## Algebraic Kan extensions along morphisms of internal algebra classifiers.(English)Zbl 1342.18017

In his treatment of algebraic theories and structure, an important fact Lawvere proved was that, for any functor $$f : A\to B$$ between small categories $$A$$ and $$B$$ with finite products, the left Kan extension $$\mathrm{Lan}_f(m) : B\to \mathrm{Set}$$ of any finite-product-preserving functor $$m : A\to \mathrm{Set}$$ is also finite product preserving. The special properties of finite product and $$\mathrm{Set}$$ which make this work are: (a) there is a canonical comparison $$f(a_1\times \dots \times a_n)\to f(a_1)\times \dots \times f(a_n)$$; (b) for any set $$x$$, the endofunctors $$x\times -$$ and $$-\times x$$ of $$\mathrm{Set}$$ preserve colimits; and (c) the left Kan extension is constructible pointwise by the usual colimit formula.
After the Introduction, in Section 2 the present paper centres attention on showing how the above result about Kan extensions can be lifted to the context where finite products are replaced by pseudo-algebra structure for a 2-monad $$T$$ on a 2-category $$\mathcal{K}$$. Item (a) is now replaced by the condition that $$f : A\to B$$ should be a colax morphism of pseudo-$$T$$-algebras. Items (b) and (c) require the introduction of a number of new pseudo-$$T$$-algebra concepts: algebraic left extension for pseudomorphisms, algebraic cocompleteness with respect to a morphism, exactness of a square containing a 2-cell, and exactness for a colax morphism.
The main theorem of the paper is stated in Section 3, with examples, while the proof is postponed until Section 5.7. This theorem involves three 2-monads $$T$$, $$R$$ and $$S$$ on respective 2-categories $$\mathcal{K}$$, $$\mathcal{M}$$ and $$\mathcal{L}$$. The idea is that $$T$$ describes the ambient structure, while $$R$$ and $$S$$ describe types of structure which can be defined internal to a pseudo-$$T$$-algebra, where $$S$$ structure is richer than $$R$$ structure as expressed using an adjunction $$G: (\mathcal{M},R)\to (\mathcal{L}, S)$$. There is a forgetful functor $$U^G_A: S\text{-}\mathrm{Alg}(A)\to R\text{-}\mathrm{Alg}(A)$$ between the categories of algebras internal to each pseudo-$$T$$-algebra $$A$$. Conditions are developed under which a left adjoint for $$U^G_A$$ can be constructed via algebraic left extension. The main theorem states that these conditions are met when dealing with polynomial monads in $$\mathrm{Cat}$$. For background, see the author’s paper [Theory Appl. Categ. 30, 1659–1712 (2015; Zbl 1375.18058)] and his other two papers in that volume.
The paper contains a wealth of new results for working within a 2-category. The merit of working at this level is reinforced by the diversity of the applications. For example, further insight is gained on situations studied by R. M. Kaufmann and B. C. Ward [“Feynman categories”, arXiv:1312.1269] and by M. Batanin and C. Berger [“Homotopy theory for algebras over polynomial monads”, arXiv:1305.0086].

### MSC:

 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 18D20 Enriched categories (over closed or monoidal categories) 55P48 Loop space machines and operads in algebraic topology 18D50 Operads (MSC2010)

Zbl 1375.18058
Full Text: