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11
votes
3answers
1k views

Grothendieck's Homotopy Hypothesis - Applications and Generalizations

Grothendieck's homotopy hypothesis, is, as the $n$lab states: Theorem: There is an equivalence of $(∞,1)$-categories $(\Pi⊣|−|): \mathbf{Top} \simeq \mathbf{\infty Grpd}$. What are the ...
3
votes
1answer
255 views

Does Ind-completion commute with finite limits?

The broad and vague question is in the title. The more precise question is: Say $\{\mathcal{C}_i\}$ is a finite diagram of (essentially small) stable $\infty$-categories and exact functors with ...
43
votes
4answers
2k views

Is there an accepted definition of $(\infty,\infty)$ category?

For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with ...
10
votes
1answer
431 views

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 ...
18
votes
1answer
339 views

What is higher equivariant homotopy?

In Lurie's "Survey of elliptic cohomology" it is claimed that there exists some mystical "2-equivariant homotopy theory" for elliptic cohomology. The classical equivariant elliptic cohomology is ...
14
votes
1answer
465 views

Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an ...
5
votes
1answer
192 views

Model independent proof of colimit formula for left Kan extensions

I am interested in finding a proof of the colimit formula for left Kan extensions $(\infty,1)$-categories which does not rely on a chosen model. (By a model inpendent proof I mean a proof which uses ...
3
votes
0answers
171 views

Operadic Lift of Lurie's Relative Tensor Product

In Section 4.4 of his book Higher Algebra, Lurie introduces, for a monoid object $A$ of a monoidal quasicategory $C$, and right and left $A$-modules $M,N$, the relative tensor product $M\otimes_AN$. ...
7
votes
0answers
58 views

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 ...
13
votes
3answers
897 views

Does the classification diagram localize a category with weak equivalences?

Let $(C,W)$ be a category equipped with a subcategory of weak equivalences. Its "classification diagram" or "bisimplicial nerve" $N(C,W)$ is a bisimplicial set, for which $N(C,W)_n$ is the nerve of ...
11
votes
1answer
346 views

Examples of differential cohomology in cohesive $\infty$ topos

I might direct this question to Urs Schreiber directly, but just in case someone else has some interesting examples, I'll make the question public. The formulation of differential cohomology in ...
2
votes
0answers
148 views

stackifickation of BG

Let $Man$ be the category of smooth manifold. Fix a $M\in Man$ and let $G$ be a group acting smoothly on $M$. The nerve of the group action on $M$ by $G$ defines a simplicial presheaf $X\: : \:Man\to ...
13
votes
1answer
548 views

The category theory of $(\infty, 1)$-categories

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent ...
3
votes
1answer
73 views

explicit description of the cosimplicial simplicial set $Q^{\bullet}$

I'm struggling to understand the explicit description of the cosimplicial simplicial set $Q^{\bullet}$ on page 76 (section 2.2.2) of Lurie's book Higher Topos Theory, and would be grateful if someone ...
7
votes
0answers
121 views

A completeness criterion for $\infty$-categories

We all know that for ordinary categories $\mathscr{C}, \mathscr{D}$ (with $\mathscr{C}$ small) the limit of a functor $F:\mathscr{C} \to \mathscr{D}$, if it exists, can be constructed by using ...
13
votes
2answers
759 views

Non-unique splittings of homotopy idempotents

By a homotopy idempotent I mean a map $f:X\to X$, where $X$ is a space, equipped with a homotopy $f\circ f \sim f$. In contrast to the situation in stable homotopy theory (where $X$ would be a ...
6
votes
0answers
254 views

Item (4) in Lurie's definition of the class of marked anodyne morphisms

I have a question concerning Remark 3.1.1.3 in Lurie's "Higher Topos Theory" about the definition of the class of marked anodyne morphisms. There, it is mentioned that "it suffices to allow $K$ to ...
2
votes
0answers
47 views

Cancellation property of groupoidal cartesian fibrations

I have an issue concerning a property of "left cancellation" for groupoidal cartesian fibrations of ∞-cosmoi (but everything works fine in a 2-category as well). A 1-cell $p: E \to B$ is called ...
6
votes
0answers
140 views

A “universally non Hypercomplete” $\infty$-topos?

My question is : Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ? What I mean by $\infty$-connected ...
11
votes
2answers
255 views

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 ...
2
votes
0answers
98 views

Is the dg-nerve functor a Quillen equivalence?

Lurie defines the dg-nerve $N_{dg}(\mathcal{C})$ of a dg-category $\mathcal{C}$ in Higher Algebra Construction 1.3.1.6: for each $n \geq 0$, we define $N_{dg}(\mathcal{C})_n\simeq ...
6
votes
2answers
489 views

On combinatorial and cellular model categories and infinity categories

I am looking for a counterexample. Let me first give the set-up. When you work with model categories, it is extremely common to assume they are cofibrantly generated. For me, this means the definition ...
9
votes
0answers
153 views

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 ...
3
votes
0answers
236 views

Quasicategorical Construction of a Cosimplicial Map of Rognes

In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose ...
1
vote
1answer
307 views

Construction of Highly Structured Quotient Groups in Quasicategories

Suppose we have a map of $E_n$-spaces $X\to Y$. Then there is a highly structured action of $X$ on $Y$, $X\wedge Y\to Y\wedge Y\to Y$, using the multiplication of $Y$. As such, I believe that there ...
3
votes
1answer
94 views

Segal maps for Segal precategories

A Segal precategory is just a simplicial space $X:\Delta^{op} \to sSet$ such that its $0$-th space is discrete (i.e. constant). A Segal category is defined everywhere in the literature as a Segal ...
6
votes
1answer
252 views

Compact objects in undercategories and filtered colimits

Let $\mathcal{C}$ be a compactly generated presentable $(\infty, 1)$-category. Consider the functor $$ \Phi: \mathcal{C} \to \mathrm{Cat}_\infty, \quad x \mapsto (\mathcal{C}_{x/})^\omega,$$ that ...
4
votes
1answer
418 views

Why care about Fourier-Mukai partners?

Two (smooth, projective, complex?) varieties are called Fourier-Mukai partners if they have equivalent derived categories of coherent sheaves. On the other hand, my general impression is that cool ...
5
votes
0answers
371 views

is there a moduli of stable infinity categories?

I know there exists a groupoid-valued prestack parameterizing (connected) triangulated dg-categories (ie the points of this moduli are not objects of a fixed dg-category, but rather dg-categories ...
7
votes
0answers
253 views

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 ...
1
vote
0answers
182 views

Monoidal Forgetful/Free Adjunction for $E_2$-algebras

Suppose I am given two $E_2$-ring spectra $A$ and $B$ and a morphism of $E_2$-rings $\phi:A\to B$. Then I have $E_1$-monoidal categories of modules $LMod_A$ and $LMod_B$. Moreover I have morphisms ...
5
votes
1answer
221 views

Difference between coherent nerve of simplical model category and simplicial category

Suppose I have a simplicial model category $M$. Then I can take the homotopy coherent nerve of $M$ to obtain a quasicategory. This, however, only depends on the fact that $M$ is a category enriched in ...
8
votes
1answer
293 views

When is a quasicategory over $N(\Delta)^{op}$ a planar $\infty$-operad?

In Lurie's DAG II, a notion of monoidal $\infty$-category is given that differs from the notion given in his later book Higher Algebra. In the former, the relevant structure is a cocartesian ...
4
votes
1answer
235 views

Is the infinity-groupoid of a finite CW complex finitely-presented?

An infinity-groupoid is finitely-presented when it is equivalent to the free infinity-groupoid on a finite family of generators, possibly of different dimensions. Is the infinity-groupoid of a finite ...
12
votes
1answer
418 views

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 ...
3
votes
1answer
546 views

Opposite Symmetric Monoidal Structure on an Infinity Category

Given an $\infty$-category (in the sense of Lurie) $C$, and a symmetric monoidal structure on $C$ associated to a coCartesian fibration $p:C^\otimes\to N(Fin_\ast)$, Lurie says in Remark 2.4.2.7 of ...
7
votes
1answer
393 views

Generalized Thom spectra

I am currently trying to wrap my head around all kinds of different definitions of the notion of (generalized) Thom spectrum. My setup is as follows: Suppose I have a commutative (symmetric) ring ...
2
votes
0answers
190 views

A Cartesian model structure (and straightening for) on $n$-trivial simplicial sets

A pair $(X,tX)$, with $X$ a simplicial set and $tX$ a collection of simplices of $X$, is said to be stratified if no $0$-simplex is in $X$ and all degenerate simplices of $X$ are in $tX$. Recall a ...
32
votes
1answer
14k views

If I want to study Jacob Lurie's books “Higher Topoi Theory”, “Derived AG”, what prerequisites should I have?

I've been told that it's important to know modern physics, Differential Geometry and Algebraic Topology for understanding higher structures. Is there any other prerequisite for understanding Lurie's ...
2
votes
1answer
240 views

When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories?

Let $\mathrm{Quillen}$ be the model category of simplicial sets with the Quillen model structure, and $\mathrm{Joyal}$ the model category of simplicial sets with the Joyal model structure. As is ...
2
votes
1answer
192 views

Showing left module actions are highly structured

For my own convenience I'll work in $\infty$-categories, feel free to answer in whatever framework best suits you. My question is essentially how to show, given an $E_\infty$-ring object $R$ in an ...
1
vote
0answers
173 views

Under which conditions the inclusion of a sub-simplicial set of the nerve of a category is a Joyal equivalence?

Let $i: X \to \mathrm{N}\mathcal C$ be a monomorphism in the category of simplicial sets, with $\mathcal C$ a category and $\mathrm{N}\mathcal C$ its nerve. I am looking for sufficient conditions (and ...
4
votes
1answer
220 views

Bar Construction Model of Ring Spectrum Quotient

Suppose I am given a morphism $f:BG\to BGL_1(R)$ for $R$ some at least $E_1$-ring spectrum and $G$ a loop space. Then This corresponds, I believe, to an action of $G$ on $R$, coming from a morphism ...
6
votes
1answer
886 views

$(\infty, 1)$-Yoneda embedding via the Grothendieck construction

Let $C$ be a quasi-category. Then there is an imbedding $$ C^{op} \times C \to \mathrm{Kan}$$ where $\mathrm{Kan}$ is the quasi-category of Kan complexes. This is essentially constructed in Lurie's ...
3
votes
0answers
161 views

Twisting of the power functor

Let $k$ be a field of characteristic $p$ and $D^b(k)$ be the infinity (equivalently, DG) category of perfect complexes over $k$. Let $C_p(=\mathbb{Z}/p)$ be the cyclic group on $p$ elements. For a ...
3
votes
0answers
239 views

Straightening for $\infty$-operads

There is this straightening/unstraightening procedure of Jacob Lurie's which takes a symmetric monoidal $\infty$-category (which is the data of a coCartesian morphism of simplicial sets $C^\otimes\to ...
6
votes
0answers
462 views

A model category for descent?

Recall that an $(\infty,1)$-category $C$ is said to have descent if for any small diagram $X:I\to M$ with (homotopy) colimit $\overline{X}$, the adjunction between $C/\overline{X}$ and "equifibered" ...
2
votes
1answer
164 views

Is the discrete nerve of a small category a complete Segal space?

While reading Rezk's paper "A model for the homotopy theory of homotopy theory", I found a remark which contradicts a guess of mine, but I can't see where I am wrong (perhaps it might be a silly ...
6
votes
2answers
200 views

Localizations of model categories and $\infty$-categories

I am interested in the relation between Bousfield localizations of model categories and localizations of $(\infty,1)$-categories. According to Hirschhorn's book we can form the left Bousfield ...
3
votes
1answer
358 views

unbounded derived category of a $\infty$-topos

In HTT(Higher Topos Theory) Remark7.3.1.19, it it sketched that the proper base change theorem for $\infty$-topos implies the usual proper base change theorem in (unbounded) derived category. However, ...