# ct.category-theory's questions - English 1answer

3.748 ct.category-theory questions.

### 3 Product-preserving fibrant replacement functor for the Joyal model structure

There are a few fibrant replacement functors for the Quillen model structure on simplicial sets that preserve cartesian products, namely $\operatorname{Ex}^\infty$ and $\operatorname{Sing}(|\cdot|)$. ...

### -1 Adjunctions between Groupoids and Hilbert spaces

I am interested in any adjunctions between any of the familiar categories of Groupoids and the category of finite dimensional Hilbert spaces. Do any exist? Are there any well know monads on the ...

### To check if a stack is coming from a manifold

Let $\mathcal{D}$ be a stack. An atlas for stack $\mathcal{D}$ is a smooth manifold $X$ and a map of stacks $p:\underline{X}\rightarrow \mathcal{D}$ such that, for any manifold $M$ and a map of ...

### 7 Simplicial nerve functor commutes with opposites

There are two "opposite" functors: $$op_\Delta\colon sSet\to sSet$$ and $$op_s\colon sCat\to sCat.$$ The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite ...

### 8 Balls in Lawvere metric spaces

Let $V$ be the monoidal category $[0,\infty)$ (as a poset) with $+$ and $0$. Lawvere shows that $V$-enriched categories are a more natural generalisation of the notion of a metric space (note no ...

### 68 Has incorrect notation ever led to a mistaken proof?

In mathematics we introduce many different kinds of notation, and sometimes even a single object or construction can be represented by many different notations. To take two very different examples, ...

### 10 The category of hypergraphs as a topos

It seems known that the category of hypergraphs is a topos. I am looking for any reference here, or just a statement of this in the literature, but can't find anything. There is one paper A ...

### 1 Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]

Is that correct $R^2\cong R$ as measurable spaces? If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?

### 7 Bicategory of bimodules over internal monoids

In a monoidal category $(C,\otimes)$ one can consider internal monoids and bimodules over these monoids (provided some additional requirements like existence of coequalizers hold). These monoids ...

### 8 Is every presentable category a limit of locally finitely-presentable categories and finitary left adjoints?

Let $Pr^L$ be the category of presentable categories and left adjoint functors (probably this should be at least a (2,1)-category; anyway ultimately I'm interested in the $(\infty,1)$-setting). For ...

### 18 Eckmann-Hilton argument / Grothendieck

In terse terms, the “Eckmann-Hilton argument” says that a monoid in the category of monoids is commutative. It is essentially used, for example, to prove that homotopy groups of topological groups are ...

### 4 Exponential law w.r.t. compact-open topology

It is well-known that if a topological space $Y$ is locally compact (not necessarily Hausdorff), then the map $$\operatorname{Hom}(X \times Y, Z) \to \operatorname{Hom}(X, Z^Y)$$ (here we use the ...

### 9 What are the reflective subcategories of the category of presentable categories?

I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well. A nice property of presentable $\infty$-categories is that if ...

### 11 On a surprising property of free theories

Yesterday I observed (and proved) the following odd fact, which I found very surprising. I'm very curious to know if this was known by some people, or if it follows from some other more general fact, ...

### 1 Domain Monad on Density Operators Using Spectral Order

The spectral order for density operators is given in this paper Coecke Martin 2010. I won't give the full definition here. Essentially, it allows for a partial order of density matrices that forms a ...

### 33 What is… A Grossone? [closed]

Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this ...

### 14 Why “modding out the homeomorphism” in the category Top makes no rigorous sense?

We can rigorously talk about Top, the category of all topological space, and also FTop, the category of all finite topological space. So I thought, we can define a category FTop', where we “mod out by ...

### 1 (Co)Monads with a mixed distributive law on the 2-Category of Groupoids

I am looking for containers on the 2-Category of Groupoids. In particular, though, I would like my container to be both a monad and a comonad with a mixed distributive law. Can someone provide one ...

### 10 Set-theoretical multiverses and their representation as functors? Why *the* multiverse?

In some related MO questions like The set-theoretic multiverse as a (bi)category it is discussed how one might represent the multiverse (see The set-theoretic multiverse) in a category theoretic way, ...

### 4 What are morphisms between regularity structures?

In Hairer's notes A Theory of Regularity Structures he defines automorphisms of a regularity structure on page 28. I will recall the definition here: Is there any way of extending this to morphisms ...

### 62 whence commutative diagrams?

It seems that commutative diagrams appeared sometime in the late 1940s -- for example, Eilenberg-McLane (1943) group cohomology paper does not have any, while the 1953 Hochschild-Serre paper does. ...

### 12 What is a tensor category?

A monoidal category is a well-defined categorical object abstracting products to the categorical setting. The term tensor category is also used, and seems to mean a monoidal category with more ...

### 5 Frobenius monads and groupoids

For a while, I was looking for a Frobenius monad on Set. It doesn't exist as pointed out here. I am now looking at the 2-category of groupoids. Does the 2-category of groupoids admit a Frobenius ...

### 29 Tannaka formalism and the étale fundamental group

For quite a while, I have been wondering if there is a general principle/theory that has both Tannaka fundamental groups and étale fundamental groups as a special case. To elaborate: The theory of ...