Delta lenses were introduced in 2011 in [https://link.springer.com/chapter/10.1007/978-3-642-24485-8_22] as a framework for bidirectional transformations [F. Abou-Saleh et al., Lect. Notes Comput. Sci. 9715, 1–28 (2018; Zbl 1476.68034)]. Johnson and Rosenbrugh [https://eceasst.org/index.php/eceasst/article/view/2090] initiated the study of delta lenses using category theory.
This paper considers a way of comparing delta lenses and split opfibrations: the class of functors that they lift against. These are the twisted coreflections and split coreflections, respectively.
The synopsis of the paper goes as follows.

§ 1

recalls the concepts required to state the definition of an algebraic weak factorization system (AWFS).

§ 2

recalls the notion of delta lense [https://link.springer.com/chapter/10.1007/978-3-642-24485-8_22; P. J. Freyd and G. M. Kelly, J. Pure Appl. Algebra 2, 169–191 (1972; Zbl 0257.18005)], constructing a thin double category \(\mathbb{L}\mathrm{ens}\)of categories, functors and delta lenses [https://bryceclarke.github.io/other/the-double-category-of-lenses.pdf]. It is shown (Theorem 2.6) that the double category of delta lenses is isomorphic to the double category \(\mathbb{R}\mathrm{LP}\left( \mathbb{J}\right) \)of right lifts for a small double category \(\mathbb{J}\). It is also shown that has tabulators, which is exploited to show that a delta lens is equivalent to a commutative diagram of functors \[ \begin{array} [c]{ccc} X & \overset{\varphi}{\longrightarrow} & A\\ & \underset{f\varphi}{\searrow} & \downarrow_{f}\\ & & B \end{array} \] where \(\varphi\)is bijective-on-objects and \(f\varphi\)is a discrete opfibration.

§ 3

introduces a special kind of split coreflection, called a twisted coreflection, constructing the double category \(\mathbb{T}\mathrm{wCoref}\)of categories, functors and twisted coreflections. It is shown (Theorem 3.19) that a split coreflection is a twisted coreflection iff it satisfies a certain pushout condition. It is also shown that a twisted coreflection is equivalent to a pushout square in \(\mathcal{C}\mathrm{at}\) \[ \begin{array} [c]{ccc} A_{0} & \overset{\iota_{A}}{\longrightarrow} & A\\ f^{\prime}\downarrow & & \downarrow f\\ X & \underset{\pi}{\longrightarrow} & B \end{array} \] where \(A_{0}\)is a discrete category , \(\iota_{A}\)is identity-on-objects, and \(f^{\prime}\)is an initial functor.

§ 4

establishes that twisted coreflections and delta lenses form an algebraic weak factorization system on \(\mathcal{C}\mathrm{at}\). The proof is divided into three steps.

(1)

The author describes a lifting operation on the following cospan of double functors \[ \mathbb{T}\mathrm{wCoref}\overset{U}{\mathrm{\longrightarrow}}\mathbb{S} \mathrm{q}\left( \mathcal{C}\mathrm{at}\right) \overset{V}{\longleftarrow }\mathbb{L}\mathrm{ens} \]

(2)

It is shown that each functor admits a factorization as a cofree twisted coreflection followed by a free delta lens.

(3)

It is shown that the induced double functors \begin{align*} \mathbb{T}\mathrm{wCoref} & \longrightarrow\mathbb{L}\mathrm{LP}\left( \mathbb{L}\mathrm{ens}\right) \\ \mathbb{L}\mathrm{ens} & \longrightarrow\mathbb{R}\mathrm{LP}\left( \mathbb{T}\mathrm{wCoref}\right) \end{align*} are invertible.

§ 5

outlines directions for future work.