Questions tagged [infinity-categories]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4 votes
1 answer
170 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
117 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
  • 10k
3 votes
1 answer
105 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
1 vote
0 answers
130 views

Computing nonabelian derived functors on fibrant-cofibrant objects

I am learning the process of "Animation" from Cesnavicius and Scholze's paper Purity for flat cohomology. In my understanding the animation of a category/functor is simply the nonabelian ...
Yebo Peng's user avatar
5 votes
1 answer
139 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
  • 1,049
7 votes
1 answer
150 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
110 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
  • 603
6 votes
1 answer
177 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
133 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
  • 317
5 votes
0 answers
74 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
135 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
369 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
  • 5,008
5 votes
0 answers
345 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
259 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
211 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
262 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
  • 5,895
8 votes
0 answers
217 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
251 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
0 answers
111 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
248 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
195 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
2 votes
1 answer
80 views

Reference request-Natural equivalence detected pointwise for complete Segal spaces

I am looking for a reference for the following elementary assertion on complete Segal spaces: Let $A$ be a bisimplicial set and let $W$ be a complete Segal space. A morphism of $W^A$ is an ...
Ken's user avatar
  • 1,739
9 votes
1 answer
429 views

Is there a shape-independent definition of (∞,1)-categories?

For all definitions of $\infty$-categories I am aware of, an $(\infty,1)$-category is defined via reference to some shape, be it simplices in a form of a quasi-category or a cubical analogue of a ...
lemmanade's user avatar
6 votes
1 answer
378 views

A possible alternative model for $\infty$-groupoids

I've been studying homotopy theory on myself for quite some time now, and it is to my understanding that there's still no generally accepted definition for $\infty$-groupoids. The closest to this is ...
XiaohuWang's user avatar
3 votes
0 answers
124 views

$\mathbf{E}_n$-algebras in nerves of 2-categories

In Example 5.1.2.4 of Higher Algebra, Lurie explains how there is a bijection between equivalence classes braided monoidal structures on one-category $\mathcal{C}$ and $\mathbf{E}_2$-algebra ...
Pulcinella's user avatar
  • 5,506
5 votes
1 answer
143 views

Simplicial objects in quasicategory which come from homotopy coherent nerve

Let $\mathcal{C}$ be a simplicially enriched category whose Hom-objects are all Kan complexes. Denote by $N\mathcal{C}$ the homotopy-coherent nerve of $\mathcal{C}$, which is a quasicategory. Suppose ...
K. Strong's user avatar
  • 335
9 votes
1 answer
194 views

Does $\infty$-categorical localization commute with taking directed fibered products?

Suppose we are given categories $\mathsf{C},\mathsf{D},\mathsf{E},$ equipped with collections of weak equivalences $\mathcal{W}_{\mathsf{C}},\mathcal{W}_{\mathsf{D}},$ and $\mathcal{W}_{\mathsf{E}},$ ...
Stahl's user avatar
  • 1,049
3 votes
1 answer
116 views

Reference request: the free adjunction being free as an $(\infty, 2)$-category?

This question is a particular case of Tim Campion's question. Let $\mathrm{Adj}$ be the strict $2$-category corepresenting adjunctions, i.e., the free strict $2$-category generated by two objects $x, ...
nrkm's user avatar
  • 424
4 votes
0 answers
91 views

Is the pushforward of an exponentiable fibration along an exponentiable fibration again exponentiable?

Recall that functor $p\colon \mathcal{C} \to \mathcal{D}$ of $\infty$-categories is said to be an exponentiable fibration if the following equivalent conditions hold: The pullback functor $p^*\colon \...
Yonatan Harpaz's user avatar
4 votes
1 answer
183 views

Model categories: "equivalence" of finite limits and finite colimits

I am needing a reference for the following statement (in case it is true): Quillen functor between stable model categories preserve finite limits iff it preserves finite colimits. For stable $\infty$-...
Alexey Do's user avatar
  • 636
5 votes
1 answer
190 views

Are $\infty$-categories functorially colimits of their simplices?

Let $\mathcal C$ be an $\infty$-category. If $C$ is a quasicategory modeling $\mathcal C$, then we have a coend decomposition $$\mathcal C = \int^{[n] \in \Delta} \Delta[n] \times C_n.$$ This allows ...
Tim Campion's user avatar
  • 60.5k
6 votes
0 answers
203 views

Higher categories using just simplicial sets

Is there a definition of $(\infty, n)$-category using just simplicial sets? This is the case for $n \leq 2$. Is the forgetful functor from saturated $n$-trivial complicial sets to simplicial sets an ...
Daniel Bruegmann's user avatar
7 votes
1 answer
271 views

From the *usual* nerve of topological categories to $\infty$-categories

It is standard from work of Joyal and Lurie that there is a Quillen equivalence between the model category of simplicially enriched categories $Cat_\Delta$ and $\mathcal{S}\text{et}_\Delta$ with the ...
Xin Jin's user avatar
  • 337
5 votes
1 answer
278 views

Commuting homotopy colimits and arbitrary products in spaces

Let $X : D \rightarrow Spc$ be a diagram with values in the $\infty$-category of spaces and $I$ some (discrete) set, not necessarily finite. ($D$ can be a 1-category if that makes statements easier, ...
Georg Lehner's user avatar
  • 1,933
10 votes
2 answers
562 views

Effective epimorphisms and 0-truncations (HTT, 7.2.1.14)

In Proposition 7.2.1.14 of Higher Topos Theory, Lurie asserts the following: Let $\mathcal{X}$ be an $\infty$-topos and let $\tau_{\leq0}:\mathcal{X}\to\tau_{\leq0}\mathcal{X}$ denote a left adjoint ...
Ken's user avatar
  • 1,739
3 votes
1 answer
178 views

A fiber-like method to show equivalence of infinity categories

Suppose I have a functor of quasi-categories $f: \mathcal{C} \to \mathcal{D}$. I want to show a criterion like: "$f$ is an equivalence of $\infty$-categories if the homotopy fiber of $f$ ...
Andrea Marino's user avatar
11 votes
1 answer
427 views

Recasting straightening/unstraightening equivalence as $(\infty, 2)$-adjunction

This is a vague set of questions that relies on (possibly non-existent) generalizations of low-dimensional results, mostly because I don't know many of the technical details underlying the ...
Lorenzo Riva's user avatar
4 votes
0 answers
104 views

Equalizer-product formula for $(\infty, 1)$-limits

If $F : K \to C$ is a functor of ordinary categories and $C$ has products and equalizers, then there is an isomorphism \begin{equation*} \lim F \cong \mathrm{eq} \left( \prod_{k_0 \in K_0} F(k_0) \...
Lorenzo Riva's user avatar
4 votes
2 answers
447 views

Categorical equivalences vs. categories of simplices

Let $j: K\to K′$ be a categorical equivalence of simplicial sets. By [HTT, Remark 2.1.4.11], we have a Quillen equivalence (with the covariant model structures) $$ j_!:\mathsf{sSet}_{/K}\...
Lao-tzu's user avatar
  • 1,856
5 votes
1 answer
212 views

Connectedness of truncated version of cosimplicial indexing category

Let $F:\mathbf{\Delta}\to\mathcal{S}_{\leqslant n-1}$ be a cosimplicial object in the $\infty$-category of $(n-1)$-truncated spaces. Is it always a right Kan extension of its restriction along $\...
Lao-tzu's user avatar
  • 1,856
28 votes
5 answers
5k views

What is the motivation for infinity category theory?

To my understanding, most mathematical theories can be simply understood in the view point of Category theory and its derivative theories. But what exactly is the motivation to study infinity category ...
tryst with freedom's user avatar
3 votes
1 answer
116 views

$n$-truncation of a Simplicial Model Category

I'm working in the category of rational $CDGAs$ and trying to find a reference/construction of a natural $2$-categorical structure via truncation of the mapping spaces. In my head, the key point is ...
kelly maggs's user avatar
1 vote
1 answer
347 views

Limits of infinity categories and mapping spaces

Let $p:I\to Cat_{\infty}$ be a diagram of infinity categories, where $I$ is a small Kan complex. Let $C:=\lim p$ be the limit of $p$. For any two objects $x,y\in C$ and $i\in I$, let $x_i,y_i\in C_i=p(...
Kim's user avatar
  • 505
2 votes
2 answers
292 views

Does the homotopy category of finite spectra act on stable homotopy categories?

Assume that C is a stable infinity category; $SH_{fin}$ is the homotopy category of finite spectra. Is there a canonical bi-functor (action? module structure?) $SH_{fin}\times hC \to hC$? Is there any ...
Mikhail Bondarko's user avatar
5 votes
1 answer
220 views

Cofinal maps from posets (HTT, 4.2.3.16)

I do not understand the proof of Variant 4.2.3.16 of Higher Topos Theory by Jacob Lurie, and I need help. Variant 4.2.3.16 asserts the following: ($\diamond$) Let $K$ be a finite simplicial set. ...
Ken's user avatar
  • 1,739
2 votes
1 answer
166 views

Is the symmetric monoidal product on the $\infty$-category of $R$-modules unique?

In Higher Algebra 4.2.8.19, Lurie shows that the symmetric monoidal structure on spectra is uniquely defined (on the $\infty$-category level) by the following properties: The sphere spectrum is the ...
Colin Aitken's user avatar
5 votes
2 answers
355 views

Monomorphisms of diagrams in an $\infty$-category

Let $f,g\colon K\to \mathcal{C}$ be diagrams in a nice $\infty$-category $\mathcal{C}$. I have two general questions: If I have a natural transformation $\eta\colon f\Rightarrow g$ which is a ...
Jonathan Beardsley's user avatar
4 votes
0 answers
94 views

Localization and space of morphisms

I have a question regarding the proof of Proposition 2.19 of Factorization homology of topological manifolds by Ayala and Francis. In the final paragraph of the proof (more specifically, in the second ...
Ken's user avatar
  • 1,739
9 votes
2 answers
379 views

Simplicial sets with horn filling conditions up to some fixed degree

Let $X_\bullet$ be a simplicial set such that some horn filling condition (inner horns fill/inner horns fill uniquely/all horns fill) holds up to dimension $n$ (i.e. for $\Lambda_i[p]$ for all $p\leq ...
Tim's user avatar
  • 1,237
4 votes
1 answer
167 views

Homotopy totalization and chains - reference

Simple case. Take $X_{\bullet}$ a cosimplicial space. Is it true that the chain complex of its homotopy totalization is quasi-isomorphic to the homotopy totalization of its chain complex? Because of ...
Andrea Marino's user avatar

1
2 3 4 5
12