Questions tagged [infinity-categories]

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4
votes
2answers
163 views

The correct homotopically relevant notion of ideals of dg-algebras (or $\mathbb E_1$-rings)

I'm trying to figure out what an ideal of a, say, dg-algebra (or, if you prefer, $\mathbb E_1$-ring) $R$ is in a homotopically relevant fashion, but I can't actually figure it out. I can assume that $...
9
votes
2answers
253 views

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 ...
5
votes
0answers
152 views

The $\infty$-category of natural transformations as an end

Let $\mathcal{C}$ be an $\infty$-category viewed as a fibrant scaled simplicial set with all 2-simplices thin and let $\mathfrak{C}\!at_{\infty}$ be the $\infty$-bicategory of $\infty$-categories. A ...
10
votes
0answers
337 views

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 ...
5
votes
1answer
133 views

Can a locally presentable category have a proper class of accessible localizations?

Question: What is an example of a locally presentable category $\mathcal C$ such that there exists a proper class of accessible localizations $(\mathcal C \to \mathcal D_i)_{i < ORD}$? In other ...
13
votes
0answers
205 views

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 $...
5
votes
2answers
304 views

Can conservativity depend on the universe?

Question 1: Let $F: C \to D$ be a conservative, $\kappa$-cocontinuous functor between small, $\kappa$-cocomplete categories. Is the induced functor $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ ...
1
vote
0answers
82 views

What is the meaning of lifting property? and some question in $\infty$-category

When I learn model category, the important compute tool is the lifting property between $(\operatorname{Cof}, \operatorname{Fib} \cap W)$ and $(\operatorname{Cof} \cap W, \operatorname{Fib}) $, where $...
8
votes
0answers
232 views

Does any 'logical' theory have a bounded ∞-pretopos as syntactic category?

Stone duality may be understood as providing a duality between syntax and semantics for propositional logic, so that a theory may be recovered from its models. In order to do likewise for first-order ...
6
votes
1answer
202 views

Descent for $K(1)$-local spectra

For odd primes, we have an equalizer diagram for the $K(1)$- local sphere given by $$L_{K(1)}S \rightarrow K{{ \xrightarrow{\Psi^g}}\atop{\xrightarrow[i_K ] {}}} K$$ where $g$ is a topological ...
11
votes
0answers
336 views

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 ...
12
votes
0answers
211 views

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 ...
9
votes
0answers
191 views

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 $X$, ...
2
votes
1answer
186 views

Abelian versions of straightening and unstraightening functors

Let $X$ be a quasi-category (an inner Kan complex), let $\mathfrak{C}(X)$ be its rigidification (its associated simplicial category). J. Lurie in "Higher Topos Theory" proved the following theorem 2.1....
4
votes
1answer
162 views

descent implies hyperdescent

My question concerns the Propsition 5.12 of the paper of Bhatt-Mathew on arc-topology where they claim that the functor $X\mapsto D^b_{\text{cons}}(X,\Lambda)^{[-n,n]}$ which assigns to a qcqs scheme $...
1
vote
1answer
112 views

Morphisms on fibre products

Let $X$ and $Y$ be two quasi compact, separated schemes over $k$, and consider the fibre product $X \times Y$. If we call $p_1$ and $p_2$ the two projections, and we take perfect complexes $F_1, F_2 \...
4
votes
1answer
71 views

On equivalences of cartesian fibrations

Let $p:X^{\natural} \to S$, $q:Y^{\natural} \to S$ be two cartesian fibrations of simplicial sets (where the marking is given by cartesian edges) and assume that we are given an equivalence of ...
4
votes
0answers
108 views

Is there a model-independent characterization of the gaunt strict $n$-categories amongst the weak $(\infty,n)$-categories?

Recall that a strict $n$-category $C$ is called gaunt if every $k$-morphism in $C$ with a weak inverse is an identity, for all $k$; let $Gaunt_n$ denote the strict 1-category of gaunt $n$-categories. ...
2
votes
0answers
51 views

A name in literature for a certain kind of 2-categories

Let $tr_2: \mathrm{sSet} \to \mathrm{sSet}_{\le 2} $ be the 2-truncation functor. Let $C$ be a 2-truncated simplicial set such that every horn $tr_2( \Lambda^2_1) \to C$ extends to $tr_2(\Delta_2) \...
5
votes
1answer
149 views

Which free strict $\omega$-categories are also free as weak $(\infty,\infty)$-categories?

There are a number of formalisms available for presenting free strict $\omega$-categories -- Street's parity complexes, Steiner's directed complexes, computads, polygraphs,... Typically one has a ...
8
votes
1answer
390 views

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 ...
6
votes
1answer
144 views

Is the inclusion of its 2-skeleton into the walking idempotent homotopy cofinal?

Let $Idem = Idem^{(\infty)}$ be the walking idempotent [1], and let $Idem^{(n)}$ be its n-skeleton. Note that $Idem$ has one nondegenerate simplex in each dimension. Let $\iota_n^m: Idem^{(n)} \to ...
9
votes
2answers
269 views

When is the homotopy category of an accessible $\infty$-category accessible?

Let $\mathcal C$ be an accessible $\infty$-category, and let $ho(\mathcal C)$ be its homotopy category. I can think of two "trivial" reasons for $ho(\mathcal C)$ to be accessible: $ho(\mathcal C) = \...
7
votes
1answer
228 views

When is the model structure on functors correct, i.e. when does localization commute with taking functor categories?

Let $C$ be a small category and $M$ a model category. Then there are various "global" model structures (projective, injective, Reedy) on the category $Fun(C,M)$ of functors from $C$ to $M$, all with ...
8
votes
3answers
480 views

Can filtered colimits be computed in the homotopy category?

For $\mathcal{S}$ the $(\infty,1)$-category of spaces its homotopy category $h\mathcal{S}$ does not have pushouts or pullbacks. Even if it does, they won't always agree with the (homotopy) pushouts or ...
5
votes
1answer
124 views

$k$-linear $\infty$ stable categories and dg categories

This question is related to this question, where I asked about the relation between the derived category of a fiber product $Y \times_Z W$ and the push out of the diagram of derived categories one ...
4
votes
0answers
97 views

Adjunctions between $\mathcal{A}_{\infty}$-categories

Is there a theory of homotopy coherent adjunctions between $\mathcal{A}_{\infty}$-categories? By this I mean at least a definition of what an adjunction is and a construction of the corresponding ...
6
votes
0answers
252 views

An adjunction between monads on $\mathcal{C}$ and presentable categories under $\mathcal{C}$

Fix a regular cardinal $\kappa$ and let $\mathcal{C}$ be a $\kappa$-presentable $\infty$-category (comments about the 1-categorical case are welcome as well!). I'm looking for a reference for the ...
1
vote
0answers
103 views

1-connected infinity groupoids, groupoids and 1-connected spaces

I am exploring a bit the world of groupoids. What I have in mind is that infinity groupoids correspond to spaces. So my first question is the following: Consider the model category $\infty-Grpd$ of ...
4
votes
1answer
189 views

Homotopy pullbacks and fibers

It is true that in the category of spaces there exists a characterization of homotopy pullbacks in terms of homotopy fibers (Proposition 4.1). I want to know a category (or $\infty$-category) where I ...
1
vote
0answers
117 views

Is this a model of $(\infty, \infty)$?

Please pardon me if this idea is silly. But could we produce a model for $\infty$ categories by allowing simplices with compatible orientation? Let me elaborate. One can think of simplicial sets as ...
5
votes
0answers
172 views

Why do we need straightening?

The method of straightening and unstraightening produces, for instance, an equivalence between left fibration and copresheaves. The construction seems very intricate. But what is the need for this ...
8
votes
1answer
375 views

Is the de Rham complex in characteristic $p$ a CDGA?

In the paper by Bhatt and Scholze on prismatic cohomology (https://arxiv.org/pdf/1905.08229.pdf), it is stated that the de Rham comparison theorem for prismatic cohomology can be lifted to an ...
5
votes
0answers
141 views

The notion of $\infty$-Cooperads for which Bar-Cobar duality is an equivalence

In the paper Bar-Cobar Duality by Michael Ching, he proves that the category of operads in spectra is equivalent via the Bar-Cobar adjunction to some model category of co-operads defined in the paper. ...
14
votes
0answers
221 views

Well pointed endofunctors on $\infty$-categories

In $1$-category theory, a well pointed endofunctor of a category $C$, is an endofunctor $F:C \rightarrow C$ endowed with a natural transformation $\sigma : Id \rightarrow F$ such that the two natural ...
4
votes
3answers
376 views

The homotopy category of the category of enriched categories

We know that if $\mathcal C$ is a combinatorial monoidal model category such that all objects are cofibrant and the class of weak equivalences is stable under filtered colimits, then $\mathsf{Cat}_{\...
7
votes
0answers
162 views

Duality of Hopf algebras and duality of spectra

Let $S$ be the sphere spectrum, and for $X$ a topological space, let $S(X)$ be the mapping spectrum from the free loop spectrum on $X$ to the sphere spectrum. This is an $E_\infty$ ring spectrum (also ...
7
votes
0answers
149 views

Are (complete) 2-Segal spaces the same as Span-enriched infinity categories?

The question is basically in the title. More generally, I would like to know if this, or any reasonable variant of it, is true. Or perhaps, to understand better the gap between 2-Segal spaces and Span-...
6
votes
0answers
208 views

Uniqueness of the $(2,2)$-category theory of $(\infty,1)$-categories?

The question, as in the title, may be very simply stated as follows: Main Question: Can the homotopy $(2,2)$-category of $(\infty,1)$-categories be characterized as the unique $2$-category upto ...
2
votes
0answers
89 views

Infinity-categorical exceptional push-forward

Classically, if $f:X\to Y$ is a map of locally compact Hausdorff topological spaces, one can define the exceptional push-forward functor $f_!:Sh(X;k)\to Sh(Y;k)$ among $k$-valued sheaves for, say, a ...
4
votes
0answers
126 views

Generators and colimit-closures in higher categories

In ordinary categories $\mathcal{C}$, there are nice conditions under which a generator is also a colimit-generator for $\mathcal{C}$. In other words, under suitable conditions, if $\mathcal{C}$ ...
4
votes
2answers
230 views

Pullbacks and fibers in the $\infty$-category of spaces

Suppose given a commutative diagram in the $\infty$-category of spaces, as the one depicted below, where all but the bottom-right squares are pullbacks. Is it true that $H$ is (equivalent to) the ...
7
votes
1answer
181 views

Is the simplicial nerve a localization?

Given a simplicial category $\mathcal{C}_{\ast}$ (if necessary, you may assume it's fibrant), denote as $\mathcal{C}$ its underlying ordinary category, and as $\mathcal{W}$ the class of all ...
7
votes
1answer
249 views

Higher-dimensional version of the “Magic Cube Lemma” for homotopy pushouts/pullbacks

The "Magic Cube Lemma" is a surprising (to me) relationship between (homotopy) pushouts and (homotopy) pullbacks of spaces: Consider a cubical diagram $I^3\to \mathcal{S}$ in the $\infty$-category of ...
22
votes
4answers
851 views

How should I think about presentable $\infty$-categories?

Let me start out with a confession. I have never cared much for set-theoretic size issues, for they seem not to cause much trouble in my day-to-day mathematical life. Despite that, I have always been ...
1
vote
0answers
89 views

L-infinity algebra of deformations of an L-infinity algebra?

From Schlessinger-Stasheff we know that a deformation problem should come with an associated $L_\infty$-algebra, so that gauge-equivalence classes of solutions to its Maurer-Cartan equation (the "MC ...
5
votes
0answers
132 views

What are the effective epimorphisms of presentable $\infty$-categories?

Let $\mathcal C$ be a sufficiently nice $\infty$-category, and let $f: U \to X$ be a morphism in $\mathcal C$. Recall that $f$ is said to be an effective epimorphism if the induced map $|U^{\times_X (\...
6
votes
0answers
140 views

Is there an $\infty$-topos of monochromatic spaces?

Fix (a prime $p$ and) a chromatic height $h$. Recall that the Bousfield-Kuhn functor $\Phi_h: \mathcal M_h^f \to Sp_{T(h)}$ is monadic, where $\mathcal M_h^f \subseteq Top_\ast$ is a certain ...
6
votes
0answers
81 views

Reference request: retracts are summand inclusions in additive $\infty$-categories

Suppose that $\mathcal{A}$ is an additive $\infty$-category. By this I mean that $\mathcal{A}$ is pointed, semi-additive (i.e., admits biproducts, which I will call direct sums and denote by $\oplus$),...
1
vote
0answers
159 views

A categorial PCF theory?

I'm not an expert in PCF theory, so please forgive me if this question makes no sense. I'm looking for a categorial version of PCF theory. Specifically, if we replace $Set$ with another category, ...

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