# Questions tagged [infinity-categories]

The infinity-categories tag has no usage guidance.

580
questions

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### How to prove the classical definition of addition map for additive ordinary categories is commutative and associative up to coherent homotopy?

Could somebody please help me with this?
We know if $C$ is an ordinary additive category, then the addition of maps $f,g:x \rightarrow y$ coincides with the composition
$$x \xrightarrow{\bigtriangleup}...

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0
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### Motivic stable homotopy categories of ind-schemes

In the work A Motivic Snaith Decomposition, Viktor Kleen extends the notion of motivic stable homotopy categories $\mathbf{SH}$ to smooth ind-schemes over a base $S$ (colimit of smooth $S$-schemes) by ...

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### Generalized $\infty$-operads are an analog of ??? in $\infty$-category theory

In Section 2.3.2 of Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads. This is a functor $p:\mathcal{O}^\otimes \to \mathcal{F}\mathrm{in}_\ast$ of $\infty$-categories, where ...

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### Equivalence of two definitions of relative limits

This is a question on seemingly equivalent definitions of relative limits, formulated in the language of quasi-categories. I will use notations from Higher Topos Theory.
Let $p:\mathcal{C}\to\mathcal{...

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1
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### Homotopy (co)limits in oo-categories vs model categories

In $\infty$-category theory one can define limits and colimits by analogues of the usual universal properties, but stated in terms of mapping spaces and homotopy equivalences instead of mapping sets ...

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### $\infty$-category of spectra and cofibrancy

I have two options for the $\infty$-category of spectra. I would like to know they are equivalent as $\infty$-categories.
Premise: by work of Dwyer and Kan, if we have a simplicial model category, the ...

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### Is there a synthetic approach to (symmetric) monoidal infinity-categories?

Recent work of Riehl and Verity (e.g. the book "Elements of $\infty$-category theory") has established a "synthetic" / model-independent approach to the study of $\infty$-...

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### Is strictness decidable?

Let $\mathcal C$ be an $\infty$-category. We can ask:
Q: Is $\mathcal C$ a 1-category?
That is, are the hom-spaces of $\mathcal C$ essentially discrete?
Roughly, my question is:
Proto-Question: Is Q ...

6
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### Universal property of category of categories

As discussed here, Using the universal property of spaces, the $(\infty,1)$-category of spaces has a universal property: it is the free $\infty$-categorical cocompletion of the terminal category $*$. ...

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+50

### Descent vs effective descent for morphisms of ring spectra

Define a homomorphism $\varphi : A \to B$ of commutative discrete rings or commutative ring spectra to be a (effective) descent morphism if the comparison functor from $\mathsf{Mod}_A$ to the category ...

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### Quasi-equivalent vs. homotopy equivalent functors in $A_\infty$ categories

Suppose that $\mathcal{A}, \mathcal{B}$ are strictly unital $A_\infty$ categories, and $\mathcal{F}, \mathcal{G}: \mathcal{A} \rightarrow \mathcal{B}$ are (strictly unital) functors.
On one hand, we ...

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### Two definitions of a monad on an ∞-category

In the literature on $\infty$-categories (quasi-categories) I found two different definitions of a monad on an $\infty$-category, and I don't understand the relation between them.
The first ...

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### The infinity category of dg-categories is bicomplete

We can define the $\infty$-category of dg-categories $dgCat_\infty$ as the definition of the $\infty$-category of $\infty$-categories which given gy the section.3 of J.Lurie "Higher Topos Theory&...

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### A fibration for the functor category functor

Consider the (2,1) category $Cat$ of ($1$-)categories. There is a functor
$$
Cat^{op}\times Cat\to Cat
$$
sending $(C,D)$ to the functor category $Fun(C,D)$.
This gives rise to a fibration $F\to Cat^{...

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1
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### $\operatorname{Fun}(\mathcal{C},\mathcal{D})^n$ is a subcategory of $\operatorname{Fun}(\mathcal{C}^n,\mathcal{D}^n)$

Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$-categories (by which I mean quasicategories, though I suspect that it hardly matters), and let $n\geq 1$ be an integer. There is a functor
$$\theta:\...

6
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1
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### Reference: the category of derived affine schemes is extensive

The category (that is, $(\infty, 1)$-category) of derived affine schemes is the opposite category of the localization of simplicial commutative rings in weak equivalences.
See extensive category. Does ...

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### How to get an $E_\infty$-ring from a commutative differential graded ring?

I want to figure out the following question: How to get an $E_\infty$-ring from a commutative differential graded ring?
More precisely, let $\operatorname{cdga}$ be the ($1$-)category of cdgas, let $...

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0
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### Treatment of classes of mono/epi morphisms in $(\infty, 1)$-categories

In the classical theory of $(1, 1)$ categories, the chain of classes of mono/epi morphisms is well known: plain $\leftarrow$ strong $\leftarrow$ effective $\leftarrow$split ((I assume that the ...

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### What do we know about effective epimorphisms of derived affine schemes/manifolds?

By default, all terms are understood in the infinity sense (“category” means “(∞,1)-category”, etc.)
Recall that the morphism $X \to Y$ is an effective epimorphism if the Čech diagram
$$ ... \to X \...

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### Double categories and fibrations

Is there a way in which Conduche fibrations can lead to completeness in double categories? I know that Conduche conditions on functors play a role in completeness or cocompleteness in pseudo-double ...

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### When do the different notions of homotopy inside a general simplicial set agree?

$\newcommand{\defeq}{\overset{\mathrm{def}}{=}}\newcommand{\rmR}{\mathrm{R}}\newcommand{\rmL}{\mathrm{L}}\newcommand{\End}{\mathrm{End}}\newcommand{\Hom}{\mathrm{Hom}}$This question is a sequel to my ...

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### Homotopy coherent transformation and totalization

Let $C$ be the category of chain complexes over a field $F$ and $C^\prime$ be the subcategory of chain complexes with zero differentials. If $X:I\to C$ is a functor, there is an induced "homology&...

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1
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### Conservative cocompletion of categories of geometric shapes for homotopy theory

The recent paper
Calin Tataru, Partial orders are the free conservative cocompletion of total orders. arXiv:2404.12924
has shown that the conservative cocompletion of the simplex category $\Delta$ ...

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0
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### Isbell duality for simplicial sets

$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Nat}{\mathrm{Nat}}$Isbell duality sets up an adjunction (see here for a short abstract summary)
$$\mathsf{O}\dashv\...

4
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1
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### Why is the universal $n$-gerbe universal? (HTT, 7.2.2.26)

Let $\mathcal{X}$ be an $\infty$-topos and let $A$ be an abelian group object of the category $\operatorname{Disc}(\mathcal{X})$ of discrete objects of $X$. Recall that a morphism $f:\widetilde{X}\to ...

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### "$X$ is $n$-truncated $\iff$ $\Omega X$ is $(n-1)$-truncated" for connected pointed $X$. (HTT, 7.2.2.11)

In the proof of Lemma 7.2.2.11 of Higher Topos Theory, Lurie makes the following claim:
($\ast$) Let $n\geq1$ be an integer, let $\mathcal{X}$ be an $\infty$-topos, and let $1\to X$ be a pointed ...

4
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1
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### Left Kan extension and finite product preserving

Let ${\rm Ani(Ring)}$ be the $\infty$-category of animated commutative rings, ${\rm Ani(Ring)^{\leq 0}}$ be the category of discrete animated rings and ${\rm Ani}$ be the $\infty$-category of spaces . ...

2
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1
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### Lax Gray tensor product and opposite categories:

For any two strict (infinity, infinity)-categories $A,B$ let $A \otimes B $ be the lax Gray tensor product of $A$ and $B$. Let $A^{op}$ be the opposite (infinity,infinity)-category, where morphisms in ...

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### Canonical comparison between $\infty$ and ordinary derived categories

This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...

13
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1
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### On Lemma 5.5.16 of Cisinski's "Higher Categories and Homotopical Algebra"

I have a question regarding Section 5 of Cisinski's
"Higher Categories and Homotopical Algebra".
Let us write $\mathbf{sSet}$ and $\mathbf{bisSet}$ for the categories of
simplicial sets and ...

4
votes

1
answer

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### Presentability rank of tensor product of presentable categories

In this post category means $(\infty, 1)$-category.
Let $X, Y$ be two presentable categories. We can then form their tensor product $X \otimes Y \cong \operatorname{ContFun}(X^{\mathrm{op}}, Y)$. Can ...

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1
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### Co/limits and 2-co/limits of categories in the $\infty$- and $(\infty,2)$-category of $\infty$-categories

Recently, in a conversation with Gabriel, the following question came up:
Question. Do co/limits of categories taken in the $\infty$-category of $\infty$-categories agree with the usual co/limits ...

3
votes

1
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### Different notions of equivalences of $\mathcal{O}$-monoidal $\infty$-categories

I am currently reading Higher Algebra by Jacob Lurie and I have a question regarding equivalences of $\mathcal{O}$-monoidal categories.
Let $\mathcal{O}$ be an $\infty$-operad. Suppose that I have two ...

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1
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### Completeness of comma $\infty$-categories

Let $\mathsf{A},\mathsf{B},$ and $\mathsf{C}$ be (ordinary) categories and $F : \mathsf{A}\to\mathsf{C}$ and $G : \mathsf{B}\to\mathsf{C}$ be functors such that
$\mathsf{A}$ and $\mathsf{B}$ are ...

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### Reference request for equivalences between different models of lax limits

There are several models for lax limits of model categories/ $\infty$-categories in the literature. For example, within the realm of $\infty$-categories one can construct them using coCartesian ...

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### Hypercube of chain complexes as functor from (Δ^1 )^n to ∞-category of chain complexes

A hypercube of chain complexes consists of $\mathbb{Z}$-graded vector spaces $C_\epsilon$ for $\epsilon\in\{0,1\}^n$ and maps $D_{\epsilon,\epsilon^\prime}:C_{\epsilon}\to C_{\epsilon^\prime}$ for $\...

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### Finitely presentable objects in the categories of algebras of $\infty$-algebraic theories

By default, all terms are understood in the infinity sense (“category” means “$(\infty, 1)$-category”, etc.).
An object $A$ in a category is said to be finitely presentable (or compact) if the functor ...

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1
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### How to prove a 1-localization of a 1-category is already an $(\infty,1)$-localization?

I don't even know this fits in here or in Mathematics Stack Exchange, but let me ask. I'm new to simplicial stuff, so a good reference would be quite helpful.
Let's say $C$ is a certain category, and ...

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### Tensor product of modules in model vs. infinity categories

Let $C$ be a combinatorial symmetric monoidal model category and let $A$ be a associative algebra object in $C$, that is cofibrant as an object in $C$. In Higher Algebra 4.3.3.17, Lurie proves an ...

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### Examples of $\ast$-autonomous $(\infty,1)$-categories

A $\ast$-autonomous category is a biclosed monoidal category together with a dualizing object. An object $\bot$ in a biclosed monoidal category $(\mathcal{C},\otimes)$ with left internal hom $[-,-]$ ...

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2
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### Natural ways to make a functor adjoint

Let $F: C \to D$ be a functor between two categories without a right adjoint. What are some natural ways to create a right adjoint for $F$?
Of course, this does not make sense on the nose. One needs ...

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### Comparing notions related to $(\infty,2)$-categories

I am trying to understand two related notions:
$(\infty,2)$-category as in Definition 5.5.1.3, Kerodon
weak $\infty$-bicategory as in Definition 4.1.1 in "$(\infty,2)$-Categories and the ...

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### Comparing the Stacks Project Homotopy limit with limits in the $\infty$-category

In the Stacks project Tag 08TC, there is a definition of a homotopy limit in a derived category, and I expect it to compare with a limit in the $\infty$-categorical enhancement. I guess this is also ...

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2
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### Adjunctions and inverse limits of derived categories

Consider a tower $\dots\to A_{2}\to A_{1}$ of rings. This gives rise to a diagram $\mathbb{N}^{\text{op}}\to\text{Cat}_{\infty}$ of $\infty$-categories (confusing $\mathbb{N}^{\text{op}}$ with its ...

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### Homotopy theory of differential objects

In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...

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### What is the exact definition of the $\infty$-topos of sheaves on a localic $\infty$-groupoid?

The category $\mathrm{Locale}$ is equivalent to the category $0\text{-}\mathrm{Topos}$ .
The 2-category $\mathrm{LocalicGroupoid}$ (with suitable localization) is equivalent to the 2-category $1\text{...

5
votes

1
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### Do finitely presentable $\infty$-groupoids precisely correspond to the finite cell complexes?

In the Higher Topos Theory, Example 1.2.14.2 says “finitely presentable $\infty$-groupoids correspond precisely to the finite cell complexes” But, for example, $K(\mathbb{Z}, 2)$ is seems finitely ...

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### Cocartesian fibration classifying $\mathrm{Fun}(F,G)$

Throughout this question we consider $\infty$-categories.
Fix a cartesian fibration $p : \mathcal{F} \to \mathcal{C}$ and a cocartesian fibration $q : \mathcal{G} \to \mathcal{C}$ which straighten to $...

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1
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### Filtered homotopy colimits of spectra

Let $\mathcal{I}: \mathbb{N} \to \operatorname{Sp}$ be a diagram in the infinity category of spectra. Let $\pi_0(\mathcal{I})$ denote the corresponding $1$-categorical diagram (i.e. compose $\mathcal{...

1
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1
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### Proof in Higher Algebra that $\mathcal{C}at(\mathcal{K})$ is presentable

In Higher Algebra Lemma 4.8.4.2, Lurie shows that for $\mathcal{K}$ a small set of simplicial sets, the $\infty$-category $\mathcal{C}at(\mathcal{K})$ of small $\infty$-categories with $K$-shaped ...