# 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.

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### Quantum groups at $q=-1$

For a Drinfeld--Jimbo quantized enveloping algebra $U_q(\frak{g})$, it is standard knowledge that the categories of modules are very different in the $q$ a root of unity, and $q$ not a root of unity ...

**4**

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**1**answer

121 views

### In a fibration, how do properties of arrows downstairs affect the base-change functors?

Fix a fibration of categories. Suppose $f:A\to B$ is an arrow in the base.
What are the relations between the following pairs?
$$f\text{ epi}\qquad f^\ast \text{ faithful}$$
$$f\text{ mono}\qquad f^...

**7**

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166 views

### Testing for equivalences of $\infty$-categories on strictifications?

It is in general not too hard to show that maps between finite $CW$-complexes/finite simplicial sets are homotopy equivalences.
Question : Can we do something similar for:
quasi-categorical ...

**6**

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**3**answers

219 views

### Constructive proof of exponential law in a category

I'm trying to write a constructive proof of the isomorphism ${Z}^{X \times Y} \equiv (Z^Y)^X$ in a category with exponential objects.
I can construct a map $(Z^Y)^X \rightarrow {Z}^{X \times Y}$, but ...

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161 views

### Tannaka duality for $DG$ indschemes

In Lurie's paper on Tannaka duality for geometric stacks he proves that there is a natural isomorphism
$$\operatorname{Hom}(X,Y) \cong \operatorname{Hom}(\mathbf{QC}(Y), \mathbf{QC}(X)$$
where $X$ and ...

**3**

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**1**answer

67 views

### sequences of iterated orthogonals (lifting property) in a category

I am looking for examples of properties of morphisms defined by taking orthogonals with respect to the Quillen lifting property.
For example, several iterated orthogonals of $ \emptyset\...

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**1**answer

91 views

### Equivalent conditions to be a lax idempotent contravariant monads

$\require{AMScd}$As already said, a contravariant monad is "like a monad, but a contravariant functor":
the multiplication and unit are dinatural transformations
an algebra is a map $a : TA\to A$ ...

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**1**answer

129 views

### Cellularity of anodyne extensions?

Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts)
If not, is there a known counterexample?
Similarly, does ...

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**1**answer

270 views

### Explicit Left Adjoint to Forgetful Functor from Cartesian to Symmetric Monoidal Categories

There is a forgetful functor from the category of (small) Cartesian monoidal categories (a symmetric monoidal category in which the tensor product is given by the categorical product) to the category ...

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**1**answer

163 views

### Finally dense implies dense

I am reading the article "A convenient category for directed homotopy" by Fajstrup and Rosicky and I have a doubt about the proof of Proposition 3.5. The setting is the following:
let $\cal{C}$ be a ...

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203 views

### A serendipitous connection between Isbell duality and Yoneda structures

Another questions about unidetermined monads.
EDIT: Here a note with a few more details on what they are: link
Let $T : C^o \to C$ be such a monad, so that the multiplication $\mu_A$ is determined ...

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**1**answer

234 views

### A monad that unions sets

Suppose we have a monad that maps types of some kind to other types (see below) , and let types be sets. Let $\alpha, \beta$ be types, $\rightarrow$ denote a function between types, and let $a : \...

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313 views

### What is a spectrum object in $\infty$-topoi?

For any spectrum $E$, there is a "discrete" topos spectrum $(Spaces / E_n)_n$. And I believe any topos spectrum is a localization of a "discrete" one. Are there any "non-discrete" topos spectra?
To ...

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117 views

### Categorifying Hyperoperations

Is there some categorical version of tetration or higher hyperoperations?
This is motivated by the fact that coproducts categorify addition of finite cardinals, and products/exponentials categorify ...

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70 views

### The multi-set monad and modules

I am trying to analyze the category of algebras for the finite free commutative monoid monad, aka the finite multiset monad. This monad is frequently described as having a multiplication that takes a ...

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**1**answer

294 views

### Intuition for pseudo-points and the inductive step in Johnstone's proof of Deligne's completeness theorem

In Johnstone's Topos Theory appears the following lemma.
7.41 Lemma. Let $P$ be a pseudo-point of $\mathsf C$, $X$ a $J$-sheaf on $\mathsf C$, and $x,y$ two distinct element of $P(X)$. Let $(V_j\to V)...

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### What is a category of “Lepagean equivalent” or “variation problem”?

I get to know about it form Mark Gotay's work An exterior differential system approach to the Cartan form, in that paper he defined the canonical Lepagean equivalent. The following is cited from it:
...

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264 views

### $P = [-°,Set]$ is a contravariant co/lax idempotent monad, whose multiplication is determined by the unit

A unidetermined contramonad is a 2-monad $T : {\cal C}\to \cal C$ such that
$T$ is contravariant, i.e. a contravariant endofunctor;
the multiplication $\mu_A : TTA \to TA$ is determined as $T\eta_A = ...

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45 views

### Localization at the left edge of a coherent left horn inclusion

Let $\Lambda^n_0 \hookrightarrow \Delta^n$ be a left horn inclusion for $n>1$. Then consider $\mathfrak{C}(\Lambda^n_0)\hookrightarrow \mathfrak{C}(\Delta^n)$, and we have a cofibration $[1]_{\...

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296 views

### When is $\Omega^1$ an equivalence?

Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{...

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**1**answer

117 views

### Is the category of inclusion prespectra bicomplete?

Working in compactly generated weak Hausdorff spaces, is the category of inclusion prespectra bicomplete?
I should probably specify that by inclusion prespectra, I mean prespectra such that the ...

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**1**answer

142 views

### Codensity monad is idempotent?

Let $j: A \to B$ be a fully faithful functor.
When $j$ has a left adjoint $L$, the codensity monad $\text{Ran}_jj$ will coincide with the monad $jL$ and thus will be idempotent, because $A$ is ...

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62 views

### Codensity monad preserves some colimit?

Let $j: A \to B$ be a functor.
When $j$ has a left adjoint $L$, the codensity monad $\text{Ran}_jj$ will coincide with the monad $jL$. Since a left adjoint preserves all colimits, it is easy to ...

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652 views

### What does it mean for a category to admit direct integrals?

Given an infinite countable group $G$, the category of unitary representations of $G$ admits direct integrals.
Namely, given a measure space $(X,\mu)$ and a measurable family of unitary $G$-reps $(H_x)...

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136 views

### Universal Enveloping algebra of a L$_\infty$ algebra

In their paper Strongly homotopy Lie algebras, Lada and Markl first show, that there is a symmetrization functor $(-)_L:\mathcal{A}(m)\rightarrow \mathcal{L}(m)$ from the category of $A(m)$-algebras ...

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**1**answer

391 views

### Enhancing Grothendieck's universes and Grothendieck's axiom: Feferman's universe

A Grothendieck's universe is such a set $U$ so that
$\forall x \in U, x \subseteq U$,
$\forall x,y \in U, \{x,y\} \in U$,
$\forall x \in U, \mathcal{P}(x) \in U$,
given a family $(X_i)_{i \...

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323 views

### Is there any significance to Bousfield localization in the non-derived context?

The term "Bousfield localization" of a category $C$ is used in roughly two different ways:
There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...

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148 views

### Is $Ind(N_{dg}(\mathcal{C})) \simeq N_{dg}(Ind(\mathcal{C}))$ for an additive category $\mathcal{C}$?

Let $\mathcal{C}$ be an additive category and let $N_{dg}(\mathcal{C})$ be the differential graded nerve of the differential graded category $Ch(\mathcal{C})$. This is a stable $\infty$-category.
...

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**1**answer

230 views

### How to define 0-sphere in a category with zero object?

The 0-sphere $S^0$ is the coproduct of two points,
$$S^0 \simeq \ast \coprod \ast$$
How to define 0-sphere in a category with zero object?
Let $\mathcal{C}$ be a category. A cylinder, $\mathbf{I}$, ...

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228 views

### What are the monomorphisms of ($\infty$-)toposes?

There are standard notions of "surjections" and "embeddings" of toposes. However, not every surjection is an epimorphism, and not every regular monomorphism is an embedding. (EDIT: as Alexander ...

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423 views

### Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?

The name pretty much says it all: there's a well-defined operation on categories equivalent to modules over some ring: if $\mathcal{C}_1=A\operatorname{-mod}$ and $\mathcal{C}_2=B\operatorname{-mod}$, ...

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133 views

### Behaviour of direct limit with matrices

I am trying to understand direct limits in the category of $C^*$-algebras by self reading. My last question was also related to direct limits. Here is another of my doubts:
Let $(A_n,f_n)$ be a ...

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**1**answer

181 views

### Dualizable presheaves with respect to Day convolution

This question was posted on MSE and got very little attention, so I'm also posting it here.
Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, \...

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77 views

### Axiomatization of the shuffle decomposition

I am trying to figure out an axiomatization for Reedy categories such that the product of representables admits a shuffle decomposition. Cisinski suggested imposing a condition, which we will state ...

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140 views

### Does each monotonic endofunctor on the category of sets and relations preserve conversion?

Consider a functor $F : \mathbf{Rel} \to \mathbf{Rel}$ that is monotonic (for all relations $R$ and $S$ with $R \subseteq S$ we have $FR \subseteq FS$). Does such a functor always preserve conversion ...

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180 views

### Products, coproducts and equalizers in category of lattices

Background: Let $\mathbf{Lat}$ be the 2-category of lattices which can be viewed as a subcategory of the 2-cateogry of posets $\mathbf{Pos}$, that is, objects in $\mathbf{Pos}$ that have all finite ...

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266 views

### Are there continua in $\infty$-topoi?

If topology were invented for algebraic geometry or logic, in ignorance of Euclidean space, we might reasonably regard connected compact Hausdorff spaces as pathological, or even doubt their existence....

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### Koszul duality between QLS algebras and cdg algebras

A Quadratic-Linear-Constant (QLC) algebra $U$ is an algebra which can be written as $T(V)/P$ where
$T(V) = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$ is the tensor algebra
and $P \subseteq k \oplus ...

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221 views

### 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 ...

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**1**answer

530 views

### What is the correct definition of localisation of a category?

Disclaimer: I wasn't sure if this was an appropriate question for MathOverflow, and so I've also asked this on StackExchange.
There appears to be a discrepancy in the literature regarding the ...

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145 views

### Virtual generators

Let $\mathcal{K}$ be a category.
Prop 1. If $\mathcal{K}$ has a (strong) generator then there is a faithful (and conservative) functor $U: \mathcal{K} \to \text{Set}$ preserving connected limits.
...

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### Fundamental Theorem of Category Theory appropriate for undergraduates?

I have been self-studying category theory (mostly from Lawvere and Schanuel’s Conceptual Mathematics). In my experience in undergraduate mathematics, many one or two quarter classes in mathematics ...

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246 views

### Why must the essential image break the principle of equivalence?

I'm having trouble understanding why the "essential image" is defined the way it is.
The nlab article gives the following definition:
(A concrete realization of) the essential image of a functor $...

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### Cencov's “categories of figures”

In his 1982 book Statistical Decision Rules and Optimal Inference, N. N. Cencov studies statistical models (parametrized families of probability distributions) from an unconventional category-...

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### sub relative cell complex

This is a question concerning the proposition 11.4.8 of P.Hirschhorn's book Model categories and their localizations.
The proposition 11.4.8 is an analogous, in my opinion, to the well known ...

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644 views

### What is the geometric significance of fibered category theory in topos theory?

Often in topos theory, one starts with a geometric morphism $f: \mathcal Y \to \mathcal X$, but quickly passes to the Grothendieck fibration $U_f: \mathcal Y \downarrow f^\ast \to \mathcal X$, which ...

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### Homotopy pullbacks of simplicial sets; Joyal vs Kan-Quillen model structures

I am interested in comparison of homotopy pullback squares in the category of simplicial sets with respect to Joyal' model structure and Quillen's one.
Suppose we are given a (strict) pullback square
...

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### What is the value of a polynomial form for a data structure, aka a Container

Data structures like Lists and Trees are often referred to as Containers. They can be given as monads and containers are polynomial functors. The List monad is well known and can be given as a ...

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227 views

### Measures on sites

Let $\mathcal{C}$ be a category equipped with a Grothendieck topology $\tau$, i.e., a site.
Under which conditions on $\mathcal{C}$ can one construct a Borel $\sigma$-algebra, $\sigma_\tau$, for $\...

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243 views

### Is there a citeable source for generators and relations of simplicial sets?

Simplicial sets can be specified using generators and relations,
in complete analogy with groups, rings, etc.
More precisely, a system of generators of relations for a simplicial set
consists of a ...