Diagrammatic category theory

In category theory, the use of string diagrams is well known to aid in the intuitive understanding of certain concepts, particularly when dealing with adjunctions and monoidal categories. We show that string diagrams are also useful in exploring fundamental properties of basic concepts in category theory, such as universal properties, (co)limits, Kan extensions, and (co)ends. For instance, string diagrams are utilized to represent visually intuitive proofs of the Yoneda lemma, necessary and sufficient conditions for being adjunctions, the fact that right adjoints preserve limits (RAPL), and necessary and sufficient conditions for having pointwise Kan extensions. We also introduce a method for intuitively calculating (co)ends using diagrammatic representations and employ it to prove several properties of (co)ends and weighted (co)limits. This paper proposes that using string diagrams is an effective approach for beginners in category theory to learn the fundamentals of the subject in an intuitive and understandable way

Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions

Consider a diagram of quasi-categories that admit and functors that preserve limits or colimits of a fixed shape. We show that any weighted limit whose weight is a projective cofibrant simplicial functor is again a quasi-category admitting these (co)limits and that they are preserved by the functors in the limit cone. In particular, the Bousfield-Kan homotopy limit of a diagram of quasi-categories admit any limits or colimits existing in and preserved by the functors in that diagram. In previous work, we demonstrated that the quasi-category of algebras for a homotopy coherent monad could be described as a weighted limit with projective cofibrant weight, so these results immediately provide us with important (co)completeness results for quasi-categories of algebras. These generalise most of the classical categorical results, except for a well known theorem which shows that limits lift to the category of algebras for any monad, regardless of whether its functor part preserves those limits. The second half of this paper establishes this more general result in the quasi-categorical setting: showing that the monadic forgetful functor of the quasi-category of algebras for a homotopy coherent monad creates all limits that exist in the base quasi-category, without further assumption on the monad. This proof relies upon a more delicate and explicit analysis of the particular weight used to define quasi-categories of algebras.Comment: 33 pages; a sequel to arXiv:1306.5144 and arXiv:1310.8279; v3: final journal version with updated internal references to the new version of "Homotopy coherent adjunctions and the formal theory of monads

Coends of higher arity

We specialise a recently introduced notion of generalised dinaturality for functors $T : (\mathcal{C}^\text{op})^p \times \mathcal{C}^q \to \mathcal{D}$ to the case where the domain (resp., codomain) is constant, obtaining notions of ends (resp., coends) of higher arity, dubbed herein $(p,q)$-ends (resp., $(p,q)$-coends). While higher arity co/ends are particular instances of "totally symmetrised" (ordinary) co/ends, they serve an important technical role in the study of a number of new categorical phenomena, which may be broadly classified as two new variants of category theory. The first of these, weighted category theory, consists of the study of weighted variants of the classical notions and construction found in ordinary category theory, besides that of a limit. This leads to a host of varied and rich notions, such as weighted Kan extensions, weighted adjunctions, and weighted ends. The second, diagonal category theory, proceeds in a different (albeit related) direction, in which one replaces universality with respect to natural transformations with universality with respect to dinatural transformations, mimicking the passage from limits to ends. In doing so, one again encounters a number of new interesting notions, among which one similarly finds diagonal Kan extensions, diagonal adjunctions, and diagonal ends.Comment: produced with codi https://www.ctan.org/pkg/commutative-diagram

Double Homotopy (Co)Limits for Relative Categories

We answer the question to what extent homotopy (co)limits in categories with weak equivalences allow for a Fubini-type interchange law. The main obstacle is that we do not assume our categories with weak equivalences to come equipped with a calculus for homotopy (co)limits, such as a derivator.Comment: 34 page

Behavior of Quillen (co)homology with respect to adjunctions

This paper aims to answer the following question: Given an adjunction between two categories, how is Quillen (co)homology in one category related to that in the other? We identify the induced comparison diagram, giving necessary and sufficient conditions for it to arise, and describe the various comparison maps. Examples are given. Along the way, we clarify some categorical assumptions underlying Quillen (co)homology: cocomplete categories with a set of small projective generators provide a convenient setup.Comment: Minor corrections. To appear in Homology, Homotopy and Application

Totally distributive toposes

A locally small category E is totally distributive (as defined by Rosebrugh-Wood) if there exists a string of adjoint functors t -| c -| y, where y : E --> E^ is the Yoneda embedding. Saying that E is lex totally distributive if, moreover, the left adjoint t preserves finite limits, we show that the lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes, studied by Johnstone and Joyal. We characterize the totally distributive categories with a small set of generators as exactly the essential subtoposes of presheaf toposes, studied by Kelly-Lawvere and Kennett-Riehl-Roy-Zaks.Comment: Now includes extended result: The lex totally distributive categories with a small set of generators are exactly the injective Grothendieck toposes; Made changes to abstract and intro to reflect the enhanced result; Changed formatting of diagram