Coherence laws when composing 2-monads

To have the composition of two monads be a monad itself, we need a distributive law natural transformation satisfying certain coherence laws. I’m interested in the strict 2-monad case, i.e. a strict 2-functor equipped with unit and counit natural transformations that satisfy the zig-zag equations on the nose. I presume in such a case it’s … Read more

Checking a monad is idempotent

I have a monad T:C→C on a (Grothendieck) abelian category which preserves filtered colimits and direct sums (but is not exact). There is a finite collection G of compact, projective objects which generate C in the sense that every object X is presented as a cokernel R1→R0→X→0 of objects R1 and R0 which are (possibly … Read more

Monads which are monoidal and opmonoidal

Do monads which are monoidal and opmonoidal have a name? (Bimonoidal?) In case they have already been studied, who can point me to a reference? More in detail. Let $(C,\otimes)$ be a symmetric (or braided) monoidal category. Let $(T,\eta,\mu)$ be a monad on $C$ such that: $T:C\to C$ is a bilax monoidal functor (compatible lax … Read more

What is the category of algebras for the finitely supported measures monad?

In this post, I was introduced to the monad of finitely supported measures. HX is the set of finitely supported measures on X, with monad structure defined as for the Giry monad. I have three questions. This monad, H, is presented on Set in the post, but I am wondering if the category of groupoids … Read more

Coherence for pseudomonads and their pseudoalgebras

Let K be a bicategory. For every pseudomonad T:K→K, does there exist a 2-monad S:C→C, where C is a 2-category biequivalent to K, such that the bicategory of pseudoalgebras T–Alg is biequivalent to the 2-category of pseudoalgebras S–Alg? That is, can we always choose to work with 2-monads rather than pseudomonads, so long as we … Read more

Original reference for the correspondence between commutative algebraic theories and commutative monads

Commutative algebraic theories were introduced by Linton in the 1966 paper Autonomous Equational Categories. Commutative monads were introduced by Kock in the 1970 paper Monads on symmetric monoidal closed categories. The correspondence between (finitary) algebraic theories and (finitary) monad is known to specialise to a correspondence between (finitary) commutative algebraic theories and (finitary) commutative monads. … Read more

Kan condition for bar construction

Let T be a monad on a concrete category C, and A an algebra over T. The bar construction is a simplicial object in the category CT of algebras which we can think of a sort of “resolution” of A. Some of the arrows look like the following diagram: ⋯TTA⇉ (I unfortunately cannot draw more … Read more

Free monad or monad defined from an adjunction.

My first question here. Accordingly to M. Barr “Coequalizers and free triples” by a free triple (or free monad) generated by an endofunctor R:X→X we mean a triple T=(T,η,ν) and a natural transformation p:R→T such that if T′=(T′,η′,ν′) is another triple and p′:R→T′ is a natural transformation, then p′=τp, where τ:T→T′ is a map of … Read more

Transporting algebraic structure along adjoint equivalences

I have two questions, one general and the other particular to the case I am interested in. The ‘homotopically correct’ notion of equivalence of categories is an adjoint equivalence (from one point of view, at least). Given an (edit: equivalence invariant) algebraic structure on a groupoid G, say given by some 2-monad M on the … Read more