nLab lax extranatural transformation

Contents

Context

Category theory

Enriched category theory

Limits and colimits

Contents

Idea

A lax extranatural transformation is to a lax transformation as an extranatural transformation is to a natural transformation.

Definition

Let 𝒜\mathcal{A}, \mathcal{B}, 𝒞\mathcal{C}, and 𝒟\mathcal{D} be bicategories and P:𝒜× op×𝒟P\colon\mathcal{A}\times\mathcal{B}^\mathsf{op}\times\mathcal{B}\to\mathcal{D} and Q:𝒜×𝒞 op×𝒞𝒟Q\colon\mathcal{A}\times\mathcal{C}^\mathsf{op}\times\mathcal{C}\to\mathcal{D} be pseudofunctors (we are going to use the invertibility of their lax functoriality constraints below). The following is [Corner 2017, Definition 2.2], but laxified:

Definition

A lax extranatural transformation PQP\overset{\bullet}{\Rightarrow}Q from PP to QQ consists of

  1. Components. For each AObj(A\in\mathrm{Obj}(\mathcal{B} and each CObj(C)C\in\mathrm{Obj}(C), a lax transformation

    α (),B,C:P (),B BQ (),C C,\alpha_{(-),B,C}\colon P^B_{(-),B}\Rightarrow Q^{C}_{(-),C},

    called the component of α\alpha at (B,C)(B,C).

  2. Lax Extranaturality Constraints I. For each 11-morphism g:BBg\colon B\longrightarrow B' of \mathcal{B}, a 22-morphism

    α g:α A,B,CP id A,g id Bα A,B,CP id A,id B g\alpha_{g}\colon\alpha_{A,B',C}\circ P^{\mathsf{id}_{B'}}_{\mathsf{id}_{A},g}\Rightarrow\alpha_{A,B,C}\circ P^{g}_{\mathsf{id}_{A},\mathsf{id}_{B}}

    of 𝒟\mathcal{D} as in the diagram

    called the lax extranaturality constraint of α\alpha at gg.

  3. Lax Extranaturality Constraints II. For each 11-morphism h:CCh\colon C\longrightarrow C' of 𝒟\mathcal{D}, a 22-morphism

    α h:Q biid A,h biid Cα A,B,CQ biid A,biid C hα A,B,C\alpha_{h}\colon Q^{\biid_{C}}_{\biid_{A},h}\circ\alpha_{A,B,C}\Rightarrow Q^{h}_{\biid_{A},\biid_{C'}}\circ\alpha_{A,B,C'}

    of 𝒟\mathcal{D} as in the diagram

    called the lax extranaturality constraint of α\alpha at hh.

【…】

References

  • Alexander Corner, A Universal Characterisation of Codescent Objects (arXiv).

Last revised on July 8, 2020 at 20:14:45. See the history of this page for a list of all contributions to it.