Questions tagged [infinity-categories]
The infinity-categories tag has no usage guidance.
551
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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 ...
5
votes
1
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117
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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
answer
105
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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 ...
1
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0
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130
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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 ...
5
votes
1
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139
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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 ...
7
votes
1
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150
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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 ...
2
votes
0
answers
110
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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 $\...
6
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1
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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 ...
7
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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 ...
5
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0
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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 $[-,-]$ ...
7
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2
answers
369
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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 ...
5
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0
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345
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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 ...
6
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1
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259
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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 ...
3
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2
answers
211
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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 ...
7
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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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251
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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 ...
4
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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 $...
2
votes
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
vote
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 ...
2
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1
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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 ...
9
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1
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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 ...
6
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1
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378
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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 ...
3
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0
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124
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$\mathbf{E}_n$-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 ...
5
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1
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143
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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 ...
9
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1
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194
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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 $\mathcal{W}_{\mathsf{E}},$ ...
3
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1
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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, ...
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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 \...
4
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1
answer
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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 $\infty$-...
5
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1
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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 ...
6
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0
answers
203
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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 ...
7
votes
1
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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 ...
5
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1
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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, ...
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2
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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 ...
3
votes
1
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178
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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 $f$ ...
11
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1
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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 ...
4
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0
answers
104
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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) \...
4
votes
2
answers
447
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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}\...
5
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1
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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 $\...
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What is the motivation for infinity category theory?
To my understanding, most mathematical theories can be simply understood in the view point of Category theory and its derivative theories. But what exactly is the motivation to study infinity category ...
3
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1
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$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 ...
1
vote
1
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347
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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(...
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2
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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 ...
5
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1
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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. ...
2
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1
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166
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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 ...
5
votes
2
answers
355
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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 ...
4
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0
answers
94
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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 ...
9
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2
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379
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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 ...
4
votes
1
answer
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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 ...