The formal theory of monads II

By Stephen Lack, Ross Street. 2002

383 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/44ce1b824d7546d6e6e8b761ffc000f89c5ed3bb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Homotopy coherent adjunctions and the formal theory of monads (2013, neutral): “The “formal theory of monads” plays homage to Ross Street’s paper [29], which develops a formal 2-categorical theory of monads and their associated Eilenberg–Moore and Kleisli constructions.”
Algebraic weak factorisation systems I: Accessible AWFS☆ (2014, background): “α,1): R′F2 ⇒ F2Rand (1,α): L′F2 ⇒ F2Lare respectively a lax monad morphism R→ R′ and a lax comonad morphism L→ L′ over F2: C2 → D2 (i.e., a monad functor and a comonad opfunctor in the terminology of [48]). A transformation (F,α) ⇒ (G,β) between lax morphisms is a natural transformation γ: F⇒ Grende”
From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories (2001, background): “Now, a monad (comonad) in a general 2-category F is most naturally de3ned [61] as an object C in F (the basis) together with a monoid (resp.”
Metric, topology and multicategory—a common approach (2003, neutral): “Hence, here we - do not discuss monads and related notions in the general context of bicategories or 2categories (see [2] and, for a recent account, [16]) but restrict ourselves to presenting them ad hoc as needed - present the 2-categorical structure of the category of reflexive, transitive (T,V)-a”
Polynomials in categories with pullbacks (2011, background): “Lax and colax morphisms of monads were called “monad functors” and “monad opfunctors” in [26] respectively.”
Frobenius algebras and ambidextrous adjunctions (2005, positive): “Using a wealth of results from category theory, especially the formal theory of monads [32], we extend Street’s work and prove this result for Frobenius objects in a generic monoidal category M .”
A convenient category for higher-order probability theory (2017, background): “λx.q >>= λy.f(x,y) equals q λy.p λx.f(x,y). 3) The faithful functor L: QBS →Meas with L(X,M X) = (X,Σ M X ) extends to a faithful functor Kleisli(P) →Kleisli(G), i.e. (L,l) is a monad opfunctor [36]. 4) When (X,Σ X) is a standard Borel space, the map l X of Eq. (3) is a measurable isomorphism.”
Mapping F_1-land:An overview of geometries over the field with one element (2009, background): “ Xto RX, the free R-module generated byX. The ring multiplication and unit translate into properties of this functor, making it into a monad that commutes with ﬁltered direct limits (cf. for instance [39] or [28, Chapter VI]). Motivated by this fact, Durov deﬁnes a generalized (commutative) ring as ”
Enhanced 2-categories and limits for lax morphisms (2011, background): “obtain EilenbergMoore and Kleisli objects for comonads. If (A,t) and (B,s) are monads in K , a lax monad morphism is a morphism f: A→ Btogether with a 2-cell f: sf → ftsatisfying suitable axioms. (In [Str72] these were called monad functors.) These are the lax morphisms of algebras for a suitable 2-”
Quantum categories, star autonomy, and quantum groupoids (2003, background): “ On top of the already discussed diverse properties and rich structure enjoyed by Mod( )V , we also have the property that all right liftings and right extensions exist. Despite the terminology (from [St1] for example), these concepts are very familiar in the usual theory of modules. Suppose M and M”
Polynomial functors and polynomial monads (2009, background): “From the point of view of the formal theory of monads [53], a polynomial monad is a monad in the 2-category PolyFunE .”