Recent zbMATH articles in MSC 18N65https://zbmath.org/atom/cc/18N652024-03-13T18:33:02.981707ZWerkzeugLax monoidal adjunctions, two-variable fibrations and the calculus of mateshttps://zbmath.org/1528.180222024-03-13T18:33:02.981707Z"Haugseng, Rune"https://zbmath.org/authors/?q=ai:haugseng.rune"Hebestreit, Fabian"https://zbmath.org/authors/?q=ai:hebestreit.fabian"Linskens, Sil"https://zbmath.org/authors/?q=ai:linskens.sil"Nuiten, Joost"https://zbmath.org/authors/?q=ai:nuiten.joost-jakobThis paper provides a calculus of mates for functors to the \(\infty\)-category of \(\infty\)-categories, extending Lurie's unstraightening equivalences [\textit{J. Lurie}, ``(Infinity,2)-Categories and the Goodwillie Calculus I'', Preprint, \url{arXiv:0905.0462}] to show that (op)lax natural transformations correspond to maps of (co)cartesian fibrations that do not necessarily preserve (co)cartesian edges. As the technical heart of the paper, various new types of fibrations over a product of two \(\infty\)-categories are studied, particularly showing how they can be dualized over one of the two factors and how they encode functors out of the Gray tensor product of \(\left( \infty,2\right) \)-categories. As a sample application, an equivalence between lax symmetric monoidal structures on right adjoint functors and oplax symmetric monoidal structures on the left adjoint fnctors between symmetric monoidal \(\infty \)-categories that is compatible with both horizontal and vertical composition of such structures is obtained.
The synopsis of the paper goes as follows.
\begin{itemize}
\item[\S 2] introduces curved orthofibrations and Gray fibrations in more detail, establishing their basic properties. It is shown as Theorem 2.5.1 that
Theorem. There is a natural equivalence of \(\infty\)-categories
\begin{align*}
\mathrm{Gray}\left( A,B\right) & \simeq\mathrm{CrvOrtho}\left( A^{\mathrm{op}},B\right) \\
\mathrm{OpGray}\left( A,B\right) & \simeq\mathrm{CrvOrtho}\left( A,B^{\mathrm{op}}\right)
\end{align*}
where \(\mathrm{Gray}\left( A,B\right) \)\ and \(\mathrm{OpGray}\left( A,B\right) \)\ are the \(\infty\)-category of Gray fibrations and op-Gray fibrations over \(A\times B\), respectively, whereas \(\mathrm{CrvOrtho}\left( A^{\mathrm{op}},B\right) \)\ is the \(\infty\)-category of curved orthofibrations over \(A^{\mathrm{op}}\times B\) and in both cases, the morphisms are required to preserve the defining (co)cartesian morphisms.
An equivalence of this kind was first established by \textit{D. Stevenson} [``Model Structures for Correspondences and Bifibrations'', Preprint, \url{arXiv:1807.08226}] by different means.
\item[\S 3] introduces and studies parametrized adjunctions in fibrational form. It is established as Theorem 3.1.11 that
Theorem. Let \(B\)\ be an \(\infty\)-category. Then there is an equivalence of \(\left( \infty,2\right) \)-categories
\[
\mathrm{Cocart}^{\mathrm{lax,R}}\left( B\right) \simeq\left( \mathrm{Cart} ^{\mathrm{opl,L}}\left( B^{\mathrm{op}}\right) \right) ^{\left( 1,2\right) \mathrm{-op}}
\]
extracting adjoints fiberwise, where the left-hand side denotes the \(\left( \infty,2\right) \)-category with cocartesian fibrations over \(B\)\ as objects, fiberwise right adjoint functors as 1-morphisms, and natural transformations between these 2-morphisms, while the right-hand side is defined dually, using cartesian fibrations and fiberwise right adjoints, with the directions of 1- and 2-morphisms reversed by the superscript. Furthermore, these equivalences are natural in pulling back along the base.
It is established as Theorem 3.4.7 that
Theorem. For any \(\infty\)-operad \(O\), the extraction of adjoints gives a canonical equivalence of \(\left( \infty,2\right) \)-categories
\[
\mathrm{MonCat}_{O}^{\mathrm{lax,R}}\simeq\left( \mathrm{MonCat} _{O}^{\mathrm{opl,L}}\right) ^{\left( 1,2\right) \mathrm{-op}}
\]
where the left-hand side denotes the \(\left( \infty,2\right) \)-category of \(O\)-monoidal \ \(\infty\)-categories, lax \(O\)-monoidal functors admitting (objectwise) left adjoints and \(O\)-monoidal transformations, while the right-hand side is defined dually using oplax \(O\)-monoidal functors admitting right adjoints. Furthermore, these equivalences are natural in pulling back along operad maps in the base.
A special case of this theorem was established by \textit{V. Hinich} [Doc. Math. 20, 879--926 (2015; Zbl 1339.18013)].
It is established as Corollary 3.4.8 that
Corollary. Given two symmetric monoidal \(\infty\)-categories \(C\)\ and \(D\), the extraction of adjoints gives inverse equivalences between the \(\infty\)-category of lax symmetric monoidal right adjoints \(C^{\otimes}\rightarrow D^{\otimes}\)\ and the opposite of the \(\infty\)-category of oplax monoidal left adjoints \(C_{\otimes}\rightarrow D_{\otimes}\).
\item[\S 4] addresses units and counits for parametrized adjunctions, deriving the functoriality of the passage to adjoint morphisms in the parametrized context.
\item[\S 5] establishes the connection to lax natural transformations, establishing the following theorem as a combination of Corollary 5.2.10 and Theorem 5.3.1 thanks to Lurie's locally cocartesian unstraightening equivalence [\textit{J. Lurie}, ``(Infinity,2)-Categories and the Goodwillie Calculus I'', Preprint, \url{arXiv:0905.0462}].
Theorem. There are natural equivalences of \(\infty\)-categories
\[
\mathrm{Gray}\left( A,B\right) \simeq\left( A\boxtimes B,\boldsymbol{Cat} \right)
\]
together with natural equivalences of \(\left( \infty,2\right) \)-categories
\begin{align*}
\mathrm{Cocart}^{\mathrm{lax}}\left( B\right) & \simeq\mathrm{Fun} ^{\mathrm{lax}}\left( B,\boldsymbol{Cat}\right) \\
\mathrm{Cart}^{\mathrm{opl}}\left( B\right) & \simeq\mathrm{Fun} ^{\mathrm{opl}}\left( B^{\mathrm{op}},\boldsymbol{Cat}\right)
\end{align*}
given on objects by straightening of (co)cartesian fibrations.
The following reformulation of Theorem 3.1.11 is obtained as Theorem 5.3.5 thanks to the above theorem.
Theorem. Extracting adjoints gives an equivalence of \(\left( \infty,2\right) \)-categories
\[
\mathrm{Fun}^{\mathrm{lax,R}}\left( B,\boldsymbol{Cat}\right) \simeq\left( \mathrm{Fun}^{\mathrm{opl,L}}\left( B,\boldsymbol{Cat}\right) \right) ^{\left( 1,2\right) \mathrm{-op}}
\]
for every \(\infty\)-category \(B\), where the superscript \(\mathrm{Fun} ^{\mathrm{lax}}\left( B,\boldsymbol{Cat}\right) \)\ denotes the locally full (or 1-full) sub-2-category of \ spanned by those lax natural transformations admitting pointwise left adjoints, and dually for the right-hand side. Furthermore, these equivalences are natural for restriction in \(B\).
\end{itemize}
Theorems 1.1, 1.2 and 1.3 from [\textit{R. Haugseng} et al., Proc. Lond. Math. Soc. (3) 127, No. 4, 889--957 (2023; Zbl 07785229)] are now contained Theorem 5.3.5, Corollary 3.4.8 and Proposition 3.4.9, respectively, while Theorems A, B and C from [\textit{R. Haugseng} et al., Forum Math. Sigma 11, Paper No. e111, 70 p. (2023; Zbl 07781651)] are now part of Corollary 3.4.8, Theorem 2.5.1 and Theorem 3.4.7, respectively.
Reviewer: Hirokazu Nishimura (Tsukuba)