# Questions tagged [infinity-categories]

The infinity-categories tag has no usage guidance.

399
questions

**2**

votes

**0**answers

159 views

### Why is $\operatorname{Hom}^R_{\mathcal{C}}(X, Y )$ the fiber of $\mathcal{C}_{\backslash Y} \to \mathcal{C}$ (Lurie's HTT)

Reading Jacob Lurie's Higher Topos Theory I not understand the proof of the "only if" part in Proposition 1.2.12.4. It states
Proposition 1.2.12.4. Let $\mathcal{C}$ be an $\infty$-category ...

**2**

votes

**1**answer

68 views

### Fibrations of fibrant marked simplicial sets

Let $\mathrm{sSet}^+ = \mathrm{sSet}^+_{/ \Delta^0}$ be the model category of marked simplicial sets over the point. By Theorem 3.1.5.1 in Higher Topos Theory, this model category is Quillen ...

**4**

votes

**0**answers

79 views

### $\mathbb{E}_\infty$-refinements of the graded tensor product of $\mathbb{Z}$-graded spectra

The category
$$\mathsf{Gr}_\mathbb{Z}\mathsf{Mod}_R\overset{\mathrm{def}}{=}\mathsf{Fun}^\otimes(\mathbb{Z}_\mathsf{disc},\mathsf{Mod}_R)$$
of $\mathbb{Z}$-graded $R$-modules has a natural monoidal ...

**2**

votes

**1**answer

186 views

### Corepresentability of involutory objects in monoidal $\infty$-categories

The group $\mathbb{Z}/2$ corepresents the functor $\mathrm{Inv}\colon\mathsf{Mon}\to\mathsf{Sets}$ sending a monoid $A$ to its set of involutory elements (those satisfying $a^2=1_A$).
A similar story ...

**1**

vote

**0**answers

129 views

### Involutions in $\infty$-categories

$\newcommand{\id}{\mathrm{id}}$An involution in a category is a functor $\mathbf{B}\mathbb{Z}/2\to\mathcal{C}$, corresponding precisely to an object $X$ of $\mathcal{C}$ together with a $\mathbb{Z}/2$-...

**1**

vote

**0**answers

93 views

### Delooping monoidal $(\infty,1)$-categories into $(\infty,2)$-categories

This is the one categorical level higher version of the question Delooping monoidal $\infty$-groupoids into $\infty$-categories.
The classical, bicategorical, setting.
Given a monoidal category $(\...

**2**

votes

**1**answer

127 views

### Delooping monoidal $\infty$-groupoids into $\infty$-categories

The classical setting.
Given a monoid $A$, there's a category $\mathbf{B}A$, called the delooping of $A$, having a single object $\star$ and satisfying $\mathrm{Hom}_{\mathbf{B}A}(\star,\star)\overset{...

**2**

votes

**1**answer

121 views

### A question about cofiber diagrams in stable $\infty$-categories

My question is as follows say I have a commutative diagram
$\require{AMScd}$
\begin{CD}
X @>f>> Y @>g>> Z\\
@V \alpha V V @VV \beta V @VV \gamma V\\
X’ @>>f’> Y @>>g’&...

**6**

votes

**1**answer

515 views

### What is the homotopy category of the sphere spectrum?

Is there a known explicit description of the abelian $2$-group $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)\cong\Pi_{\leq1}(QS^0)$?

**3**

votes

**1**answer

78 views

### Weak composition rule for simplicial categories

Informally, an $\infty$-category should be the following data:
A collection of objects
A space of morphisms between any two objects
Weak associativity rules: Coherent homotopies between all of the ...

**2**

votes

**0**answers

97 views

### Base-change theorems for stable $\infty$-categories

Omitting some technicalities, the base-change theorem for quasicoherent sheaves says that if we have the following diagram of (derived) schemes
$\require{AMScd}$
\begin{CD}
X \times_S Y @>\pi_2>&...

**3**

votes

**0**answers

268 views

### Reconstructing an analytic ring from its module category

When reading Lectures on Analytic Geometry, I found that in the data of an analytic ring, the underlying ring seems unnatural and the module category should be the soul, but in fact one needs the ...

**3**

votes

**0**answers

101 views

### $(-n-1)$-connected spectra vs. reduced excisive functors from $n$-truncated pointed spaces

It's possible to view nonconnectivity for spectra as arising from enlarging Segal's category $\Gamma^\mathsf{op}\overset{\mathrm{def}}{=}\mathsf{Sk}(\mathrm{FinSets}_*)$ to the $\infty$-category of ...

**2**

votes

**0**answers

73 views

### Symmetric monoidal structure on categorical nerves

There are several notions of nerves, including nerves of categories, $2$-categories, and simplicial categories. These define functors
\begin{align*}
\mathrm{N} &\colon \mathrm{Cats}_{(2,1)} ...

**1**

vote

**0**answers

134 views

### Do all $\mathbb{E}_{k}$-comonoids in $\mathcal{C}_*$ come from “freely-pointed” $\mathbb{E}_{k}$-comonoids on $\mathcal{C}$?

In Coalgebras in symmetric monoidal categories of spectra, Péroux and Shipley prove the following (Lemma 2.4):
Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint ...

**9**

votes

**1**answer

336 views

### Tensor products of $\mathbb{E}_\infty$-spaces

In the $\infty$-world, connective spectra play the role of abelian groups, while $\mathbb{E}_\infty$-spaces play that of commutative monoids. This may be rephrased by saying that we may identify the $\...

**8**

votes

**0**answers

173 views

### Classifying spaces of monoidal categories and deloopings

$\newcommand{\abs}[1]{|#1|}$The classifying space $\abs{\mathcal{C}}$ of a category $\mathcal{C}$ is the geometric realisation $\abs{\mathrm{N}_{\bullet}(\mathcal{C})}$ of its nerve $\mathrm{N}_\...

**7**

votes

**0**answers

244 views

### Coherent objects in a hypercomplete $\infty$-topos

In Lurie's "Spectral Algebraic Geometry", Proposition A.6.6.1 (2) shows that for $\mathcal{X}$ an $\infty$-topos that is both locally coherent and hypercomplete, the full subcategory $\...

**10**

votes

**1**answer

397 views

### What is the free symmetric monoidal $\infty$-category on one object?

It is well-known that the free symmetric monoidal category on one object is the category $\mathbb{F}$ of finite sets and bijections. This is supposed to be the categorification of the monoid of ...

**11**

votes

**1**answer

333 views

### Intermediate notions of bilinearity in higher algebra

It is well-known that when passing to $\infty$-categories the notion of commutativity gets replaced by an infinite array of notions of commutativity: $\mathbb{E}_{1}$, $\mathbb{E}_{2}$, ..., $\mathbb{...

**4**

votes

**1**answer

142 views

### Is there essentially unique notion of module over monoidal stable $\infty$-categories?

There is this (folklore?) fact: for a commutative ring $R$, the category of $R$-modules is equivalent to the category of internal abelian groups in the slice category $\operatorname{Commutative rings}/...

**1**

vote

**1**answer

188 views

### Equivalence relations, Segal groupoids and groupoid objects in an infinity category

There are three forms of "equivalence relations are effective" as part of Giraud's axioms in $1$-Grothendieck topoi, Model topoi and Infinity topoi. I am trying to understand how they relate ...

**3**

votes

**0**answers

90 views

### A notion of "generalized nerve" of categories enriched over a presheaf

Let $\mathcal{C}$ be a small category, $p : \mathcal{C} \to \mathsf{Cat}$ a functor, and $\mathsf{P} = \mathrm{PSh}(\mathcal{C})$ the category of presheaves over $\mathcal{C}$ valued in sets. The ...

**11**

votes

**1**answer

360 views

### Making the ($\infty$-categorical) Bar construction valued in (bi)-modules

In Lurie's Higher Algebra, construction 4.4.2.7 presents a Bar construction in the setting of $\infty$-categories. The construction in 4.4.2.7 takes as input an $\...

**3**

votes

**0**answers

220 views

### Is hammock localization a localization in the sense of Lurie?

In a series of papers ([1], [2] and [3]), Dwyer and Kan introduced the hammock localization [2] as an effective technique to compute the simplicial localization of a model category [1]. This is meant ...

**3**

votes

**0**answers

141 views

### For which categories $D$ is a $D^{\vartriangleleft\vartriangleright}$-shaped diagram in a stable $\infty$-category a limit iff it is a colimit?

Throughout, I'll omit the "$\infty$" from the term "$\infty$-category".
It is well-known (and sometimes even included in the definition, although not by Lurie) that pushouts and ...

**4**

votes

**1**answer

75 views

### Non-enriched Bousfield localizations

We know that whenever we have a Bousfield localization between two simplicial model categories, this gives rise to a reflective subcategory in $\infty$-categories (or coreflective, depending on the ...

**2**

votes

**0**answers

73 views

### On $\mathbb{E}_{n-k}$-monoidal structures on $\mathbb{E}_{n-m}$-algebras in $\mathbb{E}_{n}$-monoidal $\infty$-categories

For ordinary categories, the assignment $\mathcal{C}\mapsto\mathsf{Mon}(\mathcal{C})$ defines a functor $\mathsf{Mon}\colon\mathsf{Alg}_{\mathbb{E}_{k}}(\mathsf{Cats})\to\mathsf{Alg}_{\mathbb{E}_{k-1}}...

**4**

votes

**1**answer

117 views

### Homotopy coherent space maps induces homotopy coherent chain complex morphisms

It is an elementary fact that a map $f:X \to Y$ between spaces induces a chain complex morphism $f_* : C_*(X) \to C_*(Y) $. This allows one to transfer 1-category theoretic arguments from spaces to ...

**1**

vote

**1**answer

100 views

### n-truncation/n-connected factorization in an $\infty$-topoi

I want to prove that given an $\infty$-topos $\mathscr{C}$ and a morphism $f: X \to Y$ in $\mathscr{C}$, for each $k \geq -1$, there exists a factorization $X \xrightarrow{\eta_f} E_k^f \xrightarrow{\...

**2**

votes

**0**answers

68 views

### Diagrammatic model for free product in monad infinity category

$\newcommand{\C}{\mathcal{C}}$ Suppose $M$ is a monad in an $\infty$-category $\C,$ and $A, B$ are two algebras over $M$. I'm willing to assume any reasonable "niceness" conditions on $\C$, $...

**19**

votes

**1**answer

756 views

### Why stable $\infty$-categories?

I begin by saying that while I understand what a triangulated / derived category is pretty well, I know nothing about Higher Algebra stuff and not even $\infty$-categories.
I've heard some people say ...

**2**

votes

**1**answer

134 views

### (Local) Homotopy dimension of $\infty$-topoi on paracompact spaces

I have a question concerning the proof of Corollary 7.3.6.5 in Luries "Higher Topos Theory" (the same issue also occurs in the proof of 7.3.6.10, but it is clearer here). Given is a ...

**5**

votes

**1**answer

131 views

### Join as a bifunctor

I have been reading these great notes by Charles Rezk, and one thing that has been bothering me is the join construction. To solve lifting problems in quasicategory theory we use the Leibniz ...

**3**

votes

**2**answers

349 views

### Model categories with uniqueness

I've been learning about the construction of $(\infty,1)$-categories from simplicial sets, and more generally about the model category structure on simplicial sets, defined in terms of lifting ...

**2**

votes

**0**answers

89 views

### Mapping spaces of simplicial model categories and quasicategories

Let $M$ be a simplicial model category, $M^o$ its full subcategory of bifibrant objects. The axioms of a simplicial model category guarantee that $M^o$ is enriched in Kan complexes. Thus the homotopy ...

**9**

votes

**2**answers

306 views

### Is the $\infty$-category $N_{dg}(\mathrm{Ch}(\mathcal{A}))$ presentable?

(See Jacob Lurie's "Higher Algebra", section 1.3.5 for context.)
Let $\mathcal{A}$ be a Grothendieck abelian category. Then the stable $\infty$-category $\mathcal{D}(\mathcal{A})$ is a ...

**6**

votes

**1**answer

164 views

### Monochromatic infinity operads as algebras over the "operad operad"

In the "ordinary" operad category, it is known that there is a colored operad $Op$ with set of colors $\mathbb{N}$ corresponding to "degrees" of vertices and with operations ...

**6**

votes

**2**answers

457 views

### Deformation of a diagram preserve the homotopy limit

I have been a bit sloppy in the title, but let me be specific. I stepped again into the subtle difference between homotopy limit and limit in the homotopy category, in the following version.
Suppose ...

**6**

votes

**1**answer

269 views

### $\infty$-natural transformations and adjunctions

I'm having troubles proving these two related statements, which are immediate for 1-categories and should of course be true for $\infty$-categories:
Given a natural transformation $\alpha: f \...

**2**

votes

**0**answers

64 views

### Identifying discrete points in derived hom spaces

Let M be a model category presenting an ∞-category $\mathcal{M}$, and let $f : X \to Y$ and $g : Y \to Z$ be arrows of M. Consider the following propositions:
The connected component of $f$ in $\...

**10**

votes

**1**answer

424 views

### Modern proofs for simplicial localizations

I know that the references usually regarded as standard for simplicial localizations are the Dwyer and Kan's three articles from the 80's. I would be interested in a more modern approach to the ...

**7**

votes

**1**answer

689 views

### Derived category of abelian sheaves on a site equivalent to sheaves on the derived category of abelian groups

Reading Scholze's notes on Condensed Mathematics it is mentioned that when considered as $\infty$-categories,
$$ D(\operatorname{Cond(Ab)}) \cong \operatorname{Cond}(D(\operatorname{Ab}))$$
and that ...

**6**

votes

**1**answer

255 views

### Using the universal property of spaces

The $\infty$-category of spaces has the following properties:
It is the $\infty$-category obtained from the (ordinary) category of finite sets by freely adding sifted colimits. (See e.g. Cesnavicius-...

**2**

votes

**1**answer

268 views

### Counterexamples concerning $\infty$-topoi with infinite homotopy dimension

In "Higher Topos Theory", Lurie introduces three different notions of dimension for an $\infty$-topos $\mathcal{X}$, namely:
Homotopy dimension (henceforth h.dim.), which is $\leq n$ if $n$-...

**13**

votes

**1**answer

708 views

### $\infty$-topoi versus condensed anima

Let $ExDisc_\kappa$ denote the category of $\kappa$-small extremally disconnected topological spaces (for now fix a strong limit cardinal $\kappa$). There's a functor $ExDisc_\kappa \to \mathsf{RTop}$ ...

**7**

votes

**1**answer

539 views

### Functorial kernel in derived category

By the work of Verdier, we know that cones in a triangulated category $\mathcal{T}$ are functorial if and only if $\mathcal{T}$ is semisimple abelian. However, in these notes, it is said that
In the ...

**8**

votes

**0**answers

165 views

### Symmetric monoidal structure(s) on the $\infty$-category of dg-categories

Let $k$ be a commutative ring with $1$, and let $\mathsf{dgCat}_k$ be the category of $k$-linear dg-categories, as defined in [1, Section 2]. We may equip $\mathsf{dgCat}_k$ with the Morita model ...

**7**

votes

**2**answers

228 views

### What is the functoriality of the $\infty$-categorical slice construction?

The "slice" construction associates to a functor $f\colon C\to D$ a category $C_{f/}$, whose objects are pairs $(d,f\xrightarrow{\gamma}\kappa_d)$, where $d$ is an object of $D$, and $\...

**3**

votes

**0**answers

125 views

### On coalgebras and comodules in slice $\infty$-categories

Given a presentable Cartesian symmetric monoidal $\infty$-category $C$, every object is a cocommutative comonoid and for a fixed $Z\in C$ there is an equivalence $C_{/Z}\simeq LCoMod_{Z}(C)$ where the ...