# Questions tagged [ct.category-theory]

Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

**7**

votes

**1**answer

111 views

### 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**

votes

**0**answers

101 views

### 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 ...

**4**

votes

**1**answer

159 views

### 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 ...

**1**

vote

**0**answers

39 views

### 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 ...

**2**

votes

**0**answers

76 views

### Isomorphism of functors from isomorphism of of their endomorphisms

I am currently stuck at the following problem, and I was hoping that some might know of some literature or known results that might enable me to tackle it.
Let $F,G:\mathcal{C}\rightarrow \mathsf{...

**11**

votes

**1**answer

309 views

### 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, ...

**2**

votes

**0**answers

115 views

### Have you seen this extension property?

I'm looking for a reference for the following extension property:
A functor $F: C \to D $ such that for any arrow $f: x \to y$ in $C$ there is an extension $f = \hat{f} \circ i$ with $i:x \...

**24**

votes

**2**answers

625 views

### Another notion of exactness: how to refine it, and where does it fit?

There are many notions of "exactness" in category theory, algebraic geometry, etc. Here I offer another that generalizes the category of frames, the notion of valuation (from probability theory), and ...

**13**

votes

**1**answer

2k views

### 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 ...

**6**

votes

**1**answer

80 views

### Is a weak functor which strictly preserves horizontal composition and which runs between strict bicategories automatically strict?

Let $\mathbf{B}$ and $\mathbf{B'}$ be strict bicategories and $F: \mathbf{B} \to \mathbf{B'}$ a weak functor which preserves horizontal composition strictly (i.e. $Ff * Fg = F(f * g)$ natural in f and ...

**10**

votes

**1**answer

465 views

### 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, ...

**1**

vote

**0**answers

37 views

### (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 ...

**2**

votes

**2**answers

247 views

### Understanding definition of gerbe over a stack

I am reading Differentiable Stacks and Gerbes by Kai Behrend and Ping Xu.
They define gerbe over a stack as follows.
Let $\mathfrak{X}$ be a differentiable stack. An $\mathfrak{S}$-stack $\...

**11**

votes

**2**answers

433 views

### 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 ...

**12**

votes

**2**answers

776 views

### 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 ...

**8**

votes

**0**answers

124 views

### 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 ...

**0**

votes

**0**answers

49 views

### Comments/references on an obscure category of “rudimentary representations”

Consider a finite toset (cool word) $A=\{a_1<\ldots<a_N\}$. Let an ordered monomial be a finite formal product $b_1\ldots b_M$ with $b_i\in A$ and $b_i\le b_{i+1}$.
Consider the following ...

**4**

votes

**1**answer

114 views

### 2-morphisms for Bord(n)

I am currently reading in Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, and have a (I guess) pretty basic question for my understanding of the (...

**4**

votes

**1**answer

78 views

### Equivariant non symmetric operads

The definition of a symmetric $G$-Operad is basically a $G$ object in the category of symmetric operads. As far as I understand there is not a good notion of the non symmetric case. I would like to ...

**12**

votes

**0**answers

395 views

### What's wrong with the obvious argument that the unstable motivic category is an $\infty$-topos?

Fix a base scheme $S$, and let $Sm_S$ be the (ordinary) category of smooth schemes over $S$.
Denote by $Psh(Sm_S)$ the $\infty$-category of $\infty$-presheaves on $Sm_S$.
Let $Sh_{Nis}(Sm_S)\...

**5**

votes

**1**answer

97 views

### Products of representables are regular on a regular skeletal Reedy category?

The definition of a (pre)regular skeletal Reedy category in the sense of Cisinski generalizes the intuition behind the Eilenberg-Zilber factorization for simplicial sets, specifically the property ...

**6**

votes

**1**answer

263 views

### Is there a relationship between norms/transfers in equivariant homotopy theory and norms in the Tate construction / ambidexterity?

In homotopy theory, the word "norm" is commonly used in two different ways (well, surely there are other ways, but these two have a particular familial resemblence).
Let $G$ be a finite group. A $G$-...

**3**

votes

**0**answers

111 views

### Is the simplicial objects functor a comonad?

Let $T$ be the functor of simplical objects $[\Delta^{\mathrm {op}},-]:\mathrm{Cat} \to \mathrm{Cat}$. I am trying to construct counit and comultiplication maps $\eta$ and $\mu$ to make $(T,\eta,\mu)$ ...

**4**

votes

**0**answers

81 views

### 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 ...

**-3**

votes

**1**answer

544 views

### What is the big picture of algebraic geometry? [closed]

I am trying to understand a big-picture for Algebraic Geometry:
Given a category of commutative rings $\mathrm{CRing}$, we can create objects that locally look like these objects called schemes. This ...

**4**

votes

**0**answers

82 views

### Classification of combinatorial model categories presentable by simplicial presheaves on a Reedy category

Dan Dugger proved that every combinatorial model category can be obtained up to Quillen equivalence as the localization of a model structure on simplicial presheaves on a small category $C$.
Is there ...

**13**

votes

**1**answer

279 views

### What is the coskeleton tower of a quasi-category?

I was giving a talk in a seminar, and I mistakenly said that the coskeleton tower of a quasi-category was its Postnikov tower. Someone corrected me, but a discussion then ensued about what, precisely,...

**5**

votes

**1**answer

136 views

### Model category of diagrams with the colimit detecting the weak equivalences

Let $I$ be a small category and $\mathcal{K}$ be a combinatorial model category.
Is it known a model category structure on the functor category
$\mathcal{K}^I$ such that a map of diagrams $D\to ...

**6**

votes

**1**answer

208 views

### Understanding two proofs in Dwyer and Kan article “Simplicial Localizations”

I can't understand the proofs of propositions 2.6 and 4.2 in https://www3.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf
We have a category $C$ and a family of maps $W$, and we define the ...

**12**

votes

**3**answers

600 views

### General principles which lead to good questions in many concrete situations [closed]

I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such ...

**15**

votes

**2**answers

310 views

### How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?

If $\mathcal{C}$ is a symmetric monoidal $(\infty,1)$-category with duals, then there should be a functor
$$
d: \mathcal{C} \longrightarrow \mathcal{C}^{op}
$$
such that $d(x)$ is dual to $x$ for ...

**6**

votes

**3**answers

257 views

### The skew monoidal structure induced by a functor

$\require{AMScd}$Let $\cal K$ be a 2-category, and $j : A\to B$ one of its 1-cells. Assume that the induced map
$$
j^* : {\cal K}(B,B)\to {\cal K}(A,B)
$$ precomposing with $j$ has a left adjoint $j_!$...

**5**

votes

**0**answers

91 views

### References on categorical TVS theory

A survey by Castillo lists results on the category of Banach spaces and on Banach space constructions, such as:
Existence of limits in Banach spaces or suitable subcategories
Demonstrations of ...

**3**

votes

**1**answer

111 views

### The binary product of two presentable objects

The binary product of two $\lambda$-presentable objects (in a locally presentable category) is $\mu$-presentable for some regular cardinal $\mu \geq \lambda$ (because all objects are $\mu$-presentable ...

**1**

vote

**1**answer

152 views

### What are the internal categories in an endofunctor category

Take a category $C$, and take all endofunctors of $C$, so the set $E= \{ M| M: C \rightarrow C \}$. $E$ forms the objects of a category with morphisms given by all natural transformations $\mu : M \...

**4**

votes

**1**answer

160 views

### What is the “free symmetric monoidal category” 2-monad?

I have come across an n-category cafe post where someone describes a monad that generates symmetric monoidal categories. Can someone give details, like what is the base category, what exactly is the ...

**2**

votes

**0**answers

84 views

### Transformation from the Bag monad to the List monad

The bag monad, sometimes called the multiset monad or free commutative monoid monad is a functor on Set that takes a set to its set of bags. These bags are like strings written in the elements of the ...

**4**

votes

**0**answers

41 views

### What are the special properties of adjunctions that generate polynomial monads

The subject of polynomial monads is well trodden. We know that every monad is generated by an adjunction. What are the special properties of any adjunction that generates a polynomial monad?
Take a ...

**0**

votes

**0**answers

48 views

### What are the compact elements in the domain of functions on the Interval Domain, [D,D]?

Take the interval domain $D = \{[a,b]| a \le b, \in \mathbb{R}\}$, ordered by reverse inclusion.
Next, take the domain of functions $D' = \{f| f:D \rightarrow D \}$, ordered as follows $f \le g$ if $...

**1**

vote

**2**answers

190 views

### Are there ever non-universal cones to the identity functor?

An initial object in e in C is also a universal cone to the identity functor, but can there ever be a category where an object k in C exists with a cone to the identity functor, but k is not initial. ...

**1**

vote

**1**answer

128 views

### Computing a factorization of a monad

Given a monad, $(M, \mu, \eta)$, where $M: C \rightarrow C$ for some category $C$, there is a category of factorizations, $F\cdot G = M$ where $F: X \rightarrow C$, $G: C \rightarrow X$. Though this ...

**2**

votes

**0**answers

132 views

### Why does the following construction describe the Serre functor?

In the book "Spectral Agebraic Geometry" that Jacob Lurie is currently writing, he gives a construction (11.1.5.1), which describes the Serre functor: it has already been shown that any proper $A$-...

**14**

votes

**1**answer

632 views

### Homotopy pullback of a homotopy pushout is a homotopy pushout

Let's assume that we have a cube of spaces such that everything commutes up to homotopy.
The following holds:
- The right square is a homotopy pushout and
- all the squares in the middle are ...

**2**

votes

**1**answer

72 views

### Is the site for cubical sets with connections equivalent to a full subcategory of posets?

Cubical sets with connection form a presheaf category on some category $C$. Is $C$ just the full subcategory of the category of posets whose objects are products of the interval $\Delta[1]$?

**18**

votes

**0**answers

479 views

### 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 ...

**2**

votes

**0**answers

127 views

### Multiset or Bag monad on Finite-Dimensional Hilbert Spaces

Edit: I will be happy if someone can get me the Bag monad on a 2-category of groupoids, regardless of any reference to Hilbert Spaces. (It's a fire sale!!)
I am trying to create the quantum ...

**2**

votes

**1**answer

95 views

### Specifying complexes in quasicategories via squares

Let $J$ be an interval of integers viewed as a linearly ordered set, and let $I \subseteq \mathbf{N}(J)$ be the subsimplicial set given by the union of the elementary edges $(x, x+1)$.
The inclusion $...

**2**

votes

**0**answers

160 views

### What is an ambient isotopy categorically?

Let $\mathcal T$ be a category of "nice" topological spaces (CW?) and continuous maps between them. We can construct the homotopy category $\mathrm{Ho}\mathcal{T}$ which gives for any two objects $X,Y$...

**12**

votes

**1**answer

334 views

### Which motivic spectra are dualizable?

Let $S$ be a scheme, and $SH(S)$ the stable motivic category over $S$. Which objects of $SH(S)$ are dualizable with respect to the smash product?
All I can find on this question is an old abstract of ...

**3**

votes

**1**answer

209 views

### Definition of $(\infty,1)$-category in HoTT [duplicate]

Reading Lurie's Higher topos theory got me thinking, if we can think of $(\infty,1)-$categories as (1-)categories enriched over $\infty$-groupoid, then wouldn't the internal definition of an $(\infty,...