[Paper Review] A double-dimensional approach to formal category theory

Seerp Roald Koudenburg|arXiv (Cornell University)|Nov 12, 2015

Homotopy and Cohomology in Algebraic Topology36 references6 citations

TL;DR

This paper introduces an augmented virtual double category framework to formalize formal category theory, particularly extending the Yoneda lemma to algebraic contexts such as monoidal categories. It establishes that the Yoneda embedding into presheaves lifts to T-algebras under suitable conditions, and proves that the resulting presheaf category is the free small cocompletion in the algebraic setting, recovering Day convolution and Im-Kelly's result on free monoidal cocompletion.

ABSTRACT

Whereas formal category theory is classically considered within a $2$-category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of augmented virtual double categories, a notion extending that of virtual double category by adding cells with nullary target. [...] After this the notion of `weak' Kan extension within an augmented virtual double category is considered, together with three strengthenings. [...] The notion of yoneda embedding is then considered in an augmented virtual double category, and compared to that of a good yoneda structure on a $2$-category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding $y \colon A o \hat A$ defines $\hat A$ as the free small cocompletion of $A$, in a suitable sense. In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad $T$ on an augmented virtual double category $\mathcal K$ several augmented virtual double categories $T ext-\mathsf{Alg}_{(v, w)}$ of $T$-algebras are associated, [...]. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors $T ext-\mathsf{Alg}_{(v, w)} o \mathcal K$. The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in $\mathcal K$ lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in $T ext-\mathsf{Alg}_{(v, w)}$. As a first example we apply the previous to monoidal structures on categories, hence recovering Day convolution of presheaves and Im-Kelly's result on free monoidal cocompletion, as well as obtaining a "monoidal Yoneda lemma".

Motivation & Objective

• To generalize the classical Yoneda lemma to settings with algebraic structures such as monoidal categories.
• To formalize the monoidal Yoneda lemma via a double-categorical framework that handles both lax monoidal functors and profunctors.
• To characterize when the Yoneda embedding into presheaves lifts along forgetful functors from T-algebras to the base category.
• To prove that the lifted Yoneda embedding defines the free small cocompletion in the category of T-algebras.
• To recover and generalize classical results such as Day convolution and Im-Kelly’s theorem on free monoidal cocompletion.

Proposed method

• The paper develops the theory of augmented virtual double categories, which extend virtual double categories by allowing nullary target cells.
• It introduces three notions of weak Kan extensions—generalizing Borceux-Kelly, Street’s pointwise extensions, and a combined notion—within this framework.
• It defines a good yoneda structure in augmented virtual double categories and characterizes when a yoneda embedding induces free small cocompletion.
• It constructs T-algebras for a monad T on an augmented virtual double category, forming categories T-Alg(v,w) of T-algebras with specified coherence types.
• It proves that forgetful functors T-Alg(v,w) →K create Kan extensions and that the Yoneda embedding lifts along them under suitable conditions.
• It applies the framework to monoidal categories, showing that Day convolution arises naturally and that (bA, b⊗) is the free monoidal cocompletion of (A, ⊗).

Experimental results

Research questions

• RQ1 How can the Yoneda lemma be generalized to settings with algebraic structures such as monoidal categories?
• RQ2 Under what conditions does the Yoneda embedding into presheaves lift to the category of T-algebras for a monad T?
• RQ3 When does the lifted Yoneda embedding in T-Alg(v,w) still define a free small cocompletion?
• RQ4 What is the relationship between lax monoidal profunctors and lax monoidal functors induced by the Yoneda embedding in the algebraic setting?
• RQ5 How can the theory of Kan extensions in augmented virtual double categories be used to recover classical results like Day convolution?

Key findings

• The Yoneda embedding y: A → bA lifts to a yoneda embedding in T-Alg(c,l) and T-Alg(c,ps,lbc) when the base category K satisfies appropriate conditions.
• The lifted Yoneda embedding in T-Alg(c,l) defines bA as the free S(c,l)-cocompletion of A, where S(c,l) is the ideal of left extension diagrams.
• The same holds for T-Alg(c,ps,lbc), showing that the free cocompletion is preserved under the algebraic structure.
• The paper establishes a monoidal Yoneda lemma: for any lax monoidal profunctor J, the induced functor J^λ is lax monoidal and satisfies bA(y–, J^λ–) ≅ J as lax monoidal profunctors.
• The construction recovers Day convolution as the monoidal structure on bA induced by the Yoneda embedding.
• For any small V-cocomplete monoidal V-category N, there is a natural equivalence MonCat(l,cocts)(bM, N) ≃ MonCatl(M, N), recovering Theorem 5.1 of [IK86].