBmaYTSMEG17nluYvyMKsOZwFmpn2pMKJvQEfYslTRC7WeLQ2fkwipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDGz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m5INwVt9eZ20r6qeW/Wa15X6bR5HEc7gHC7BgxrU4R4a0AIGCM/wCm/Oo/PivDsfy9aCk8+cwh84nz/epYz3
AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0mkoMeiF49V7Ae0oWy2m3bpZhN2J0IJ/QdePCji1X/kzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6xEnC/YgOlQgFo2ilB3PeL1fcqjsHWSVeTiqQo9Evf/UGMUsjrpBJakzXcxP0M6pRMMmnpV5qeELZmA5511JFI278bH7plJxZZUDCWNtSSObq74mMRsZMosB2RhRHZtmbif953RTDaz8TKkmRK7ZYFKaSYExmb5OB0JyhnFhCmRb2VsJGVFOGNpySDcFbfnmVtC6rnlv17muV+k0eRxFO4BQuwIMrqMMdNKAJDEJ4hld4c8bOi/PufCxaC04+cwx/4Hz+AD8njSg=
7 !
7 !
AAAB8nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBxEByhL3NXrJkb/fYnRPCkZ9hY6GIrb/Gzn/jJrlCEx8MPN6bYWZelEph0fe/vdLa+sbmVnm7srO7t39QPTxqW50ZxltMS206EbVcCsVbKFDyTmo4TSLJH6Px7cx/fOLGCq0ecJLyMKFDJWLBKDqp25NaDROaWtT9as2v+3OQVRIUpAYFmv3qV2+gWZZwhUxSa7uBn2KYU4OCST6t9DLLU8rGdMi7jiqacBvm85On5MwpAxJr40ohmau/J3KaWDtJIteZUBzZZW8m/ud1M4yvw1yoNEOu2GJRnEmCmsz+JwNhOEM5cYQyI9ythI2ooQxdShUXQrD88ippX9QDvx7cX9YaN0UcZTiBUziHAK6gAXfQhBYw0PAMr/DmoffivXsfi9aSV8wcwx94nz+7KpGI
7 ! AAAB8nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBxEByhL3NXrJkb/fYnRPCkZ9hY6GIrb/Gzn/jJrlCEx8MPN6bYWZelEph0fe/vdLa+sbmVnm7srO7t39QPTxqW50ZxltMS206EbVcCsVbKFDyTmo4TSLJH6Px7cx/fOLGCq0ecJLyMKFDJWLBKDqp25NaDROaWtT9as2v+3OQVRIUpAYFmv3qV2+gWZZwhUxSa7uBn2KYU4OCST6t9DLLU8rGdMi7jiqacBvm85On5MwpAxJr40ohmau/J3KaWDtJIteZUBzZZW8m/ud1M4yvw1yoNEOu2GJRnEmCmsz+JwNhOEM5cYQyI9ythI2ooQxdShUXQrD88ippX9QDvx7cX9YaN0UcZTiBUziHAK6gAXfQhBYw0PAMr/DmoffivXsfi9aSV8wcwx94nz+7KpGI
the state-based Put, we can surmise
that Put ∘ has to take at least a new
Get Get AAAB8HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRgx4r2A9pY9lsJ+3S3STsboQS+iu8eFDEqz/Hm//GbZuDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgkfYMNwIbCcKqQwEtoLR9dRvPaHSPI7uzThBX9JBxEPOqLHSww2axy7jivXKFbfqzkCWiZeTCuSo98pf3X7MUomRYYJq3fHcxPgZVYYzgZNSN9WYUDaiA+xYGlGJ2s9mB0/IiVX6JIyVrciQmfp7IqNS67EMbKekZqgXvan4n9dJTXjpZzxKUoMRmy8KU0FMTKbfkz5XyIwYW0KZ4vZWwoZUUWZsRiUbgrf48jJpnlU9t+rdnVdqV3kcRTiCYzgFDy6gBrdQhwYwkPAMr/DmKOfFeXc+5q0FJ585hD9wPn8Aq92QTw==
AAAB8nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBxEByhL3NXrJkb/fYnRPCkZ9hY6GIrb/Gzn/jJrlCEx8MPN6bYWZelEph0fe/vdLa+sbmVnm7srO7t39QPTxqW50ZxltMS206EbVcCsVbKFDyTmo4TSLJH6Px7cx/fOLGCq0ecJLyMKFDJWLBKDqp25NaDROaWtT9as2v+3OQVRIUpAYFmv3qV2+gWZZwhUxSa7uBn2KYU4OCST6t9DLLU8rGdMi7jiqacBvm85On5MwpAxJr40ohmau/J3KaWDtJIteZUBzZZW8m/ud1M4yvw1yoNEOu2GJRnEmCmsz+JwNhOEM5cYQyI9ythI2ooQxdShUXQrD88ippX9QDvx7cX9YaN0UcZTiBUziHAK6gAXfQhBYw0PAMr/DmoffivXsfi9aSV8wcwx94nz+7KpGI
AAAB8HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRgx4r2A9pY9lsJ+3S3STsboQS+iu8eFDEqz/Hm//GbZuDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgkfYMNwIbCcKqQwEtoLR9dRvPaHSPI7uzThBX9JBxEPOqLHSww2axy7jivXKFbfqzkCWiZeTCuSo98pf3X7MUomRYYJq3fHcxPgZVYYzgZNSN9WYUDaiA+xYGlGJ2s9mB0/IiVX6JIyVrciQmfp7IqNS67EMbKekZqgXvan4n9dJTXjpZzxKUoMRmy8KU0FMTKbfkz5XyIwYW0KZ4vZWwoZUUWZsRiUbgrf48jJpnlU9t+rdnVdqV3kcRTiCYzgFDy6gBrdQhwYwkPAMr/DmKOfFeXc+5q0FJ585hD9wPn8Aq92QTw==
0
view 𝑣 ′ and an old source 𝑠 as input.
′
v vv 0
! v0 v vv 0
! v AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW7Ae0oWy2k3btZhN2N4US+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikpeNUMWyyWMSqE1CNgktsGm4EdhKFNAoEtoPx/dxvT1BpHstHM03Qj+hQ8pAzaqzUmPTLFbfqLkDWiZeTCuSo98tfvUHM0gilYYJq3fXcxPgZVYYzgbNSL9WYUDamQ+xaKmmE2s8Wh87IhVUGJIyVLWnIQv09kdFI62kU2M6ImpFe9ebif143NeGtn3GZpAYlWy4KU0FMTOZfkwFXyIyYWkKZ4vZWwkZUUWZsNiUbgrf68jppXVU9t+o1riu1uzyOIpzBOVyCBzdQgweoQxMYIDzDK7w5T86L8+58LFsLTj5zCn/gfP4A4zGM+g==
AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0GPRi8cW7Ae0oWy2k3btZhN2N4US+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikpeNUMWyyWMSqE1CNgktsGm4EdhKFNAoEtoPx/dxvT1BpHstHM03Qj+hQ8pAzaqzUmPTLFbfqLkDWiZeTCuSo98tfvUHM0gilYYJq3fXcxPgZVYYzgbNSL9WYUDamQ+xaKmmE2s8Wh87IhVUGJIyVLWnIQv09kdFI62kU2M6ImpFe9ebif143NeGtn3GZpAYlWy4KU0FMTOZfkwFXyIyYWkKZ4vZWwkZUUWZsNiUbgrf68jppXVU9t+o1riu1uzyOIpzBOVyCBzdQgweoQxMYIDzDK7w5T86L8+58LFsLTj5zCn/gfP4A4zGM+g==
AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lE0GPRi8cq9gPaUDbbTbt0swm7k0IJ/QdePCji1X/kzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmjjVjDdYLGPdDqjhUijeQIGStxPNaRRI3gpGdzO/NebaiFg94SThfkQHSoSCUbTS4/i8V664VXcOskq8nFQgR71X/ur2Y5ZGXCGT1JiO5yboZ1SjYJJPS93U8ISyER3wjqWKRtz42fzSKTmzSp+EsbalkMzV3xMZjYyZRIHtjCgOzbI3E//zOimGN34mVJIiV2yxKEwlwZjM3iZ9oTlDObGEMi3srYQNqaYMbTglG4K3/PIqaV5WPbfqPVxVard5HEU4gVO4AA+uoQb3UIcGMAjhGV7hzRk5L86787FoLTj5zDH8gfP5A0O2jSs=
But 𝑣 and 𝑠 cannot be completely
AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lE0GPRi8cq9gPaUDbbTbt0swm7k0IJ/QdePCji1X/kzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmjjVjDdYLGPdDqjhUijeQIGStxPNaRRI3gpGdzO/NebaiFg94SThfkQHSoSCUbTS4/i8V664VXcOskq8nFQgR71X/ur2Y5ZGXCGT1JiO5yboZ1SjYJJPS93U8ISyER3wjqWKRtz42fzSKTmzSp+EsbalkMzV3xMZjYyZRIHtjCgOzbI3E//zOimGN34mVJIiV2yxKEwlwZjM3iZ9oTlDObGEMi3srYQNqaYMbTglG4K3/PIqaV5WPbfqPVxVard5HEU4gVO4AA+uoQb3UIcGMAjhGV7hzRk5L86787FoLTj5zDH8gfP5A0O2jSs=
AAACEXicbVA9SwNBEN3zM8avqKXNYRBThTsRtAzaWEYwH5ALYW8zSZbs7R67c5Fw3F+w8a/YWChia2fnv3HzUWjig4HHezPMzAtjwQ163rezsrq2vrGZ28pv7+zu7RcODutGJZpBjSmhdDOkBgSXUEOOApqxBhqFAhrh8GbiN0agDVfyHscxtCPal7zHGUUrdQqlwCBlQw0iDbogkHbS0egsy9JAKNnXvD9AqrV6yDqFolf2pnCXiT8nRTJHtVP4CrqKJRFIZIIa0/K9GNsp1ciZgCwfJAZiu5r2oWWppBGYdjr9KHNPrdJ1e0rbkuhO1d8TKY2MGUeh7YwoDsyiNxH/81oJ9q7aKZdxgiDZbFEvES4qdxKP2+UaGIqxJZRpbm912YBqytCGmLch+IsvL5P6edn3yv7dRbFyPY8jR47JCSkRn1ySCrklVVIjjDySZ/JK3pwn58V5dz5mrSvOfOaI/IHz+QMGnJ8F
AAACEXicbVA9SwNBEN3zM8avqKXNYRBThTsRtAzaWEYwH5ALYW8zSZbs7R67c5Fw3F+w8a/YWChia2fnv3HzUWjig4HHezPMzAtjwQ163rezsrq2vrGZ28pv7+zu7RcODutGJZpBjSmhdDOkBgSXUEOOApqxBhqFAhrh8GbiN0agDVfyHscxtCPal7zHGUUrdQqlwCBlQw0iDbogkHbS0egsy9JAKNnXvD9AqrV6yDqFolf2pnCXiT8nRTJHtVP4CrqKJRFIZIIa0/K9GNsp1ciZgCwfJAZiu5r2oWWppBGYdjr9KHNPrdJ1e0rbkuhO1d8TKY2MGUeh7YwoDsyiNxH/81oJ9q7aKZdxgiDZbFEvES4qdxKP2+UaGIqxJZRpbm912YBqytCGmLch+IsvL5P6edn3yv7dRbFyPY8jR47JCSkRn1ySCrklVVIjjDySZ/JK3pwn58V5dz5mrSvOfOaI/IHz+QMGnJ8F
unrelated. The input will only make
sense for Put ∘ if the new view 𝑣 ′ Figure 3: Input and output of Put ∘
resulted from a delta 𝛿𝑣𝑣′ applied to
an old view 𝑣 that corresponds to the old source 𝑠. This is formalised with the diagram depicted
in Fig. 3. The new view 𝑣 ′ must be the result of applying a view delta 𝛿𝑣𝑣′ to 𝑣 that must
be Get ∘ (𝑠). This is crucially important as there are settings where it is simply impossible to
get from some given 𝑣 to another 𝑣 ′ , i.e., no such corner with 𝑣 and 𝑣 ′ exists. This cannot
be expressed in a state-based framework. Furthermore, even if we are in a setting where it
is possible to get from every 𝑣 to every other 𝑣 ′ via some delta (or possibly multiple deltas),
it is still important to note that Put can treat each of these deltas as different corners. In a
state-based framework, the assumption is that it is possible to get from every 𝑣 to every other
𝑣 ′ in a unique way — think of the meaning of this delta as “discarding the original view and
changing to the target view”; therefore there is no need to bother with how to handle this
unique delta as it is fixed by the pair of connected objects.3
Example 7. Figure 4 provides a concrete example for Put ∘ . The view 𝑣 (containing J.S. Bach)
is extended by adding C.P.E. Bach to produce 𝑣 ′ . This corner is valid input for Put ∘ according
to Fig. 3, and Put ∘ produces a new source 𝑠′ consisting of J.S. Bach from the old source 𝑠 and
a new composer (C.P.E. Bach, ???, German) corresponding to the newly added (C.P.E. Bach,
German) in the view 𝑣 ′ .
s
AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkUI8FLx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbTu4XffUKleSwfzCxBP6JjyUPOqLFSSw/LFbfqLkE2iZeTCuRoDstfg1HM0gilYYJq3ffcxPgZVYYzgfPSINWYUDalY+xbKmmE2s+Wh87JlVVGJIyVLWnIUv09kdFI61kU2M6Imole9xbif14/NeGtn3GZpAYlWy0KU0FMTBZfkxFXyIyYWUKZ4vZWwiZUUWZsNiUbgrf+8ibp3FQ9t+q1apVGLY+jCBdwCdfgQR0acA9NaAMDhGd4hTfn0Xlx3p2PVWvByWfO4Q+czx/ab4zp
AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkUI8FLx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbTu4XffUKleSwfzCxBP6JjyUPOqLFSSw/LFbfqLkE2iZeTCuRoDstfg1HM0gilYYJq3ffcxPgZVYYzgfPSINWYUDalY+xbKmmE2s+Wh87JlVVGJIyVLWnIUv09kdFI61kU2M6Imole9xbif14/NeGtn3GZpAYlWy0KU0FMTBZfkxFXyIyYWUKZ4vZWwiZUUWZsNiUbgrf+8ibp3FQ9t+q1apVGLY+jCBdwCdfgQR0acA9NaAMDhGd4hTfn0Xlx3p2PVWvByWfO4Q+czx/ab4zp
AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0mkUI8FLx6r2A9oQ9lsJ+3SzSbsboQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ7dzvPKHSPJaPZpqgH9GR5CFn1FjpQV8OyhW36i5A1omXkwrkaA7KX/1hzNIIpWGCat3z3MT4GVWGM4GzUj/VmFA2oSPsWSpphNrPFpfOyIVVhiSMlS1pyEL9PZHRSOtpFNjOiJqxXvXm4n9eLzXhjZ9xmaQGJVsuClNBTEzmb5MhV8iMmFpCmeL2VsLGVFFmbDglG4K3+vI6aV9XPbfq3dcqjVoeRxHO4ByuwIM6NOAOmtACBiE8wyu8ORPnxXl3PpatBSefOYU/cD5/ADrxjRo=
(J.S. Bach, 31/03/1685, German)
AAACBnicbVDLSsNAFJ3UV62vqBtBhGARKpQ0sT66LLpQXFW0D2hDmUwn7dDJJMxMhBLqxo2/4saFIm79Bnf+jdM2C209cOFwzr3ce48bUiKkZX1rqbn5hcWl9HJmZXVtfUPf3KqJIOIIV1FAA95wocCUMFyVRFLcCDmGvktx3e1fjPz6PeaCBOxODkLs+LDLiEcQlEpq63u5a/PWfDiHqJcv2gWrWLBPSyf5S8x9yA7betYyrTGMWWInJAsSVNr6V6sToMjHTCIKhWjaViidGHJJEMXDTCsSOISoD7u4qSiDPhZOPH5jaBwopWN4AVfFpDFWf0/E0Bdi4Luq04eyJ6a9kfif14ykV3JiwsJIYoYmi7yIGjIwRpkYHcIxknSgCEScqFsN1IMcIqmSy6gQ7OmXZ0ntyLQt0745zpZ3kjjSYBfsgxywwRkogytQAVWAwCN4Bq/gTXvSXrR37WPSmtKSmW3wB9rnDwVVlYQ=
(J.S. Bach, 31/03/1685, German)
AAACBnicbVDLSsNAFJ3UV62vqBtBhGARKpQ0sT66LLpQXFW0D2hDmUwn7dDJJMxMhBLqxo2/4saFIm79Bnf+jdM2C209cOFwzr3ce48bUiKkZX1rqbn5hcWl9HJmZXVtfUPf3KqJIOIIV1FAA95wocCUMFyVRFLcCDmGvktx3e1fjPz6PeaCBOxODkLs+LDLiEcQlEpq63u5a/PWfDiHqJcv2gWrWLBPSyf5S8x9yA7betYyrTGMWWInJAsSVNr6V6sToMjHTCIKhWjaViidGHJJEMXDTCsSOISoD7u4qSiDPhZOPH5jaBwopWN4AVfFpDFWf0/E0Bdi4Luq04eyJ6a9kfif14ykV3JiwsJIYoYmi7yIGjIwRpkYHcIxknSgCEScqFsN1IMcIqmSy6gQ7OmXZ0ntyLQt0745zpZ3kjjSYBfsgxywwRkogytQAVWAwCN4Bq/gTXvSXrR37WPSmtKSmW3wB9rnDwVVlYQ=
(J.S. Bach, 31/03/1685, German)
AAACBnicbVDLSsNAFJ3UV62vqBtBhGARKpQ0sT66LLpQXFW0D2hDmUwn7dDJJMxMhBLqxo2/4saFIm79Bnf+jdM2C209cOFwzr3ce48bUiKkZX1rqbn5hcWl9HJmZXVtfUPf3KqJIOIIV1FAA95wocCUMFyVRFLcCDmGvktx3e1fjPz6PeaCBOxODkLs+LDLiEcQlEpq63u5a/PWfDiHqJcv2gWrWLBPSyf5S8x9yA7betYyrTGMWWInJAsSVNr6V6sToMjHTCIKhWjaViidGHJJEMXDTCsSOISoD7u4qSiDPhZOPH5jaBwopWN4AVfFpDFWf0/E0Bdi4Luq04eyJ6a9kfif14ykV3JiwsJIYoYmi7yIGjIwRpkYHcIxknSgCEScqFsN1IMcIqmSy6gQ7OmXZ0ntyLQt0745zpZ3kjjSYBfsgxywwRkogytQAVWAwCN4Bq/gTXvSXrR37WPSmtKSmW3wB9rnDwVVlYQ=
(C.P.E. Bach, ???, German)
AAACAnicbVDLSgMxFM34rPU16kZxEyxChTLMiKC7FovosoJ9QDuUTJppQ5PMkGSEMlQ3/oobF4q49Svc+Temj4W2HrhwOOde7r0niBlV2nW/rYXFpeWV1cxadn1jc2vb3tmtqSiRmFRxxCLZCJAijApS1VQz0oglQTxgpB70yyO/fk+kopG404OY+Bx1BQ0pRtpIbfsgX3YqzpXzcIlwr1AsFgvwmkiOxEnbzrmOOwacJ96U5MAUlbb91epEOOFEaMyQUk3PjbWfIqkpZmSYbSWKxAj3UZc0DRWIE+Wn4xeG8NgoHRhG0pTQcKz+nkgRV2rAA9PJke6pWW8k/uc1Ex1e+CkVcaKJwJNFYcKgjuAoD9ihkmDNBoYgLKm5FeIekghrk1rWhODNvjxPaqeO5zre7VmutD+NIwMOwRHIAw+cgxK4ARVQBRg8gmfwCt6sJ+vFerc+Jq0L1nRmD/yB9fkDR3KUqQ==
P ut
Get Get
AAAB8HicbVDLSgNBEOz1GeMr6tHLYBA8hV0R9Bj04jGCeUiyhtnJbDJkZnaZ6RVCyFd48aCIVz/Hm3/jJNmDJhY0FFXddHdFqRQWff/bW1ldW9/YLGwVt3d29/ZLB4cNm2SG8TpLZGJaEbVcCs3rKFDyVmo4VZHkzWh4M/WbT9xYkeh7HKU8VLSvRSwYRSc91DJ87DBhWLdU9iv+DGSZBDkpQ45at/TV6SUsU1wjk9TaduCnGI6pQcEknxQ7meUpZUPa521HNVXchuPZwRNy6pQeiRPjSiOZqb8nxlRZO1KR61QUB3bRm4r/ee0M46twLHSaIddsvijOJMGETL8nPWE4QzlyhDIj3K2EDaihDF1GRRdCsPjyMmmcVwK/EtxdlKvXeRwFOIYTOIMALqEKt1CDOjBQ8Ayv8OYZ78V79z7mrStePnMEf+B9/gDSeZBo
AAAB8XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeCBz1WsB/YxrLZbtqlm03YnQil9F948aCIV/+NN/+NmzYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsYX2d++4lrI2J1j5OE+xEdKhEKRtFKDzccH3tMaFbqlytu1Z2DrBIvJxXI0eiXv3qDmKURV8gkNabruQn6U6pRMMlnpV5qeELZmA5511JFI2786fziGTmzyoCEsbalkMzV3xNTGhkziQLbGVEcmWUvE//zuimGV/5UqCRFrthiUZhKgjHJ3icDoTlDObGEMi3srYSNqKYMbUhZCN7yy6ukdVH13Kp3V6vUa3kcRTiBUzgHDy6hDrfQgCYwUPAMr/DmGOfFeXc+Fq0FJ585hj9wPn8A3z6QVQ==
0
v
7 !
AAAB8nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBxEByhL3NXrJkb/fYnRPCkZ9hY6GIrb/Gzn/jJrlCEx8MPN6bYWZelEph0fe/vdLa+sbmVnm7srO7t39QPTxqW50ZxltMS206EbVcCsVbKFDyTmo4TSLJH6Px7cx/fOLGCq0ecJLyMKFDJWLBKDqp25NaDROaWtT9as2v+3OQVRIUpAYFmv3qV2+gWZZwhUxSa7uBn2KYU4OCST6t9DLLU8rGdMi7jiqacBvm85On5MwpAxJr40ohmau/J3KaWDtJIteZUBzZZW8m/ud1M4yvw1yoNEOu2GJRnEmCmsz+JwNhOEM5cYQyI9ythI2ooQxdShUXQrD88ippX9QDvx7cX9YaN0UcZTiBUziHAK6gAXfQhBYw0PAMr/DmoffivXsfi9aSV8wcwx94nz+7KpGI
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v0
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AAAB83icbVBNS8NAEN34WetX1aOXxSL2VBIR9Fjw4rGC/YAmlM1m2i7dbMLupFBC/4YXD4p49c9489+4bXPQ1gcDj/dmmJkXplIYdN1vZ2Nza3tnt7RX3j84PDqunJy2TZJpDi2eyER3Q2ZACgUtFCihm2pgcSihE47v535nAtqIRD3hNIUgZkMlBoIztJLvRyCR9fPJ5GrWr1TdursAXSdeQaqkQLNf+fKjhGcxKOSSGdPz3BSDnGkUXMKs7GcGUsbHbAg9SxWLwQT54uYZvbRKRAeJtqWQLtTfEzmLjZnGoe2MGY7MqjcX//N6GQ7uglyoNENQfLlokEmKCZ0HQCOhgaOcWsK4FvZWykdMM442prINwVt9eZ20r+ueW/ceb6qNWhFHiZyTC1IjHrklDfJAmqRFOEnJM3klb07mvDjvzseydcMpZs7IHzifPyT/kag=
vv 0 (J.S. Bach, German)
AAAB+3icbVDLSsNAFL2pr1pfsW4EN8EiVJCQiKDLogvFVUX7gDaUyXTSDp2ZhJmJWEL9FDcuFHHrj7jzb5w+Flo9cOFwzr3ce0+YMKq0531ZuYXFpeWV/GphbX1jc8veLtZVnEpMajhmsWyGSBFGBalpqhlpJpIgHjLSCAcXY79xT6SisbjTw4QEHPUEjShG2kgdu1i+dm/dx3OE+0eXRHIkDjt2yXO9CZy/xJ+REsxQ7dif7W6MU06Exgwp1fK9RAcZkppiRkaFdqpIgvAA9UjLUIE4UUE2uX3kHBil60SxNCW0M1F/TmSIKzXkoenkSPfVvDcW//NaqY7OgoyKJNVE4OmiKGWOjp1xEE6XSoI1GxqCsKTmVgf3kURYm7gKJgR//uW/pH7s+p7r35yUKruzOPKwB/tQBh9OoQJXUIUaYHiAJ3iBV2tkPVtv1vu0NWfNZnbgF6yPb+7FkvE=
AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0mkoMeCF49V7Ae0oWy2k3bpZhN2N4US+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikpeNUMWyyWMSqE1CNgktsGm4EdhKFNAoEtoPx3dxvT1BpHssnM03Qj+hQ8pAzaqz0OLnslytu1V2ArBMvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwJnpV6qMaFsTIfYtVTSCLWfLS6dkQurDEgYK1vSkIX6eyKjkdbTKLCdETUjverNxf+8bmrCWz/jMkkNSrZcFKaCmJjM3yYDrpAZMbWEMsXtrYSNqKLM2HBKNgRv9eV10rquem7Ve6hV6rU8jiKcwTlcgQc3UId7aEATGITwDK/w5oydF+fd+Vi2Fpx85hT+wPn8AT+AjR0=
AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeCF48t2A9oQ9lsJ+3azSbsbgol9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZW8epYthisYhVN6AaBZfYMtwI7CYKaRQI7AST+4XfmaLSPJaPZpagH9GR5CFn1FipOR2UK27VXYJsEi8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAuelfqoxoWxCR9izVNIItZ8tD52TK6sMSRgrW9KQpfp7IqOR1rMosJ0RNWO97i3E/7xeasI7P+MySQ1KtloUpoKYmCy+JkOukBkxs4Qyxe2thI2poszYbEo2BG/95U3Svql6btVr1ir1Wh5HES7gEq7Bg1uowwM0oAUMEJ7hFd6cJ+fFeXc+Vq0FJ585hz9wPn8A3vuM7A==
AAAB83icbVBNS8NAEN34WetX1aOXxSL2VBIR9Fjw4rGC/YAmlM1m2i7dbMLupFBC/4YXD4p49c9489+4bXPQ1gcDj/dmmJkXplIYdN1vZ2Nza3tnt7RX3j84PDqunJy2TZJpDi2eyER3Q2ZACgUtFCihm2pgcSihE47v535nAtqIRD3hNIUgZkMlBoIztJLvRyCR9fPJ5GrWr1TdursAXSdeQaqkQLNf+fKjhGcxKOSSGdPz3BSDnGkUXMKs7GcGUsbHbAg9SxWLwQT54uYZvbRKRAeJtqWQLtTfEzmLjZnGoe2MGY7MqjcX//N6GQ7uglyoNENQfLlokEmKCZ0HQCOhgaOcWsK4FvZWykdMM442prINwVt9eZ20r+ueW/ceb6qNWhFHiZyTC1IjHrklDfJAmqRFOEnJM3klb07mvDjvzseydcMpZs7IHzifPyT/kag=
vv 0 (J.S. Bach, German)
AAAB+3icbVDLSsNAFL2pr1pfsW4EN8EiVJCQiKDLogvFVUX7gDaUyXTSDp2ZhJmJWEL9FDcuFHHrj7jzb5w+Flo9cOFwzr3ce0+YMKq0531ZuYXFpeWV/GphbX1jc8veLtZVnEpMajhmsWyGSBFGBalpqhlpJpIgHjLSCAcXY79xT6SisbjTw4QEHPUEjShG2kgdu1i+dm/dx3OE+0eXRHIkDjt2yXO9CZy/xJ+REsxQ7dif7W6MU06Exgwp1fK9RAcZkppiRkaFdqpIgvAA9UjLUIE4UUE2uX3kHBil60SxNCW0M1F/TmSIKzXkoenkSPfVvDcW//NaqY7OgoyKJNVE4OmiKGWOjp1xEE6XSoI1GxqCsKTmVgf3kURYm7gKJgR//uW/pH7s+p7r35yUKruzOPKwB/tQBh9OoQJXUIUaYHiAJ3iBV2tkPVtv1vu0NWfNZnbgF6yPb+7FkvE=
AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0mkoMeCF49V7Ae0oWy2k3bpZhN2N4US+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikpeNUMWyyWMSqE1CNgktsGm4EdhKFNAoEtoPx3dxvT1BpHssnM03Qj+hQ8pAzaqz0OLnslytu1V2ArBMvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwJnpV6qMaFsTIfYtVTSCLWfLS6dkQurDEgYK1vSkIX6eyKjkdbTKLCdETUjverNxf+8bmrCWz/jMkkNSrZcFKaCmJjM3yYDrpAZMbWEMsXtrYSNqKLM2HBKNgRv9eV10rquem7Ve6hV6rU8jiKcwTlcgQc3UId7aEATGITwDK/w5oydF+fd+Vi2Fpx85hT+wPn8AT+AjR0=
(J.S. Bach, German) (C.P.E. Bach, German) (J.S. Bach, German)
AAAB+3icbVDLSsNAFL2pr1pfsW4EN8EiVJCQiKDLogvFVUX7gDaUyXTSDp2ZhJmJWEL9FDcuFHHrj7jzb5w+Flo9cOFwzr3ce0+YMKq0531ZuYXFpeWV/GphbX1jc8veLtZVnEpMajhmsWyGSBFGBalpqhlpJpIgHjLSCAcXY79xT6SisbjTw4QEHPUEjShG2kgdu1i+dm/dx3OE+0eXRHIkDjt2yXO9CZy/xJ+REsxQ7dif7W6MU06Exgwp1fK9RAcZkppiRkaFdqpIgvAA9UjLUIE4UUE2uX3kHBil60SxNCW0M1F/TmSIKzXkoenkSPfVvDcW//NaqY7OgoyKJNVE4OmiKGWOjp1xEE6XSoI1GxqCsKTmVgf3kURYm7gKJgR//uW/pH7s+p7r35yUKruzOPKwB/tQBh9OoQJXUIUaYHiAJ3iBV2tkPVtv1vu0NWfNZnbgF6yPb+7FkvE=
(C.P.E. Bach, German)
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Figure 4: Example for Put ∘
On our way to C- enses the cha enge is
now to work out what 𝑃 𝑢𝑡→ cou d operate
0
on How wou d a category ook ke w th “cor-
ss s̄ s̄
ner” objects shaped as the input for 𝑃 𝑢𝑡∘ in
0 P ut! ( ss̄ , v0 v̄0 )
s s
!
F g 3? How shou d we define de tas (arrows)
between these corners? As chance wou d
v̄ v̄ 0
have it categories with exact y such corner ! 0 v̄ v̄
!
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sha1_base64="G8hcDAHZIHE/stkijJXTYjkqexo=">AAAB7nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8cK9gPaUCbbTbt0swm7m0IJ/RFePCji1d/jzX/jts1BWx8MPN6bYWZemAqujed9O6WNza3tnfJuZW//4PDIPT5p6SRTlDVpIhLVCVEzwSVrGm4E66SKYRwK1g7H93O/PWFK80Q+mWnKghiHkkecorFSuxeiyiezvlv1at4CZJ34BalCgUbf/eoNEprFTBoqUOuu76UmyFEZTgWbVXqZZinSMQ5Z11KJMdNBvjh3Ri6sMiBRomxJQxbq74kcY62ncWg7YzQjverNxf+8bmai2yDnMs0Mk3S5KMoEMQmZ/04GXDFqxNQSpIrbWwkdoUJqbEIVG4K/+vI6aV3VfK/mP15X63dFHGU4g3O4BB9uoA4P0IAmUBjDM7zCm5M6L86787FsLTnFzCn8gfP5A6Duj78=
objects are a ready we -studied in category
theory This means we have a fu y worked
vv 0 0
out suggest on that we can adopt to a bx con- ! v 0 v̄ 0 v v
text (and dec de f t makes sense for us)
The bas c dea of the comma category con-
F gure 5 Input and output for Put →
struction is to create new categories by han-
d ing arrows as objects So in our case the
arrows that we want to treat as ob ects are the v ew de tas 𝛿𝑣𝑣 The construct on genera ses
th s dea by us ng functors to choose the exact arrows we are nterested n In the most genera
form one functor s used to choose the start of the arrow wh e a second functor chooses the
n ca egory heory he s a e-based se ng corresponds o cod scre e ca egor es and no d scre e ca egor es as
one m gh a se y assume
end of the arrow. Applying this idea to our “corners”, we can characterise our corner objects as
exactly those view deltas 𝛿𝑣𝑣′ , whose starting point is fixed by Get (remember Get is a functor),
and whose ending point is basically free. We can express this freeness formally by using 1𝑉
(the identity functor from 𝑉 to 𝑉 ) to “fix” the ending point of 𝛿𝑣𝑣′ .
With this we can now look up the standard definition of the comma category (Get, 1𝑉 ) and
obtain the shape for its objects and arrows as depicted in Fig. 5. An “arrow” in (Get, 1𝑉 ) is a
pair of arrows (𝛿𝑠𝑠¯ , 𝛿𝑣′ 𝑣¯′ ) between two corners (𝑠, 𝛿𝑣𝑣′ : 𝑣 → 𝑣 ′ ) and (𝑠¯, 𝛿𝑣¯𝑣¯′ : 𝑣¯ → 𝑣¯′ ) in light
grey squares. So this special “corner delta” in our category of corners is a pair of a source delta
𝛿𝑠𝑠¯ changing the old source 𝑠, and a view delta 𝛿𝑣′ 𝑣¯′ changing the new view 𝑣 ′ . As Get is a
functor, 𝛿𝑠𝑠¯ already fixes the corresponding view delta on the old view 𝑣, namely Get ∘ (𝛿𝑠𝑠¯ ). As
the bottom square must commute (according to the standard comma category construction), we
only accept such 𝛿𝑣′ 𝑣¯′ that “preserve” our corner structure, i.e., there must exist a suitable 𝛿𝑣¯𝑣¯′ .
Intuitively, think of the corner delta as a change to an input for Put ∘ that produces again
a valid input for Put ∘ , i.e., that produces a valid corner. The corner delta serves as input for
the functor Put → that now has to map the pair of source and view changes (𝛿𝑠𝑠¯ , 𝛿𝑣′ 𝑣¯′ ) to a
corresponding change Put → (𝛿𝑠𝑠 , 𝛿𝑣′ 𝑣¯′ ) of the updated source 𝑠 in the “pre-square” to obtain
the new updated source ¯𝑠′ , which completes the “post-square”. So we can think of Put → in
Fig. 5 as being able to handle a “small” perturbation of the bottom-left corner (𝑠 and 𝑣 ′ ) of a
consistent pre-square, by computing a corresponding update of the top-right part (𝑠′ ) of the
pre-square required to directly obtain a consistent post-square.
Example 8. A concrete example demonstrating input and output for Put → is depicted in
Fig. 6. The two grey rectangles contain squares that can both be interpreted as the combined
input (corner objects in (Get, 1𝑉 )) and output (objects in 𝑆) for Put ∘ . In the pre-square a
new composer (C.P.E. Bach, German) is added to the view and put back into the old source to
result in a new composer (C.P.E. Bach, ???, German) with an unknown date of birth. What the
example now demonstrates is that if the old source 𝑠 is changed by adding a composer (J.S. Bach,
08/03/1714, German) via 𝛿𝑠𝑠 , and the new view 𝑣 ′ is changed by removing (J.S. Bach, German)
via 𝛿𝑣′ 𝑣¯′ , these two changes can be passed to Put → to directly compute the delta required
to change the new source 𝑠′ to ¯𝑠′ . As Put is required to be a functor, the computed delta
Put → (𝛿𝑠𝑠 , 𝛿𝑣′ 𝑣¯′ ) must produce a source ¯𝑠′ that matches the post-square derived by applying
Put ∘ to the post corner.
While Put ∘ is clearly useful and can be intuitively understood as the straightforward delta-
based version of Put, it is not so clear if Put → is a good fit for bx. At first glance, Fig. 5 is
probably unexpected. Why do we have to handle a pair of changes to both the source and the
view? The reader’s intuition at this point is probably to map the view delta 𝛿𝑣𝑣′ to a source
delta 𝛿𝑠𝑠′ using Put → . While this would certainly be simpler, the problem is that the choice of
Put → (𝛿𝑠𝑠′ ) would not be “unique”, “special”, “minimal”, “least controversial”, or – as category
theorists would say in a general setting – universal when compared to other possible choices.
We shall see in Sect. 5 that providing the additional information of how to handle all possible
ways of changing corners (old source and corresponding new view) helps us arrive at a particular
Put behaviour that is unique in a specific sense and highly desirable in some scenarios.
(J.S. Bach, 31/03/1685, German)
s̄
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s̄0
AAAB/3icbVA9SwNBEJ2LXzF+RS1tFoNoFe5E0DJoYxnBfEByhN3NXrJkd+/Y3RPCkcK/YKu9ndj6U2z9JW6SK0zig4HHezPMzCOJ4Mb6/rdXWFvf2Nwqbpd2dvf2D8qHR00Tp5qyBo1FrNsEGya4Yg3LrWDtRDMsiWAtMrqb+q0npg2P1aMdJyyUeKB4xCm2Tmp3CdaZmZz3yhW/6s+AVkmQkwrkqPfKP91+TFPJlKUCG9MJ/MSGGdaWU8EmpW5qWILpCA9Yx1GFJTNhNrt3gs6c0kdRrF0pi2bq34kMS2PGkrhOie3QLHtT8T+vk9roJsy4SlLLFJ0vilKBbIymz6M+14xaMXYEU83drYgOscbUuogWthAiJy6UYDmCVdK8rAZ+NXi4qtRu83iKcAKncAEBXEMN7qEODaAg4AVe4c179t69D+9z3lrw8pljWID39QuEZJbr
(C.P.E. Bach, ???, German)
AAACBnicbVDLSsNAFJ3UV62vqBtBhGARKpQ0sT66LLpQXFW0D2hDmUwn7dDJJMxMhBLqxo2/4saFIm79Bnf+jdM2C209cOFwzr3ce48bUiKkZX1rqbn5hcWl9HJmZXVtfUPf3KqJIOIIV1FAA95wocCUMFyVRFLcCDmGvktx3e1fjPz6PeaCBOxODkLs+LDLiEcQlEpq63u5a/PWfDiHqJcv2gWrWLBPSyf5S8x9yA7betYyrTGMWWInJAsSVNr6V6sToMjHTCIKhWjaViidGHJJEMXDTCsSOISoD7u4qSiDPhZOPH5jaBwopWN4AVfFpDFWf0/E0Bdi4Luq04eyJ6a9kfif14ykV3JiwsJIYoYmi7yIGjIwRpkYHcIxknSgCEScqFsN1IMcIqmSy6gQ7OmXZ0ntyLQt0745zpZ3kjjSYBfsgxywwRkogytQAVWAwCN4Bq/gTXvSXrR37WPSmtKSmW3wB9rnDwVVlYQ=
AAACAnicbVDLSgMxFM34rPU16kZxEyxChTLMiKC7FovosoJ9QDuUTJppQ5PMkGSEMlQ3/oobF4q49Svc+Temj4W2HrhwOOde7r0niBlV2nW/rYXFpeWV1cxadn1jc2vb3tmtqSiRmFRxxCLZCJAijApS1VQz0oglQTxgpB70yyO/fk+kopG404OY+Bx1BQ0pRtpIbfsgX3YqzpXzcIlwr1AsFgvwmkiOxEnbzrmOOwacJ96U5MAUlbb91epEOOFEaMyQUk3PjbWfIqkpZmSYbSWKxAj3UZc0DRWIE+Wn4xeG8NgoHRhG0pTQcKz+nkgRV2rAA9PJke6pWW8k/uc1Ex1e+CkVcaKJwJNFYcKgjuAoD9ihkmDNBoYgLKm5FeIekghrk1rWhODNvjxPaqeO5zre7VmutD+NIwMOwRHIAw+cgxK4ARVQBRg8gmfwCt6sJ+vFerc+Jq0L1nRmD/yB9fkDR3KUqQ==
(J.S. Bach, 08/03/1714, German)
AAACBnicbVDLSsNAFJ34rPUVdSOIMFiECiVNtNAuiy4UVxXtA9pQJtNpO3QyCTMToYS6ceOvuHGhiFu/wZ1/47TNQlsPXDiccy/33uOFjEpl29/GwuLS8spqai29vrG5tW3u7NZkEAlMqjhggWh4SBJGOakqqhhphIIg32Ok7g0uxn79nghJA36nhiFxfdTjtEsxUlpqm4fZa+vWejhHuJ+zS3n7LO8UnULukggf8ZO2mbEtewI4T5yEZECCStv8anUCHPmEK8yQlE3HDpUbI6EoZmSUbkWShAgPUI80NeXIJ9KNJ2+M4LFWOrAbCF1cwYn6eyJGvpRD39OdPlJ9OeuNxf+8ZqS6JTemPIwU4Xi6qBsxqAI4zgR2qCBYsaEmCAuqb4W4jwTCSieX1iE4sy/Pk9qp5diWc1PIlPeTOFLgAByBLHBAEZTBFaiAKsDgETyDV/BmPBkvxrvxMW1dMJKZPfAHxucPAMmVgQ==
P ut! ( ss̄ , v0 v̄0 )
0
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
ss
v̄
AAAB7nicbVBNS8NAEJ3Ur1q/oh69LBbBU0lE0GPRi8cK9gPaUCbbTbt0swm7m0IJ/RFePCji1d/jzX/jts1BWx8MPN6bYWZemAqujed9O6WNza3tnfJuZW//4PDIPT5p6SRTlDVpIhLVCVEzwSVrGm4E66SKYRwK1g7H93O/PWFK80Q+mWnKghiHkkecorFSuxeiyiezvlv1at4CZJ34BalCgUbf/eoNEprFTBoqUOuu76UmyFEZTgWbVXqZZinSMQ5Z11KJMdNBvjh3Ri6sMiBRomxJQxbq74kcY62ncWg7YzQjverNxf+8bmai2yDnMs0Mk3S5KMoEMQmZ/04GXDFqxNQSpIrbWwkdoUJqbEIVG4K/+vI6aV3VfK/mP15X63dFHGU4g3O4BB9uoA4P0IAmUBjDM7zCm5M6L86787FsLTnFzCn8gfP5A6Duj78=
(C.P.E. Bach, German)
v̄
AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbRU0lE0GPRi8cK9gPaUDbbTbt0s4m7k0IJ/RNePCji1b/jzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmjjVjDdYLGPdDqjhUijeQIGStxPNaRRI3gpGdzO/NebaiFg94iThfkQHSoSCUbRSuxtQnY2n571yxa26c5BV4uWkAjnqvfJXtx+zNOIKmaTGdDw3QT+jGgWTfFrqpoYnlI3ogHcsVTTixs/m907JmVX6JIy1LYVkrv6eyGhkzCQKbGdEcWiWvZn4n9dJMbzxM6GSFLlii0VhKgnGZPY86QvNGcqJJZRpYW8lbEg1ZWgjKtkQvOWXV0nzsuq5Ve/hqlK7zeMowgmcwgV4cA01uIc6NICBhGd4hTfnyXlx3p2PRWvByWeO4Q+czx8ECI/w
AAAB/3icbVDNS8MwHE3n15xfVcGLl+AQPI1WBD0OvXic4D5gLSVN0y0sTUqSCqP24L/ixYMiXv03vPnfmG496OaDwOO990t+eWHKqNKO823VVlbX1jfqm42t7Z3dPXv/oKdEJjHpYsGEHIRIEUY56WqqGRmkkqAkZKQfTm5Kv/9ApKKC3+tpSvwEjTiNKUbaSIF95EWEaRTkyhMmV16Tq6II7KbTcmaAy8StSBNU6AT2lxcJnCWEa8yQUkPXSbWfI6kpZqRoeJkiKcITNCJDQzlKiPLz2f4FPDVKBGMhzeEaztTfEzlKlJomoUkmSI/VoleK/3nDTMdXfk55mmnC8fyhOGNQC1iWASMqCdZsagjCkppdIR4jibA2lTVMCe7il5dJ77zlOi337qLZvq7qqINjcALOgAsuQRvcgg7oAgwewTN4BW/Wk/VivVsf82jNqmYOwR9Ynz8/WJbi
(J.S. Bach, German)
AAAB+3icbVDLSsNAFL2pr1pfsW4EN8EiVJCQiKDLogvFVUX7gDaUyXTSDp2ZhJmJWEL9FDcuFHHrj7jzb5w+Flo9cOFwzr3ce0+YMKq0531ZuYXFpeWV/GphbX1jc8veLtZVnEpMajhmsWyGSBFGBalpqhlpJpIgHjLSCAcXY79xT6SisbjTw4QEHPUEjShG2kgdu1i+dm/dx3OE+0eXRHIkDjt2yXO9CZy/xJ+REsxQ7dif7W6MU06Exgwp1fK9RAcZkppiRkaFdqpIgvAA9UjLUIE4UUE2uX3kHBil60SxNCW0M1F/TmSIKzXkoenkSPfVvDcW//NaqY7OgoyKJNVE4OmiKGWOjp1xEE6XSoI1GxqCsKTmVgf3kURYm7gKJgR//uW/pH7s+p7r35yUKruzOPKwB/tQBh9OoQJXUIUaYHiAJ3iBV2tkPVtv1vu0NWfNZnbgF6yPb+7FkvE=
AAAB/XicbVDLSgMxFM3UV62v8bEQ3ASLUEGGGRF0WSyiywr2Ae1QMmmmDU0yQ5IR6lD9FTcuFHHrf7jzb0zbWWjrgQuHc+7l3nuCmFGlXffbyi0sLi2v5FcLa+sbm1v29k5dRYnEpIYjFslmgBRhVJCappqRZiwJ4gEjjWBQGfuNeyIVjcSdHsbE56gnaEgx0kbq2HulilN1rpzHS4T7J9dEciSOO3bRddwJ4DzxMlIEGaod+6vdjXDCidCYIaVanhtrP0VSU8zIqNBOFIkRHqAeaRkqECfKTyfXj+CRUbowjKQpoeFE/T2RIq7UkAemkyPdV7PeWPzPayU6vPBTKuJEE4Gni8KEQR3BcRSwSyXBmg0NQVhScyvEfSQR1iawggnBm315ntRPHc91vNuzYnk/iyMPDsAhKAEPnIMyuAFVUAMYPIBn8ArerCfrxXq3PqatOSub2QV/YH3+AN/0k24=
s
AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkUI8FLx5bsB/QhrLZTtq1m03Y3Qgl9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbTu4XffUKleSwfzCxBP6JjyUPOqLFSSw/LFbfqLkE2iZeTCuRoDstfg1HM0gilYYJq3ffcxPgZVYYzgfPSINWYUDalY+xbKmmE2s+Wh87JlVVGJIyVLWnIUv09kdFI61kU2M6Imole9xbif14/NeGtn3GZpAYlWy0KU0FMTBZfkxFXyIyYWUKZ4vZWwiZUUWZsNiUbgrf+8ibp3FQ9t+q1apVGLY+jCBdwCdfgQR0acA9NaAMDhGd4hTfn0Xlx3p2PVWvByWfO4Q+czx/ab4zp
AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0mkUI8FLx6r2A9oQ9lsJ+3SzSbsboQS+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikreNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoLJ7dzvPKHSPJaPZpqgH9GR5CFn1FjpQV8OyhW36i5A1omXkwrkaA7KX/1hzNIIpWGCat3z3MT4GVWGM4GzUj/VmFA2oSPsWSpphNrPFpfOyIVVhiSMlS1pyEL9PZHRSOtpFNjOiJqxXvXm4n9eLzXhjZ9xmaQGJVsuClNBTEzmb5MhV8iMmFpCmeL2VsLGVFFmbDglG4K3+vI6aV9XPbfq3dcqjVoeRxHO4ByuwIM6NOAOmtACBiE8wyu8ORPnxXl3PpatBSefOYU/cD5/ADrxjRo=
(J.S. Bach, 31/03/1685, German)
AAACBnicbVDLSsNAFJ3UV62vqBtBhGARKpQ0sT66LLpQXFW0D2hDmUwn7dDJJMxMhBLqxo2/4saFIm79Bnf+jdM2C209cOFwzr3ce48bUiKkZX1rqbn5hcWl9HJmZXVtfUPf3KqJIOIIV1FAA95wocCUMFyVRFLcCDmGvktx3e1fjPz6PeaCBOxODkLs+LDLiEcQlEpq63u5a/PWfDiHqJcv2gWrWLBPSyf5S8x9yA7betYyrTGMWWInJAsSVNr6V6sToMjHTCIKhWjaViidGHJJEMXDTCsSOISoD7u4qSiDPhZOPH5jaBwopWN4AVfFpDFWf0/E0Bdi4Luq04eyJ6a9kfif14ykV3JiwsJIYoYmi7yIGjIwRpkYHcIxknSgCEScqFsN1IMcIqmSy6gQ7OmXZ0ntyLQt0745zpZ3kjjSYBfsgxywwRkogytQAVWAwCN4Bq/gTXvSXrR37WPSmtKSmW3wB9rnDwVVlYQ=
(J.S. Bach, 31/03/1685, German)
AAACBnicbVDLSsNAFJ3UV62vqBtBhGARKpQ0sT66LLpQXFW0D2hDmUwn7dDJJMxMhBLqxo2/4saFIm79Bnf+jdM2C209cOFwzr3ce48bUiKkZX1rqbn5hcWl9HJmZXVtfUPf3KqJIOIIV1FAA95wocCUMFyVRFLcCDmGvktx3e1fjPz6PeaCBOxODkLs+LDLiEcQlEpq63u5a/PWfDiHqJcv2gWrWLBPSyf5S8x9yA7betYyrTGMWWInJAsSVNr6V6sToMjHTCIKhWjaViidGHJJEMXDTCsSOISoD7u4qSiDPhZOPH5jaBwopWN4AVfFpDFWf0/E0Bdi4Luq04eyJ6a9kfif14ykV3JiwsJIYoYmi7yIGjIwRpkYHcIxknSgCEScqFsN1IMcIqmSy6gQ7OmXZ0ntyLQt0745zpZ3kjjSYBfsgxywwRkogytQAVWAwCN4Bq/gTXvSXrR37WPSmtKSmW3wB9rnDwVVlYQ=
(C.P.E. Bach, ???, German)
AAACAnicbVDLSgMxFM34rPU16kZxEyxChTLMiKC7FovosoJ9QDuUTJppQ5PMkGSEMlQ3/oobF4q49Svc+Temj4W2HrhwOOde7r0niBlV2nW/rYXFpeWV1cxadn1jc2vb3tmtqSiRmFRxxCLZCJAijApS1VQz0oglQTxgpB70yyO/fk+kopG404OY+Bx1BQ0pRtpIbfsgX3YqzpXzcIlwr1AsFgvwmkiOxEnbzrmOOwacJ96U5MAUlbb91epEOOFEaMyQUk3PjbWfIqkpZmSYbSWKxAj3UZc0DRWIE+Wn4xeG8NgoHRhG0pTQcKz+nkgRV2rAA9PJke6pWW8k/uc1Ex1e+CkVcaKJwJNFYcKgjuAoD9ihkmDNBoYgLKm5FeIekghrk1rWhODNvjxPaqeO5zre7VmutD+NIwMOwRHIAw+cgxK4ARVQBRg8gmfwCt6sJ+vFerc+Jq0L1nRmD/yB9fkDR3KUqQ==
AAAB/XicbVDLSsNAFJ34rPUVHzs3g0XqqiQi6LLoxmUF+4AmhJvJpB06eTAzKdQQ/BU3LhRx63+482+ctllo64ELh3Pu5d57/JQzqSzr21hZXVvf2KxsVbd3dvf2zYPDjkwyQWibJDwRPR8k5SymbcUUp71UUIh8Trv+6Hbqd8dUSJbED2qSUjeCQcxCRkBpyTOPnYByBV4+rjs+iHxc1AvsmTWrYc2Al4ldkhoq0fLMLydISBbRWBEOUvZtK1VuDkIxwmlRdTJJUyAjGNC+pjFEVLr57PoCn2klwGEidMUKz9TfEzlEUk4iX3dGoIZy0ZuK/3n9TIXXbs7iNFM0JvNFYcaxSvA0ChwwQYniE02ACKZvxWQIAojSgVV1CPbiy8ukc9GwrYZ9f1lr3pRxVNAJOkXnyEZXqInuUAu1EUGP6Bm9ojfjyXgx3o2PeeuKUc4coT8wPn8AQQaVEw==
v 0 v̄ 0
AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoMeCF48t2A9oQ9lsJ+3azSbsbgol9Bd48aCIV3+SN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZW8epYthisYhVN6AaBZfYMtwI7CYKaRQI7AST+4XfmaLSPJaPZpagH9GR5CFn1FipOR2UK27VXYJsEi8nFcjRGJS/+sOYpRFKwwTVuue5ifEzqgxnAuelfqoxoWxCR9izVNIItZ8tD52TK6sMSRgrW9KQpfp7IqOR1rMosJ0RNWO97i3E/7xeasI7P+MySQ1KtloUpoKYmCy+JkOukBkxs4Qyxe2thI2poszYbEo2BG/95U3Svql6btVr1ir1Wh5HES7gEq7Bg1uowwM0oAUMEJ7hFd6cJ+fFeXc+Vq0FJ585hz9wPn8A3vuM7A==
AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0mkoMeCF49V7Ae0oWy2k3bpZhN2N4US+g+8eFDEq//Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikpeNUMWyyWMSqE1CNgktsGm4EdhKFNAoEtoPx3dxvT1BpHssnM03Qj+hQ8pAzaqz0OLnslytu1V2ArBMvJxXI0eiXv3qDmKURSsME1brruYnxM6oMZwJnpV6qMaFsTIfYtVTSCLWfLS6dkQurDEgYK1vSkIX6eyKjkdbTKLCdETUjverNxf+8bmrCWz/jMkkNSrZcFKaCmJjM3yYDrpAZMbWEMsXtrYSNqKLM2HBKNgRv9eV10rquem7Ve6hV6rU8jiKcwTlcgQc3UId7aEATGITwDK/w5oydF+fd+Vi2Fpx85hT+wPn8AT+AjR0=
v0
AAAB+3icbVDLSsNAFL2pr1pfsW4EN8EiVJCQiKDLogvFVUX7gDaUyXTSDp2ZhJmJWEL9FDcuFHHrj7jzb5w+Flo9cOFwzr3ce0+YMKq0531ZuYXFpeWV/GphbX1jc8veLtZVnEpMajhmsWyGSBFGBalpqhlpJpIgHjLSCAcXY79xT6SisbjTw4QEHPUEjShG2kgdu1i+dm/dx3OE+0eXRHIkDjt2yXO9CZy/xJ+REsxQ7dif7W6MU06Exgwp1fK9RAcZkppiRkaFdqpIgvAA9UjLUIE4UUE2uX3kHBil60SxNCW0M1F/TmSIKzXkoenkSPfVvDcW//NaqY7OgoyKJNVE4OmiKGWOjp1xEE6XSoI1GxqCsKTmVgf3kURYm7gKJgR//uW/pH7s+p7r35yUKruzOPKwB/tQBh9OoQJXUIUaYHiAJ3iBV2tkPVtv1vu0NWfNZnbgF6yPb+7FkvE=
(J.S. Bach, German)
AAAB+3icbVDLSsNAFL2pr1pfsW4EN8EiVJCQiKDLogvFVUX7gDaUyXTSDp2ZhJmJWEL9FDcuFHHrj7jzb5w+Flo9cOFwzr3ce0+YMKq0531ZuYXFpeWV/GphbX1jc8veLtZVnEpMajhmsWyGSBFGBalpqhlpJpIgHjLSCAcXY79xT6SisbjTw4QEHPUEjShG2kgdu1i+dm/dx3OE+0eXRHIkDjt2yXO9CZy/xJ+REsxQ7dif7W6MU06Exgwp1fK9RAcZkppiRkaFdqpIgvAA9UjLUIE4UUE2uX3kHBil60SxNCW0M1F/TmSIKzXkoenkSPfVvDcW//NaqY7OgoyKJNVE4OmiKGWOjp1xEE6XSoI1GxqCsKTmVgf3kURYm7gKJgR//uW/pH7s+p7r35yUKruzOPKwB/tQBh9OoQJXUIUaYHiAJ3iBV2tkPVtv1vu0NWfNZnbgF6yPb+7FkvE=
(C.P.E. Bach, German)
AAAB/XicbVDLSgMxFM3UV62v8bEQ3ASLUEGGGRF0WSyiywr2Ae1QMmmmDU0yQ5IR6lD9FTcuFHHrf7jzb0zbWWjrgQuHc+7l3nuCmFGlXffbyi0sLi2v5FcLa+sbm1v29k5dRYnEpIYjFslmgBRhVJCappqRZiwJ4gEjjWBQGfuNeyIVjcSdHsbE56gnaEgx0kbq2HulilN1rpzHS4T7J9dEciSOO3bRddwJ4DzxMlIEGaod+6vdjXDCidCYIaVanhtrP0VSU8zIqNBOFIkRHqAeaRkqECfKTyfXj+CRUbowjKQpoeFE/T2RIq7UkAemkyPdV7PeWPzPayU6vPBTKuJEE4Gni8KEQR3BcRSwSyXBmg0NQVhScyvEfSQR1iawggnBm315ntRPHc91vNuzYnk/iyMPDsAhKAEPnIMyuAFVUAMYPIBn8ArerCfrxXq3PqatOSub2QV/YH3+AN/0k24=
Figure 6: Example for Put →
4. Delta-based lens laws
Before we explain the advantages of Put → in Sect. 5, we still need to generalise the lens laws
depicted as commutative diagrams in Fig. 1. We will do so for PutGet and GetPut; as the
third law PutPut is not directly required for the rest of the paper, we shall skip it and refer the
interested reader to Johnson et al. [2, 3] for the rather technical details.
To provide delta-based versions of PutGet and GetPut we need two simple functors: 𝑄𝑉 :
(Get, 1𝑉 ) → 𝑉 to extract normal views from our comma category, and 𝜂Get : 𝑆 → (Get, 1𝑉 )
to inject sources into the comma category. As these are functors, we have to think in terms of
both objects and arrows as depicted in Fig. 7. 𝑄∘𝑉 extracts the new view 𝑣 ′ from a corner object,
while 𝑄→ ∘
𝑉 extracts the delta between views of two connected corners. 𝜂Get injects a source 𝑠
∘ → operates
into the comma category by extending it to a corner object using Get and 𝑖𝑑. 𝜂Get
∘ →
similarly on a source delta, using Get , Get and id to extend it to a pair of connected corners
(an arrow between corners).
Using 𝑄 and 𝜂Get we can now formulate delta-based versions of PutGet and GetPut as
depicted in Fig. 8. Compared with the state-based diagrams in Fig. 1, the only difference here is
that 𝑉 × 𝑆 is now generalised to our comma category (Get, 1𝑉 ). Consequently, extracting the
ss s̄
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s
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7 ! AAAB8nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5E0DJoYxnBxEByhL3NXrJkb/fYnRPCkZ9hY6GIrb/Gzn/jJrlCEx8MPN6bYWZelEph0fe/vdLa+sbmVnm7srO7t39QPTxqW50ZxltMS206EbVcCsVbKFDyTmo4TSLJH6Px7cx/fOLGCq0ecJLyMKFDJWLBKDqp25NaDROaWtT9as2v+3OQVRIUpAYFmv3qV2+gWZZwhUxSa7uBn2KYU4OCST6t9DLLU8rGdMi7jiqacBvm85On5MwpAxJr40ohmau/J3KaWDtJIteZUBzZZW8m/ud1M4yvw1yoNEOu2GJRnEmCmsz+JwNhOEM5cYQyI9ythI2ooQxdShUXQrD88ippX9QDvx7cX9YaN0UcZTiBUziHAK6gAXfQhBYw0PAMr/DmoffivXsfi9aSV8wcwx94nz+7KpGI
QV s
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0 v̄ v̄ 0
Get v ! v̄ 0 v̄ 0
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v̄
7 !
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vv 0 0
v !v
v vv 0
! v
0
v0
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v 0 v̄ 0
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AAAB/XicbVDLSsNAFJ34rPUVHzs3g0XqqiQi6LLoxmUF+4AmhJvJpB06eTAzKdQQ/BU3LhRx63+482+ctllo64ELh3Pu5d57/JQzqSzr21hZXVvf2KxsVbd3dvf2zYPDjkwyQWibJDwRPR8k5SymbcUUp71UUIh8Trv+6Hbqd8dUSJbED2qSUjeCQcxCRkBpyTOPnYByBV4+rjs+iHxc1AvsmTWrYc2Al4ldkhoq0fLMLydISBbRWBEOUvZtK1VuDkIxwmlRdTJJUyAjGNC+pjFEVLr57PoCn2klwGEidMUKz9TfEzlEUk4iX3dGoIZy0ZuK/3n9TIXXbs7iNFM0JvNFYcaxSvA0ChwwQYniE02ACKZvxWQIAojSgVV1CPbiy8ukc9GwrYZ9f1lr3pRxVNAJOkXnyEZXqInuUAu1EUGP6Bm9ojfjyXgx3o2PeeuKUc4coT8wPn8AQQaVEw==
AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbRU0lE0GPRi8cq9gPaUDbbTbt0swm7k0IJ/QdePCji1X/kzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmjjVjDdYLGPdDqjhUijeQIGStxPNaRRI3gpGdzO/NebaiFg94SThfkQHSoSCUbTS4/i8V664VXcOskq8nFQgR71X/ur2Y5ZGXCGT1JiO5yboZ1SjYJJPS93U8ISyER3wjqWKRtz42fzSKTmzSp+EsbalkMzV3xMZjYyZRIHtjCgOzbI3E//zOimGN34mVJIiV2yxKEwlwZjM3iZ9oTlDObGEMi3srYQNqaYMbTglG4K3/PIqaV5WPbfqPVxVard5HEU4gVO4AA+uoQb3UIcGMAjhGV7hzRk5L86787FoLTj5zDH8gfP5A0O2jSs=
v v̄
s̄
ss
s
!
⌘Get s̄ s id(v̄)
s ! Get ss ! = v̄ ! v̄
!
⌘Get
id(v) ! !
s =
!
v ! v G
!
id(v) G
v ! v
Figure 7 V ew extract on and n ect on
v ew n PutGet now requ res 𝑄𝑉 as a genera sat on of 𝜋0 and ft ng the source uses 𝜂Ge as a
genera sat on of the funct on (Get, 1𝑆 ) used n F g 1 When “read ng” the d agrams note that
a arrows are functors with an object and a corresponding de ta component By tracing the
commutat ve paths n the d agrams equat ons for both ob ects and de tas can be comp ed
PutGet GetPut
P ut ⌘Get
(Get, V) S S (Get, V)
= =
QV Get S P ut
V S
Figure 8 De ta-based ens aws
Examp e 9 Here we ver fy a concrete nstance of PutGet for the composers examp e by trac ng
the PutGet d agram n F g 8 — both ts ob ect and de ta parts — on the spec fic nput shown n
Fig 4 and 6 For the object part we start with the corner (𝑠, 𝛿𝑣𝑣 ) on the eft of Fig 4 which
s mapped to 𝑠 on the r ght of F g 4 by Put ∘ We then app y Get ∘ to 𝑠 y e d ng { (J S Bach,
German), (C P E Bach, German) } If PutGet ho ds we shou d a so get th s state f we app y 𝑄∘𝑉
to the nput corner n F g 4 and ndeed what 𝑄∘𝑉 does s extract the new v ew component 𝑣
which is the same as the state we arrive at through the Put ; Get path For the de ta part
the input is the pair of de tas (𝛿𝑠𝑠¯ , 𝛿𝑣 𝑣¯ ) between the corner objects (𝑠, 𝛿𝑣𝑣 ) and (𝑠¯, 𝛿𝑣¯𝑣¯ ) in
Fig 6 This de ta pair is first mapped by Put → to the source de ta Put → (𝛿𝑠𝑠¯ , 𝛿𝑣 𝑣¯ ) between
𝑠 = Put ∘ (𝑠, 𝛿𝑣𝑣 ) and ¯𝑠 = Put ∘ (𝑠¯, 𝛿𝑣¯𝑣¯ ) de et ng J S Bach Th s source de ta s then mapped
by Get → to the view delta between Get ∘ (𝑠′ ) = { (J.S. Bach, German), (C.P.E. Bach, German) }
and Get ∘ (𝑠¯′ ) = { (C.P.E. Bach, German) }, also deleting J.S. Bach. As this view delta is precisely
𝛿𝑣′ 𝑣¯′ , the result of applying 𝑄→
𝑉 to the input, the delta part of PutGet is also verified.
5. Universal updates
We are now ready to argue from a practical point of view why the delta-based bx framework
according to Johnson et al. could be a suitable formal basis for building bx tools. One point
we have left open at the end of Sect. 3 is why the apparent complexity of Put → is justified.
Indeed, instead of just requiring mappings of view deltas to source deltas as one might expect,
the framework requires information about pairs of view and source deltas as depicted in Fig. 5
and Fig. 6. One justification for this additional specification effort is that the resulting source
deltas for the provided view deltas are universal in the sense that any other valid source delta
(with respect to functoriality and the lens laws) can be obtained by composing the universal
delta with another uniquely determined delta. Due to the uniqueness condition, universal deltas
are essentially unique in the sense that they are all just different representations of the same
change, so a bx tool is free to produce any representation of a universal delta without worrying
about accidentally implementing a different change.
To explain this in more detail, let us complete the square formed by Put∘ . Up until now, we
have refrained from connecting the new 𝑠′ with the old source 𝑠 and new view 𝑣 ′ (see Fig. 3
and Fig. 4). We now fill in the missing arrows in the following (we refer the interested reader to
Johnson et al. [3] for all technical details we skip). As depicted in Fig. 9 it is helpful to start with
𝑖𝑑(𝑣)
the corner 𝑠 ↦→ 𝑣 → 𝑣. As this corner is exactly how 𝑠 is injected into the corner category
∘ , we can apply GetPut on objects and surmise that the result of applying Put ∘ to the
by 𝜂Get
corner must be 𝑠 (as indicated in the figure). As this is the same 𝑠 as before, we know that
it maps via Get ∘ to 𝑣. We now apply our actual view delta of interest 𝛿𝑣𝑣′ to 𝑣 to obtain 𝑣 ′ .
𝛿 ′
The corner 𝑠 ↦→ 𝑣 → 𝑣𝑣
𝑣 ′ maps to 𝑠′ via Put ∘ . To obtain information about the connection
′
between 𝑠 and 𝑠 we now extend 𝛿𝑣𝑣′ to a pair of deltas between corners by pairing it with
doing nothing to 𝑠, i.e., with id (𝑠) as depicted in Fig. 9. Providing this as input to Put → yields
Put → (id (𝑠), 𝛿𝑣𝑣′ ) connecting 𝑠 and 𝑠′ . After rotating the “cube” as indicated in the figure, we
obtain our actual square of interest now with the explicit connection between 𝑠 and 𝑠′ . As a
final step, we can argue with PutGet ∘ that Get ∘ applied to 𝑠′ must yield 𝑣 ′ , as this is what is
𝛿 ′
extracted via 𝑄∘𝑉 from the input corner 𝑠 ↦→ 𝑣 → 𝑣𝑣
𝑣 ′ that yielded 𝑠′ via Put ∘ . This all leads to
the final square depicted to the far right of Fig. 9. Indeed for Get and Put that obey all three
delta-based lens laws, also referred to as “C-lenses” [3], one is free to think in terms of such
squares representing how view deltas 𝛿𝑣𝑣′ are to be put to source deltas 𝛿𝑠𝑠′ computed using
the provided Put as Put → (id (𝑠), 𝛿𝑣𝑣′ ).
So why do we need to know how to handle pairs of source and view deltas to corners as
depicted in Fig. 5? Why not simply focus on the view deltas that are relevant for bx and only
provide an implementation of Put → for 𝛿𝑣𝑣′ always paired with an unchanged source, i.e., id (𝑠)?
The answer is related to the quest for constructively/computationally justifying the choices of
source deltas produced by Put. As depicted in Fig. 10, the source delta Put → (id (𝑠), 𝛿𝑣𝑣′ ) is the
id(s) s
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!
!
v ! v0
vv 0 Get =
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7 !
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Figure 9 F ng n a arrows n the square
P ut! (id(s), vv0 ) P ut! ( ss00 ,id(v 0 ))
s ! s0 ! s00
!
!
v vv 0
! v0
Figure 10 Source updates are east change updates
un versa source de ta that we (as bx deve opers) have dec ded to fix for our current cho ce of
Put This choice is justified in the sense that it sti fu y retains the possibi ities for the user
to get other va id resu ts (in unique ways) because every other va id source de ta 𝛿𝑠𝑠 (with
respect to the ens aws) that a so comp etes the square can be factored nto the un versa source
de ta and then Put → (𝛿𝑠 𝑠 , id (𝑣 )) where the atter represents exp c t y the un que add t ona
source de ta that the user can app y to the updated source 𝑠 produced by the too to get the
source 𝑠 that the user actua y wants As with any imp ementation of a universa property
the extra work of comput ng a other poss b t es prov des a computat ona “proof” for why a
certa n va d cand date s not the cho ce we have made Th s does not on y prov de for forma
c ar ty but can a so be used as the forma bas s for exp aining to users why the r synchron sers
have made certa n cho ces and not others
We shou d emphas se that un versa ty s not a ways usefu — whether and how t s usefu
depends cruc a y on the sett ng ( n part cu ar the a owed source de tas) n wh ch we nstant ate
the abstract categor ca defin t on of C- enses In some cases un versa ty may guarantee noth ng
at a whereas n some other cases un versa ty can g ve r se to a h gh y ntu t ve and des rab e
property guarantee ng that a bx too produces the east source change needed for a v ew change
One sett ng where un versa ty s not usefu s the state-based sett ng wh ch as we br efly
mentioned in Sect 3 can be seen as a specia case of the de ta-based setting where there is
exact y one de ta from any source/v ew to another whose mean ng s “d scard ng the or g na
source/v ew and chang ng to the target source/v ew” In th s sett ng any source de ta produced
by Put s un versa because wh chever source s produced t can a ways be further updated ( n
a unique way) to any other source, in particular the one that the user actually wants.4 Therefore,
what universality guarantees is already implied by the underlying assumption of the setting,
and becomes trivial.
We can argue a bit more generally that universality may become trivial when deltas can be
undone by other deltas.5 Intuitively, universality requires that a bx tool commit source changes
cautiously so that it is still possible to get to other valid sources. If committed changes can
be easily undone, however, then the tool does not need to be cautious — it can produce any
valid delta 𝛿𝑠𝑠′ and claim that it is universal, because when the user actually wants some other
delta 𝛿𝑠𝑠′′ and invokes universality to request an additional delta 𝛿𝑠′ 𝑠′′ , the tool can simply
produce a delta that undoes 𝛿𝑠𝑠′ and then applies 𝛿𝑠𝑠′′ . In the state-based setting, every delta can
be seen as undoing (discarding) the effect of all previous deltas, so the above argument applies.
On the other hand, if we turn the above argument around, we can identify settings where
universality is useful. For example, in a setting where we only allow deltas representing insertion
into a database (and not deletion or any other operations that could undo the effect of insertion),
a bx tool guaranteeing universality will have to produce exactly the absolutely necessary source
insertions, i.e., the least change — if the tool inserted some redundant entry and the user wanted
an updated database without that entry, then it would be impossible for the tool to produce a
delta to delete the entry; so, to satisfy universality, the tool must not insert the entry in the first
place. Real-world situations can be more delicate though, as the following example indicates.
Example 10. Back to the composers example, consider the instance in Fig. 4, where the view
delta 𝛿𝑣𝑣′ to be handled is an insertion. We can fill in the missing source delta 𝛿𝑠𝑠′ between 𝑠
and 𝑠′ by setting 𝛿𝑠𝑠′ = Put → (id (𝑠), 𝛿𝑣𝑣′ ), which represents the insertion of (C.P.E. Bach, ???,
German) into the source multiset. As both 𝛿𝑣𝑣′ and 𝛿𝑠𝑠′ are insertions, this instance is valid
in the insertion-only setting. Is 𝛿𝑠𝑠′ universal in this setting? Let us try some other possible
updated sources 𝑠′′ and deltas 𝛿𝑠𝑠′′ and see if 𝛿𝑠𝑠′′ can be factored through 𝛿𝑠𝑠′ . One possibility
of 𝛿𝑠𝑠′′ is redundantly inserting C.P.E. Bach twice, yielding the updated source 𝑠′′ = { (J.S. Bach,
31/03/1685, German), (C.P.E. Bach, ???, German), (C.P.E. Bach, ???, German) }. In this case 𝛿𝑠𝑠′′
can indeed be factored as 𝛿𝑠𝑠′ ; 𝛿𝑠′ 𝑠′′ where 𝛿𝑠′ 𝑠′′ represents the (redundant) insertion of the
second C.P.E. Bach. So far so good, but we will see that the situation is in fact more delicate if
we consider another possible source delta 𝛿𝑠𝑠′′′ that inserts (C.P.E. Bach, 08/03/1714, German)
which includes the specific date of birth, leading to the updated source 𝑠′′′ = { (J.S. Bach,
31/03/1685, German), (C.P.E. Bach, 08/03/1714, German) }. This time 𝛿𝑠𝑠′′′ cannot be factored
into 𝛿𝑠𝑠′ ; 𝛿𝑠′ 𝑠′′′ for any insertion delta 𝛿𝑠′ 𝑠′′′ , so 𝛿𝑠𝑠′ is in fact not universal in the insertion-only
setting. However, if we allow one additional kind of source delta that modifies a date of birth
in our setting, then we can choose 𝛿𝑠′ 𝑠′′′ to be the one that changes C.P.E. Bach’s date of birth
from ??? to 08/03/1714, making 𝛿𝑠𝑠′ universal (in this instance). The Put behaviour is not fully
determined by universality even in this amended setting though — we could have filled in any
4
In a bit more detail: suppose that, in response to a view change, Put decides to change an original source 𝑠
to a new source 𝑠′ , represented by the delta 𝛿𝑠𝑠′ (“change to 𝑠′ ”); for any other delta 𝛿𝑠𝑠′′ (“change to 𝑠′′ ”) where
𝑠′′ is also a valid result for the view change, we have 𝛿𝑠𝑠′′ = 𝛿𝑠𝑠′ ; 𝛿𝑠′ 𝑠′′ (“changing to 𝑠′′ is the same as changing
to 𝑠′ and then to 𝑠′′ ”), where 𝛿𝑠′ 𝑠′′ is automatically the unique way of updating 𝑠′ to 𝑠′′ in the state-based setting,
so 𝛿𝑠𝑠′ is universal.
5
This may-statement is deliberately conservative because there is the subtle issue about the uniqueness condi-
tion in the definition of universality, which we do not touch from now on.
specific date of birth when we inserted C.P.E. Bach into 𝑠′ and still remained universal, because
that specific date can always be changed to something else later. While determining such choices
only up to isomorphism (i.e., not caring about the specific initial date) can be advantageous, we
could fully determine the Put behaviour by restricting the date-of-birth modification deltas
to only those that change ??? to a specific date. To satisfy universality, Put must then use
??? for the dates of birth in newly-inserted entries, because ??? is the only date that can be
changed to any other dates the user actually wants. For a more general handling of such
“missing information” in structures, we refer the interested reader to Johnson et al. [16, 17] for
the categorical modelling of databases with incomplete data.
6. Related work
According to Johnson and Rosebrugh [4], there are essentially three classes of lenses: state-based
(set-based) lenses, delta-based lenses, and edit-based [18] lenses. We have discussed in this
paper how state-based lenses can be viewed as a special case of delta-based lenses (for codiscrete
categories). Edit lenses can also be represented as delta lenses via a suitable functor [4].
As state-based and edit lenses originate from the programming language community, they
typically have a strong connection to practical tooling and integration in programming lan-
guages. In the following, we discuss the situation for delta lenses and the existing work that
can be compared to our goal of bridging C-lenses and practical bx.
Diskin et al. [5] introduce D-lenses as a generalisation of the state-based bx framework to a
delta-based setting. Compared to C-lenses, D-lenses do exactly what one would expect, namely
map view deltas to source deltas. This means that C-lenses turn out to be essentially special
D-lenses with the universal property discussed in Sect. 5; every C-lens is a D-lens but not every
D-lens is a C-lens [4].
Concerning the practical relevance of D-lenses, Hermann et al. [12] have explored the relation
between D-lenses [5] and Triple Graph Grammars (TGGs) [13]. This relation is further explored
in a tutorial-like fashion by Anjorin [19] as part of the 2016 bx summer school [20]. The result
of this line of work indicates that TGGs can be regarded as an implementation of D-lenses
under a series of additional assumptions. In essence, a TGG rule can be directly interpreted
as a pair of corresponding source and target deltas. An important point is that TGG rules are
applied by checking for matches of their preconditions as graph patterns. This means that a
relatively compact specification consisting of a few TGG rules actually describes an infinite
number of concrete delta mappings for Put. While we do not (yet) see how TGG rules can
be directly used to specify C-lenses, one could take an analogous rule-based approach that
leverages graph pattern matching to enable a compact, finite specification of Get and Put. To
specify how all possible pairs of deltas are to be handled for a specific corner, amalgamated
TGG rules [21] could be explored; the fixed kernel rule would specify the universal choice, while
multiple complementary rules would be used to describe how to handle all possible source deltas.
The challenging of guaranteeing least change synchronisers has been studied in detail by
Cheney et al. [22]. While C-lenses certainly do not completely solve the least change problem,
we believe they provide a helpful formal framework in which least change requirements can be
systematically studied and precisely characterised.
Anjorin and Cheney [23] have started exploring connections between provenance and bx, with
TGGs as a concrete bx language. While they identify why-provenance with TGG correspondence
graphs, and how-provenance with the derivation sequence produced when applying TGG rules,
why-not-provenance [24] is left open. As the universal property of C-lenses can be used to
explain why a certain source delta is not produced by Put, this could be viewed as a means of
providing why-not provenance.
For state-based tools/languages developed in the programming languages community, the for-
mal guarantee about the behaviour of synchronisation programs is usually just well-behavedness,
following the precedent of Foster et al.’s lens combinators [6]. However, it is clear that well-
behavedness alone does not guarantee that users get the synchronisation behaviour they desire,
as analysed by, for example, [9]. Consequently, these tools usually strive to provide a good
range of features so as to be as expressive as possible, with one latest attempt being Matsuda
and Wang’s [10]; complementarily, Ko and Hu [9] developed a program logic with which users
can reason about and determine the bidirectional behaviour of their programs. The overall
direction is giving users adequate expressive power to write down the synchronisation pro-
grams they have in mind. For specific problem domains, however, one could argue that this
expressive power is too low-level and becomes more a burden, and that users should instead
be provided with higher-level or more declarative solutions where most, if not all, of the users’
effort is devoted to describing just the synchronisation problems/requirements, as opposed to
designing the synchronisation programs and managing the implementation details. But to be
declarative, the tools must guarantee stronger properties from which users can deduce that
the synchronisation behaviour is good enough for their purposes without looking into the
implementation details. The universality of C-lenses could be a candidate for such properties:
while universality can be trivial for e.g., codiscrete categories, it might be possible to impose
more structure and restrictions on the source category in a way similar to Ex. 10. In such a
setting where universality is meaningful, we could start all over again and work out a new
generation of lens combinators that guarantee universality compositionally.
On a final note, some indication for the ubiquity of C-lenses and their potential for practical
bx tooling can be observed by investigating existing applications that have essentially converged
to C-lenses without being aware of this (see Johnson et al. [25] for a discussion of this). For
example, the common view updating mechanisms used in the database world depend on so-
called “constant complement updates” which can be viewed in a bx context as the Puts of
C-lenses [26].
7. Conclusion and future work
In this paper, we provide a practical introduction to C-lenses for readers without any substantial
background in category theory. We start with the accessible state-based approach to bx and gen-
eralise it step-by-step to a delta-based setting using multiple examples. We provide explanations
and a practical discussion of the main concepts to help establish an intuitive understanding for
C-lenses and their potential benefit as a formal basis for a new generation of bx tools.
While we already mention some preliminary ideas for how C-lenses could be specified in a
practical bx tool, an open challenge to be addressed in the future is how the lens laws can be
automatically checked by such a tool. Finally, our primary focus in this paper was how formal
results can be leveraged in a practical implementation. Conversely, there are also extensions
to formal foundations that are yet to be worked out in full detail but are already required by
practical approaches such as handling non-determinism and resolving conflicts in the context
of concurrent synchronisation.
References
[1] K. Czarnecki, J. N. Foster, Z. Hu, R. Lämmel, A. Schürr, J. Terwilliger, Bidirectional
Transformations: A Cross-Discipline Perspective, in: R. F. Paige (Ed.), ICMT 2009, volume
5563 of LNCS, Springer, 2009, pp. 260–283.
[2] M. Johnson, R. D. Rosebrugh, Lens put-put laws: monotonic and mixed, Electron. Commun.
Eur. Assoc. Softw. Sci. Technol. 49 (2012).
[3] M. Johnson, R. D. Rosebrugh, Delta lenses and opfibrations, Electron. Commun. Eur. Assoc.
Softw. Sci. Technol. 57 (2013).
[4] M. Johnson, R. D. Rosebrugh, Unifying set-based, delta-based and edit-based lenses,
in: A. Anjorin, J. Gibbons (Eds.), Bx 2016, volume 1571 of CEUR Workshop Proceedings,
CEUR-WS.org, 2016, pp. 1–13.
[5] Z. Diskin, Y. Xiong, K. Czarnecki, H. Ehrig, F. Hermann, F. Orejas, From state- to delta-
based bidirectional model transformations: The symmetric case, in: J. Whittle, T. Clark,
T. Kühne (Eds.), MODELS 2011, volume 6981 of LNCS, Springer, 2011, pp. 304–318.
[6] J. N. Foster, M. B. Greenwald, J. T. Moore, B. C. Pierce, A. Schmitt, Combinators for
bidirectional tree transformations: A linguistic approach to the view-update problem,
ACM Transactions on Programming Languages and Systems 29 (2007) 17:1–65.
[7] A. Bohannon, J. N. Foster, B. C. Pierce, A. Pilkiewicz, A. Schmitt, Boomerang: resourceful
lenses for string data, in: G. C. Necula, P. Wadler (Eds.), POPL 2008, ACM, 2008, pp.
407–419.
[8] S. Hidaka, Z. Hu, K. Inaba, H. Kato, K. Nakano, GRoundTram: An integrated framework
for developing well-behaved bidirectional model transformations, Progress in Informatics
10 (2013) 131–148.
[9] H.-S. Ko, Z. Hu, An axiomatic basis for bidirectional programming, Proceedings of the
ACM on Programming Languages 2 (2018) 41:1–29.
[10] K. Matsuda, M. Wang, HOBiT: Programming lenses without using lens combinators, in:
ESOP, volume 10801 of LNCS, Springer, 2018, pp. 31–59.
[11] A. Anjorin, T. Buchmann, B. Westfechtel, Z. Diskin, H. Ko, R. Eramo, G. Hinkel, L. Samimi-
Dehkordi, A. Zündorf, Benchmarking bidirectional transformations: theory, implementa-
tion, application, and assessment, Softw. Syst. Model. 19 (2020) 647–691.
[12] F. Hermann, H. Ehrig, F. Orejas, K. Czarnecki, Z. Diskin, Y. Xiong, S. Gottmann, T. Engel,
Model synchronization based on triple graph grammars: correctness, completeness and
invertibility, Softw. Syst. Model. 14 (2015) 241–269.
[13] A. Schürr, Specification of graph translators with triple graph grammars, in: E. W. Mayr,
G. Schmidt, G. Tinhofer (Eds.), WG ’94, volume 903 of LNCS, Springer, 1994, pp. 151–163.
[14] P. Stevens, J. McKinna, J. Cheney, Composers v0.1 in Bx Examples Repository, http:
//bx-community.wikidot.com/examples:home, 2010. Date retrieved: July 9, 2021.
[15] M. Johnson, R. D. Rosebrugh, Symmetric delta lenses and spans of asymmetric delta lenses,
J. Object Technol. 16 (2017) 2:1–32.
[16] M. Johnson, R. Rosebrugh, Three approaches to partiality in the sketch data model, in:
CATS, volume 78 of ENTCS, Elsevier, 2003, pp. 82–99.
[17] M. Johnson, S. Kasangian, A relational model of incomplete data without NULLs, in: CATS,
volume 109 of Conferences in Research and Practice in Information Technology, Australian
Computer Society, 2010, pp. 89–94.
[18] M. Hofmann, B. C. Pierce, D. Wagner, Edit lenses, in: J. Field, M. Hicks (Eds.), POPL 2012,
ACM, 2012, pp. 495–508.
[19] A. Anjorin, An introduction to triple graph grammars as an implementation of the
delta-lens framework, in: J. Gibbons, P. Stevens (Eds.), Bidirectional Transformations -
International Summer School, Oxford, UK, July 25-29, 2016, Tutorial Lectures, volume 9715
of LNCS, Springer, 2016, pp. 29–72.
[20] J. Gibbons, P. Stevens (Eds.), Bidirectional Transformations - International Summer School,
Oxford, UK, July 25-29, 2016, Tutorial Lectures, volume 9715 of LNCS, Springer, 2018.
[21] E. Leblebici, A. Anjorin, A. Schürr, G. Taentzer, Multi-amalgamated triple graph grammars:
Formal foundation and application to visual language translation, J. Vis. Lang. Comput. 42
(2017) 99–121.
[22] J. Cheney, J. Gibbons, J. McKinna, P. Stevens, On principles of least change and least
surprise for bidirectional transformations, J. Object Technol. 16 (2017) 3:1–31.
[23] A. Anjorin, J. Cheney, Provenance meets bidirectional transformations, in: TaPP 2019,
USENIX Association, 2019.
[24] A. Chapman, H. V. Jagadish, Why not?, in: U. Çetintemel, S. B. Zdonik, D. Kossmann,
N. Tatbul (Eds.), SIGMOD 2009, ACM, 2009, pp. 523–534.
[25] M. Johnson, R. D. Rosebrugh, R. J. Wood, Lenses, fibrations and universal translations,
Math. Struct. Comput. Sci. 22 (2012) 25–42.
[26] M. Johnson, R. Rosebrugh, Constant complements, reversibility and universal view updates,
in: J. Meseguer, G. Roşu (Eds.), Algebraic Methodology and Software Technology, Springer
Berlin Heidelberg, 2008, pp. 238–252.