621 research outputs found
A Notation for Comonads
The category-theoretic concept of a monad occurs widely as a design pattern for functional programming with effects. The utility and ubiquity of monads is such that some languages provide syntactic sugar for this pattern, further encouraging its use. We argue that comonads, the dual of monads, similarly provide a useful design pattern, capturing notions of context dependence. However, comonads remain relatively under-used compared to monads—due to a lack of knowledge of the design pattern along with the lack of accompanying simplifying syntax. We propose a lightweight syntax for comonads in Haskell, analogous to the do-notation for monads, and provide examples of its use. Via our notation, we also provide a tutorial on programming with comonads
Terminal semantics for codata types in intensional Martin-L\"of type theory
In this work, we study the notions of relative comonad and comodule over a
relative comonad, and use these notions to give a terminal coalgebra semantics
for the coinductive type families of streams and of infinite triangular
matrices, respectively, in intensional Martin-L\"of type theory. Our results
are mechanized in the proof assistant Coq.Comment: 14 pages, ancillary files contain formalized proof in the proof
assistant Coq; v2: 20 pages, title and abstract changed, give a terminal
semantics for streams as well as for matrices, Coq proof files updated
accordingl
Conservative descent for semi-orthogonal decompositions
Motivated by the local flavor of several well-known semi-orthogonal
decompositions in algebraic geometry, we introduce a technique called
conservative descent, which shows that it is enough to establish these
decompositions locally. The decompositions we have in mind are those for
projectivized vector bundles and blow-ups, due to Orlov, and root stacks, due
to Ishii and Ueda. Our technique simplifies the proofs of these decompositions
and establishes them in greater generality for arbitrary algebraic stacks.Comment: Final versio
Pre-torsors and Galois comodules over mixed distributive laws
We study comodule functors for comonads arising from mixed distributive laws.
Their Galois property is reformulated in terms of a (so-called) regular arrow
in Street's bicategory of comonads. Between categories possessing equalizers,
we introduce the notion of a regular adjunction. An equivalence is proven
between the category of pre-torsors over two regular adjunctions
and on one hand, and the category of regular comonad arrows
from some equalizer preserving comonad to on
the other. This generalizes a known relationship between pre-torsors over equal
commutative rings and Galois objects of coalgebras.Developing a bi-Galois
theory of comonads, we show that a pre-torsor over regular adjunctions
determines also a second (equalizer preserving) comonad and a
co-regular comonad arrow from to , such that the
comodule categories of and are equivalent.Comment: 34 pages LaTeX file. v2: a few typos correcte
Bicategories of spans as cartesian bicategories
Bicategories of spans are characterized as cartesian bicategories in which
every comonad has an Eilenberg-Moore ob ject and every left adjoint arrow is
comonadic
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