Questions tagged [infinity-categories]
Part of higher category theory that for instance in Algebraic Topology enables us to capture finer homotopic distinctions. As in say Eilenberg-Maclane spaces.
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Algebras over the trivial $\infty$-operad
I'm learning the concept of algebras over $\infty$-operads, following Higher Algebra. The simplest case is when the operad being the trivial operad $\mathrm{Triv}^\otimes$, defined as the 1-full ...
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Are most true statements about Math unprovable (undecidable)? [duplicate]
In an essay titled: All Questions Answered, Donald Knuth states that “In fact, we now know that in some sense almost all correct statements about mathematics are unprovable.” How do we know that?
I’m ...
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Is the unbounded derived $\infty$-category of a general abelian category stable?
Let $A$ be any abelian category. Consider the stable $\infty$-category $N_{dg}(Ch(A))$ defined as the differential graded nerve of the category of chain complexes $Ch(A)$ in $A$. We can define an $\...
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Is the geometric realization of simplicial functors interesting?
While studying a completely unrelated problem, I have proved something on the following line: given a diagram of simplicial sets $X: C \to \textrm{sSet}$, some deformations of the geometric ...
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What is the group completion of the underlying multiplicative $\mathbb{E}_\infty$-monoid of the sphere spectrum?
I recently noticed the following categorical/universal way to describe the passage from $\mathbb{Z}$ to $\mathbb{Q}$:
We start with the categroy $\mathsf{Sets}^{\mathrm{actv}}_*$ of pointed sets and —...
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Is there a concise description of the $\infty$-category $\mathrm{Mod}_A^\mathcal{O}(\mathcal{C})$ of modules over an algebra over an $\infty$-operad?
[Cross-posted from this Math SE question.]
In Higher Algebra, Section 3.3 Lurie constructs the $\infty$-operads $\newcommand{\Mod}{\mathrm{Mod}}\newcommand{\cO}{\mathcal{O}}\newcommand{\cC}{\mathcal{C}...
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The assignment of derived infinity category of étale sheaf is an infinity functor?
Consider the ordinary category of schemes $Sch$, for $X\in Sch$, consider the abelian category of étale sheaf with coefficient $\wedge$ as $Mod(X_{ét},\wedge)$, then we can form the derived infinity ...
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The induced map between push outs in an exact infinity category
Let $(\mathcal{C} , \mathcal{M} , \mathcal{E})$ be an exact $\infty$-category. (I am following the definition in Higher Segal Spaces $I$ by Dyckerhoff and Kapranov). Assume that $F$ is a cofiberation ...
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Is the mapping anima functor a functor of infinity-categories?
Are two definitions of compact objects equivalent?
We refer to two definitions of compact objects in "Higher Topos Theory" (HTT) by Lurie and "Sheaves on Manifolds" (KNP) by ...
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Construction of smooth projective space in Spectral Algebraic Geometry
In section 19.2.6 of Lurie's "Spectral Algebraic Geometry," he constructs the smooth projective space, which represents the derived version of [the dual of] the usual functor of points ...
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Functoriality of infinite suspension spectrum functor on infinity groupoids!
Consider the functor $F: C \rightarrow D $ of $\infty$-groupoids. Is there any explicit proof somewhere in the literature that $\Sigma^{\infty}$ construction is functorial? I mean how do we define $\...
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Non-strictly unital functors of infinity categories?
One potential limitation of the quasicategory model of $(\infty,1)$-categories seems to be that the identity morphisms are "a part of the structure" of each quasicategory, and that morphisms ...
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Étale morphisms of derived schemes and stacks
Conventions: In the below,
unless otherwise stated, terms regarding derived algebraic geometry will follow the conventions of Yaylali.
an algebraic stack will be a stack $\mathscr{S}$ over a base ...
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Compact category which is not idempotent complete
I am interested in finding an example of a category $C$ that is a compact object in the presentable category $Cat$ of small $(\infty,1)$-categories and is not idempotent complete. A category $C$ is ...
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2-category structure on Mod(R)
Apologies for the basic question but I'm curious to know if there is an ``interesting" $2$-category structure on the category of modules over a ring $R$.
Essentially what is not clear to me if $M,...
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Singular cohomology as a sheaf of $\infty$-categories
In several expositions of $\infty$-categories, I read that singular cohomology of a topological space with integral coefficients is a sheaf valued in $D(\mathbb{Z})$, if we consider Top and $D(\mathbb{...
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Relationship between infinite suspension $\Sigma^{\infty}$ of $E_{\infty}$ grouplike space and its infinite delooping?
For an object $X$ in the infinity category of pointed space $S_{*}$, if it has an $E_{\infty}$ grouplike structure, then it give rises to a unique infinite delooping $BX$, which is a connective ...
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What is the infinity category of subspaces of $\mathbb{R}^n$?
Let $\mathcal{J}$ denote the topological category of finite-dimensional real inner product spaces with linear isometric embeddings. The space of morphisms $\mathcal{J}(\mathbb{R}^k, \mathbb{R}^n)$ is ...
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Proofs of the loop-suspension adjunction in infinity-categories
$\DeclareMathOperator{\Map}{Map}$$\DeclareMathOperator{\Fun}{Fun}$$\DeclareMathOperator{\const}{const}$$\DeclareMathOperator{\colim}{colim}$$\DeclareMathOperator{\lim}{lim}$In Elements of $\infty$-...
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Criterion for the Lurie tensor product of $\infty$-categories to commute with infinite products
The title of the question says it all. When does the Lurie tensor product of $\infty$-categories ([Higher Algebra] Section 4.8.1) commute with infinite products?
$$\mathcal{C} \otimes \prod_{i\in I}{\...
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Existence of Kan extension for the functor with codomain a complete infinity category
I am currently reading this paper on derived blow up, in definition 2.4.1, I am faced with such situation:
if we denote the infinity category of simplicial ring as $Alg$ and the 1 category of ...
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Understanding the concept of homotopy fixed points
I apologize in advance if this question is too basic for this site, I tried to search online and through the literature for a few days with no success already.
I am trying to understand the concept of ...
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On coproducts of presentably symmetric monoidal $\infty$-categories
Let $\mathcal{A}$ and $\mathcal{B}$ be presentably symmetric monoidal $\infty$-categories, i.e., symmetric monoidal $\infty$-categories whose underlying $\infty$-category are presentable and whose ...
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Does derived tensor product preserve fiber sequence？
In lemma 3.1.5 of this paper I read, there is a fiber sequence of the underlying spaces of simplicial commutative rings $A\stackrel{f}{\longrightarrow} A\rightarrow A/\!\!/f$. Here we define the "...
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Foundations and contradictions of Scholze's work: the category of presentable infinity categories contains itself
Preface: I am not an expert in the work of Scholze, or anything for that matter.
Question
Has Scholze stated what axioms he is using to develop his theory of motives and analytic geometry. In the ...
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Higher homological mirror symmetry?
The bounded derived category $D^b\mathrm{Coh}(X)$ is the homotopy category of a stable $\infty$-category $\mathbb{D}^b\mathrm{Coh}(X)$. Apparently there are reasons, such as "nonfunctoriality of ...
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The exact sequence for a derived zero locus
For a locally free sheaf of rank one $L$ on a derived scheme and a morphism $s:L\rightarrow O_{X}$, we consider the derived zero locus of $s$ defined by the following derived fiber product
$$\require{...
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How to prove $\text{Map}_C(X,Y)$ is a grouplike commutative monoid object of the $\infty$-category of spaces?
For an additive $\infty$-category $C$ every object $Y$ is a commutative grouplike object in $C$. Now my question is how we can show it is the case for any mapping space $\text{Map}_C(-,Y)$ in the $\...
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Classical schemes as derived schemes are discrete valued
$\newcommand\Spc{\mathrm{Spc}}\newcommand\SCRing{\mathrm{SCRing}}\DeclareMathOperator\Map{Map}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Spec{Spec}\DeclareMathOperator\Hom{Hom}\...
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What is the Isomorphism subspace of the mapping space in an infinity category
When $E$ is a locally free sheaf of rank n on a classical scheme $X$, there is a sheaf $Isom$ on the category $Sch_{X}$ defined as $(S\rightarrow X)\rightarrow Isom_{O_{S}}(O_{S}^{n},E)$. And this ...
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Divided power structure on $E_\infty$-algebras?
Let $A$ be a simplicial commutative ring, then it is known that the ideal of elements of degree $\ge1$ in the associated CDGA has a "DG divided power structure," which induces a divided ...
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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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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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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 ...
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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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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
$$
sending $(C,D)$ to the functor category $Fun(C,D)$.
This gives rise to a fibration $F\to Cat^{...
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$\operatorname{Fun}(\mathcal{C},\mathcal{D})^n$ is a subcategory of $\operatorname{Fun}(\mathcal{C}^n,\mathcal{D}^n)$
Let $\mathcal{C}$ and $\mathcal{D}$ be $\infty$-categories (by which I mean quasicategories, though I suspect that it hardly matters), and let $n\geq 1$ be an integer. There is a functor
$$\theta:\...
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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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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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