Questions tagged [infinity-categories]
The infinity-categories tag has no usage guidance.
558
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Dennis trace map for stable $\infty$-category, naively
I'm trying to get more intiution about higher K-theory, Hochschild homology and the trace map between by thinking about these objects from an informal $\infty$-categorical perspective, instead of ...
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The Yoneda Lemma for $(\infty,1)$-categories?
According to this page on the nLab, it is currently unclear whether or not the entire Yoneda lemma generalizes to $(\infty ,1)$-categories rather than just the Yoneda embedding. Have there been ...
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2
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sSet-enriched categories, quasi-categories and the model-independent theory
sSet-enriched categories are one possible model for $(\infty,1)$-categories, by the work of Bergner and others. They are probably the most important model from the point of view of getting actual ...
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Are n-truncated quasicategories a model for n-categories?
In Higher Topos Theory, Lurie gives a description of $n$-categories (section 2.3.4) in terms of quasicategories. In other words, he gives a definition (2.3.4.1) of an $\infty$-categorical notion of $n$...
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Describing fiber products in stable $\infty$-categories
Let $f\colon X \rightarrow Z$ and $g\colon Y \rightarrow Z$ be two morphisms in a stable infinity category $\mathcal{C}$. How does one show that the $\infty$-categorical fiber product $X \times_Z Y$ ...
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Quasicategories for non-simplicial model categories
If I have a simplicially enriched model category, then I can take the coherent nerve of the full subcategory of bifibrant opjects to obtain a quasicategory. If I have a model category that is not ...
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Is the model category of Complete Segal Spaces right proper?
Well, the title is self-explaining, I guess - I am referring to the complete Segal space model structure of Theorem 7.2 in Rezk's article "A model for the homotopy theory of homotopy theories".
Has ...
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What functors are classified by slices of $\infty$-categories?
Suppose I have a functor $f\colon C\to D$ between $\infty$-categories (I'll assume $C$ and $D$ are small.) Then I can form the slice categories and restriction functors
$$
D_{f/}\to D\qquad \text{and}...
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Infinity local systems
I have seen many references in the (geometric representation theory, symplectic geometry, etc) literature to "infinity local systems".
From what I've been told, given a good cover $\{U_i\}$ of $X$, ...
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A construction of the universal cocartesian fibration
Usually I see the universal (small) cocartesian fibration $\mathcal{Z} \to Cat_\infty$ constructed in a relatively opaque fashion, such as via the unstraightening construction.
I've stumbled on what ...
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Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?
This question asked whether $\mathrm{Sp}$ is convenient in the sense of satisfying (in the $\infty$-categorical sense) a list of desired properties of Lewis in his 1991 paper (see there).
The answer ...
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Why not a Stacks project for Homotopy Theory?
The lack of resources bridging the gap between what one finds in Hatcher's algebraic topology text and modern research on homotopy theory has been brought several times before on MathOverflow [1, 2, 3]...
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Correspondence between operads and $\infty$-operads with one object
Given a simplicial operad one can form its category of operators. This is a simplicial category with a functor to the category of finite pointed sets which is a bijection on objects and whose hom-...
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Can we use Mann's six-functor formalism with D-modules?
In a recent course in Bonn, P. Scholze explains a formalization of a six-functor formalism due to L. Mann. In this axiomatization, three of the functors $f_!,f^*,\otimes$ are "constructed" (...
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Proj construction in derived algebraic geometry
The question
My question is easy to state:
Is there a Proj construction in derived geometry, that produces a derived stack from a “graded derived algebra”?
Given the vagueness of the question, you’...
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Stable ∞-categories as spectral categories
Let C be a stable ∞-category in the sense of Lurie's DAG I. (In particular I do not assume that C has all colimits.) Then C does have all finite colimits, the suspension functor on C is an ...
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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 ...
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Comonadicity of spaces over spectra?
As connective spectra are equivalent to group-like $E_{\infty}$ algebras in spaces, the $\infty$-category of connective spectra is monadic over the $\infty$-category of spaces though the usual $\Sigma^...
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teaching higher algebra
Has anyone ever (successfully or unsuccessfully) taught a course in higher algebra (in the $\infty$-categorical sense)?
I'm asking out of curiosity (and also hoping for more resources).
The kind of ...
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Lemma 2.1.1.4 in Lurie's HTT
I have encountered a problem in understanding Lurie's proof of the following fact:
"Given a left fibration between simplicial sets $q:X \to S$, there exists a functor $$ho(S) \to Ho(sSet)$$ which is ...
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Higher Algebra, Theorem 2.4.3.18 and Remark 2.4.3.6
In his book Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads ($\S$2.3.2). Roughly speaking, a generalized $\infty$-operad is a "family" of $\infty$-operads ...
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Categorification of "Every domain embeds into a field"?
In the category of commutative rings, every domain embeds into a field. Is this true in the category of presentably symmetric monoidal stable $\infty$-categories? Here's what I mean by that.
Let $...
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A tensor product for triangulated categories?
Many triangulated categories which show up in mathematics, such as derived categories of various sorts, arise as the homotopy category of a stable $\infty$-category.
Stable $\infty$-categories give ...
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When did the Joyal model structure on simplicial sets originate?
Some of the earliest writings on the Joyal model structure on simplicial sets include Jacob Lurie's account in Higher Topos Theory from 2006,
as well as Joyal's own account in The Theory of Quasi-...
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Why Grothendieck's Homotopy Hypothesis is so difficult?
Recall that Grothendieck's Homotopy Hypothesis states that the homotopy category of weak globular $n$-groupoids (as in https://arxiv.org/pdf/1009.2331.pdf) is equivalent to the category of homotopy $n$...
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Modern proofs for simplicial localizations
I know that the references usually regarded as standard for simplicial localizations are the Dwyer and Kan's three articles from the 80's. I would be interested in a more modern approach to the ...
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Is every weak $\infty$-bicategory (à la Lurie) an $\infty$-bicategory?
In Definition 4.1.1 of $(\infty,2)$-Categories and the Goodwillie Calculus I, Lurie defines a weak $\infty$-bicategory to be a scaled simplicial set that has the extension property with respect to ...
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Extending a left fibration along an inner horn
Let $\Lambda^n_i \subseteq \Delta^n$ be an inner horn, and let $X \rightarrow \Lambda^n_i$ be a left fibration. Does there exist a left fibration $Y \rightarrow \Delta^n$ such that $X = Y \times_{\...
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How should one approach reading Higher Algebra by Lurie?
A question posed at the nForum asked for a roadmap to learn Lurie's Higher Topos Theory, including helpful sources other than HTT itself (to read along it) and information about which parts of HTT ...
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Are $\infty$-topoi determined by their localic points ?
Hello !
If $T$ is an infinity topos, then you can consider the infinity category of geometric morphism from $Sh_{\infty}(\mathcal{L})$ to $T$ for any locale $\mathcal{L}$. This associate to $T$ an ...
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Why does $Mf$ always support an $Mf$-orientation?
Let $f:X\to BGL_1(\mathbb{S})$ be a morphism of $E_n$-spaces and determine a principle $GL_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom ...
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Open conjectures and expected applications of homotopy theory to arithmetics
I hope this question is not too broad to be asked here; if it is, please feel free to close the question.
I'm currently near the end of my masters studies and subsequently search for a particular ...
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Looking for an invariant similar to algebraic K-theory
I'm wondering if there is an invariant, similar to algebraic K-theory, topological hochshild homologic, topological cyclic homology etc... that has the following properties:
a) It attach to each small ...
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Comparing derived categories of quasi-coherent sheaves in the lisse-etale and the big etale toplogy on an algebraic stack
I am trying to understand the proof of Proposition 1.4.2. in "A study of derived algebraic geometry Volume 1" by Gaitsgory-Rozenblyum. http://www.math.harvard.edu/~gaitsgde/GL/QCohBook.pdf, page 8.
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Other examples of the algebro-geometric Ran space
First off, sorry if this seems vague.
Let's recall some definition. Let $X$ be a curve over a field $k$ and $G$ an algebraic group, then the space $Ran_G(X)$ as defined by Lurie in his Tamagawa ...
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A category with weak equivalences that is not a model category
I'm only considering complete and cocomplete categories. A pair $(\mathfrak{X} , \mathfrak{W}) $ is, by definition, a category with weak equivalences if $ \mathfrak{X} $ is a category and $ \mathfrak{...
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The relation between t-structures and derived category
Let $\mathcal{D}$ be a triangulated category and a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap ...
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What does the homotopy coherent nerve do to spaces of enriched functors?
Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a ...
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The universal property of the unseparated derived category
In Appendix C of his book in progress Spectral Algebraic Geometry, Lurie defines the unseparated derived category $\check{{\cal D}}({\cal A})$ (see Definition C.5.8.2 loc.cit) associated to a ...
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Connectedness, loops and formal moduli problems
Let $k$ be an algebraically closed field of characteristic zero. Formalizing a classical folk concept, Pridham and (in a different way,) Lurie defined a formal moduli problem (over $k$) to be a ...
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Intermediate notions of bilinearity in higher algebra
It is well-known that when passing to $\infty$-categories the notion of commutativity gets replaced by an infinite array of notions of commutativity: $\mathbb{E}_{1}$, $\mathbb{E}_{2}$, ..., $\mathbb{...
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Can an enriched functor be expressed as a colimit of representable functors?
Suppose that $\mathcal C$ is an ordinary category and $F:\mathcal C^{op}\longrightarrow Set$ a functor. Then, we can form the category $\mathcal C/F$ as follows : each object is a morphism of functors ...
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A step in Lurie's treatment of $L$-theory
I am looking at Proposition 3 of Lecture 6 from Lurie's course Algebraic L-theory and Surgery (https://www.math.ias.edu/~lurie/287xnotes/Lecture6.pdf). This involves a stable $\infty$-category $\...
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What are the conjugacy classes of the category of ($\kappa$-small) sets?
$\newcommand{\unsim}{{\sim}}$The set of conjugacy classes of a group $G$ is the quotient of $G$ by the equivalence relation $\sim_1$ obtained by declaring $a\sim_1b$ if there exists some $g\in G$ such ...
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Making the ($\infty$-categorical) Bar construction valued in (bi)-modules
In Lurie's Higher Algebra, construction 4.4.2.7 presents a Bar construction in the setting of $\infty$-categories. The construction in 4.4.2.7 takes as input an $\...
user142177
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real and complex vector spaces as topological categories
Let $Vect_{\mathbb{R}}$ be the category of (say, finite dimensional) vector spaces over $\mathbb{R}$. The automorphism group of the object $\mathbb{R}^n\in Vect_{\mathbb{R}}$, is $GL_n(\mathbb{R})$. ...
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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 ...
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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\...
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Examples and non-examples of Tannakian $\infty$-categories
My definition of a Tannakian $\infty$-category is taken from this paper ("Tannaka duality over ring spectra" by James Wallbridge).
(p. $53$, definition $7.9$) Let $R$ be an $E_{\infty}$-...
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Floer cohomology from mapping spaces of $\infty$ categories
There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at ...
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