## Tagged Questions

1answer
238 views

### When do reflexive coequalizers preserve weak equivalences?

In my work I've run into the following situation. In a model category, I have two reflexive coequalizers $A_i \stackrel{\to}{\to} B_i \to C_i$ and a map of diagrams which is level …
1answer
354 views

### Reference for my monads?

I'm looking for a reference for a certain pair of monads on $Cat$. One problem is that I don't know the modern way of thinking about some basic things, so excuse me if my presentat …
1answer
189 views

### The crude monadicity theorem

In order to test the monadicity of a functor, there is a precise monadicity theorem (PM) as well as a crude monadicity theorem (CM), see the nlab. In CM, the forgetful functor shou …
1answer
170 views

### Free cocommutative commutative Hopf monoids

I have some questions about generalizations of abelian groups, relative to symmetric monoidal categories. 1) Let $C$ be a cocomplete cartesian monoidal category with equalizers. I …
1answer
252 views

### Applications of the Giry monad in probability and statistics

In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$. Will Saw …
1answer
137 views

### Seems like Reader monad composed with a strong monad produces a monad, am I right?

Take a Cartesian (or monoidal) closed category; define Reader monad for a given object $E$ as $X \mapsto X^E$; and take a strong monad $M$ (strong means preserves product or tensor …
0answers
82 views

### Compatibility between strength and costrength of a monoidal monad

Let $C$ be a closed monoidal category, and let $T : C \to C$ be a monad on the underlying category. Let $\sigma$ be a tensorial strength of $T$ and let $\sigma'$ be a cotensorial s …
0answers
236 views

### Can we describe equivariant vector bundles of free group action in terms of descent theory (Barr-Beck theorem)?

It is well known that for a compact topological group $G$ acts (say, from the right) freely on a compact space $X$. Then the category of equivariant complex vector bundles on $X$, …
2answers
484 views

### “Functors between monads”: what are these really called?

Let $(S,\eta,\mu)$ be a monad on a category $C$, and $(T,\eta,\mu)$ a monad on a category $D$. The following kind of gadget is ubiquitous: a functor $F:D\to C$, together with a na …
2answers
348 views

### When do functors induce monadic adjunctions of presheaf categories

For a category $C$, let $C-Set$ denote the category of set-valued functors $\delta\colon C\to Set$. Given categories $C$ and $D$, and a functor $F\colon C\to D$, composition with \$ …
1answer
237 views

### Coproducts of modules over an algebraic monad

Coproducts of modules over an algebraic monad $\Sigma$ are described in Section 4.16.14/15 in Durov's thesis. It is claimed there that for $\Sigma$-modules $M,N$, the set \$M \copro …
0answers
139 views

### Coalgebra for a comonad for a really tiny category [closed]

I have a category with only two objects. These objects are just sets, each set has two elements in it. I take the morphisms as all endo-maps of the sets. There are no morphisms …
1answer
361 views

### Monadicity theorem in homotopy theory.

Let $\mathbf{C}$ be a cofibrantly generated model category (assume for simplicity that all objects are fibrant) and $\mathbf{C}^{\mathrm{T}}$ the category of $\mathrm{T}$-algebras …
1answer
615 views

### The reflexive free-category comonad-resolution is a cofibrant replacement  of the discrete simplicial category associated with an ordinary category in the Bergner model structure on the category of small simplicial categories?

Let $X$ be the category of reflexive quivers, and let $Cat$ be the category of small categories. There exists an evident forgetful functor $U:Cat\to X$ sending a category $A$ to i …
3answers
768 views

### Relation between monads, operads and algebraic theories

I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. …

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