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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}...
A.karimi's user avatar
1 vote
0 answers
76 views

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 ...
Alexey Do's user avatar
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12 votes
0 answers
160 views

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 ...
Ken's user avatar
  • 2,164
4 votes
0 answers
52 views

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{...
Ken's user avatar
  • 2,164
5 votes
1 answer
228 views

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 ...
atticusw's user avatar
  • 155
5 votes
0 answers
146 views

$\infty$-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 ...
vap's user avatar
  • 412
6 votes
0 answers
114 views

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 $\infty$-...
John Nolan's user avatar
7 votes
0 answers
140 views

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 ...
Tim Campion's user avatar
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6 votes
0 answers
173 views

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 $*$. ...
user39598's user avatar
  • 499
8 votes
0 answers
401 views
+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 ...
Brendan Murphy's user avatar
5 votes
1 answer
106 views

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 ...
Tom Hockenhull's user avatar
9 votes
1 answer
394 views

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 ...
Sergei Ivanov's user avatar
3 votes
0 answers
77 views

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&...
Keima's user avatar
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3 votes
1 answer
107 views

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^{...
DamienC's user avatar
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7 votes
1 answer
210 views

$\operatorname{Fun}(\mathcal{C},\mathcal{D})^n$ 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:\...
Ken's user avatar
  • 2,164
6 votes
1 answer
236 views

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 ...
Arshak Aivazian's user avatar
6 votes
2 answers
264 views

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 $...
Yebo Peng's user avatar
6 votes
0 answers
115 views

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 ...
Arshak Aivazian's user avatar
4 votes
0 answers
190 views

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 \...
Arshak Aivazian's user avatar
2 votes
0 answers
60 views

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 ...
Siya's user avatar
  • 195
4 votes
0 answers
94 views

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 ...
Emily's user avatar
  • 11.5k
3 votes
1 answer
145 views

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&...
vap's user avatar
  • 412
8 votes
1 answer
341 views

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 $\Delta$ ...
Emily's user avatar
  • 11.5k
13 votes
0 answers
220 views

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\...
Emily's user avatar
  • 11.5k
4 votes
1 answer
237 views

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 ...
Ken's user avatar
  • 2,164
2 votes
1 answer
146 views

"$X$ 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 ...
Ken's user avatar
  • 2,164
4 votes
1 answer
119 views

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 . ...
Y.M's user avatar
  • 121
2 votes
1 answer
112 views

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 ...
Hadrian Heine's user avatar
6 votes
1 answer
565 views

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\...
Stahl's user avatar
  • 1,179
13 votes
1 answer
440 views

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 ...
Keisuke Hoshino's user avatar
4 votes
1 answer
205 views

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 ...
Brendan Murphy's user avatar
5 votes
1 answer
145 views

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 ...
Emily's user avatar
  • 11.5k
3 votes
1 answer
117 views

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 ...
YjL's user avatar
  • 41
5 votes
1 answer
164 views

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 ...
Stahl's user avatar
  • 1,179
7 votes
1 answer
167 views

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 ...
happymath's user avatar
  • 167
2 votes
0 answers
114 views

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 $\...
Faniel's user avatar
  • 653
7 votes
1 answer
220 views

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 ...
Arshak Aivazian's user avatar
7 votes
1 answer
164 views

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 ...
gksato's user avatar
  • 357
5 votes
0 answers
82 views

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 ...
Jakob's user avatar
  • 1,986
4 votes
0 answers
140 views

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 $[-,-]$ ...
Max Demirdilek's user avatar
7 votes
2 answers
385 views

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 ...
Student's user avatar
  • 5,038
5 votes
0 answers
363 views

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 ...
Balaji Subramoniam's user avatar
6 votes
1 answer
286 views

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 ...
user141099's user avatar
3 votes
2 answers
223 views

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 ...
user141099's user avatar
7 votes
0 answers
269 views

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 $S: \mathsf{C} \to \mathsf{C}$...
ಠ_ಠ's user avatar
  • 5,953
8 votes
0 answers
229 views

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{...
Arshak Aivazian's user avatar
5 votes
1 answer
266 views

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 ...
Arshak Aivazian's user avatar
4 votes
1 answer
222 views

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 $...
daniel gratzer's user avatar
2 votes
1 answer
313 views

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{...
Laurent Cote's user avatar
1 vote
1 answer
207 views

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 ...
Markus Zetto's user avatar

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