The infinity-categories tag has no wiki summary.

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### 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 ...

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### Doing some homological algebra in triangulated categories

It's well known that chain complexes are an abelian category, and in particular we can consider chain complexes of chain complexes, i.e. double complexes. Given a double complex $A^{\bullet\bullet} ...

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### Do the algebras for a $\infty$-monad form a stable $\infty$-category?

I'm wondering if a monad $T$ on a stable $\infty$-category $\cal C$ has a stable $\infty$-category of algebras, provided $T$ preserves finite limits/colimits.
Is this true?
Edit: Is something ...

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### Chromatic Spectra and Cobordism

I apologize in advance, if some of the things I've written are incorrect.
The cobordism hypothesis states that $\mathbf{Bord}^\mathrm{fr}_n$ is the free symmetric monoidal $(\infty,n)$-category with ...

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### Is there an $(\infty,2)$-category with morphisms given by $D^b\text{Coh}$?

My question is:
Has anyone constructed an $(\infty,2)$-category whose objects are (projective, maybe smooth, ...) varieties, and where the 1-morphisms from $X$ to $Y$ are given by ...

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### What is a stable $(n,1)$-category?

My question in its simplest form is whether we have any understanding of stable $(n,1)$-categories, by analogy with stable $(\infty,1)$-categories and abelian categories (the latter being like "stable ...

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### $t$-structure on modules over highly structured ring spectra

Let $Sp$ be a suitably nice model of a suitably nice category of spectra. By this I mean that $Sp$ is a symmetric monoidal closed stable model category, where we can make sense of homotopy "groups" ...

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### what is the stabilization of pointed sets?

Given a(n $\infty$-)category, there is a process called "stabilitazion" which spits out a stable $\infty$-category (as one can read about in either Higher Algebra or the nlab).
The famous example is ...

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### Higher Degree Data in a Cosimplicial Quasicategory and Delooping

If there is a short answer to this question and someone can write it here that'd be wonderful, but if it's longer, I'm also perfectly happy with a reference.
My question is regarding accessing data ...

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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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### Unbounded derived category that is not left-complete

Let me first recall some definition: Let $A$ be a Grothendieck Abelian category. Then, then category $\mathrm{Ch}(A)$ (I am using homological indexing) admits a combinatorial model structure (see for ...

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### Does simplicial localization with a 3-arrow calculus commute with functor categories?

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial ...

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167 views

### When does simplicial localization commute with functor categories?

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial ...

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134 views

### Picard-Brauer exact sequence for infinity categories

This question may be very naive, or the answer may be well-known. In any case, a good amount of googling did not bring up anything useful (maybe I'm using the wrong words?).
If $f:A\to B$ is a ...

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### 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 ...

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### Factorization systems in a triangulated/stable category

Google is of little help when questioned about "factorization systems in a triangulated category". Are there informations about them? Are there "natural" (meaning: "obviously defined") OFS/WFS in a ...

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### On a question motivating Lurie's treatment of formal moduli problems

Lurie, in his ICM 2010 proceedings paper Moduli Problems for Ring Spectra (pdf), says that one motivating problem (for him, I presume, and possibly others) for thinking about formal moduli problems is ...

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### DG categories - pre-triangulated versus small limits

A DG category can be considered as an infinity category, say by taking Dold-Kan of the coconnective part of Hom spaces, thus obtaining a simplicial category.
My question is, are the following ...

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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 ...

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249 views

### Equivariant homotopy, simplicially

It is a classic result of Kan that the homotopy categories (with appropriate model structures) of simplicial sets and of topological spaces (in fact, one could only care about CW-complexes) are ...

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### Which morphisms of ring spectra are of effective descent for modules?

There is a well understood bifibration of $\infty$-categories over the $\infty$-category of commutative ring spectra whose fiber over a ring $R$ is the category of $R$-module spectra. This is in ...

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### classifying $\infty$-toposes for topological/localic groups?

Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ?
More precisely, is there an $\infty$-topos $BG$ ...

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### Higher Descent Cohomology

Descent cohomology for a comonad is defined at degrees 0 and 1 by Mesablishvili in his paper "On Descent Cohomology" (as well as by many other authors in many other contexts). For a comonad $\bot$ on ...

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### Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology

Let $C$ be a small site with fibre products. The (injective) Čech model structure on simplicial presheaves $\operatorname{sPre}(C)$ on $C$ presents an $(\infty,1)$-topos and one may ask if this ...

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### Loops and suspensions of higher categories

Given a pointed $(\infty,n)$-category $\mathcal{C}$, one can define the suspension of $\mathcal{C}$, $\Sigma\mathcal{C}$, via the homotopy pushout of $$\ast\leftarrow \mathcal{C}\rightarrow \ast.$$ ...

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### 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 ...

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### Categories of sheaves and Kan Extensions

This is quite a broad question regarding constructions of categories of sheaves in geometry.
Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of ...

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461 views

### What is a Homotopy between $L_\infty$-algebra morphisms

A $L_\infty$-algebra can be defined in many different ways. One common way, that
gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative ...

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### Lex $\infty$-colimits

In the paper lex colimits, Garner and Lack gave a general characterization of "exactness properties" for categories (and enriched categories). A "lex-weight" is a weight for colimits whose domain ...

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### Coequalizers in stable (infinity,1)-categories

I have read it claimed in several places that in a stable $(\infty,1)$-category, the coequalizer of parallel maps $f,g:X\to Y$ can be identified with the cokernel of $f-g$ (i.e. the pushout of the map ...

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### $\infty$-categorical interpretation of type theory

One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity ...

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### 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 ...

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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 ...

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### 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 ...

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### Constructible derived category and fundamental category

Introduction (may be skipped)
Given a nice topological space $X$, the category of local systems (say over a field $k$) on it is equivalent to the category of representations of its fundamental ...

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### Directed colimits of maps in a combinatorial model category

I have the following situation. $M$ is a combinatorial model category, or if you like a locally presentable $(\infty,1)$-category. I have a set of maps $S$ and I let $C$ be the class of maps generated ...

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### 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 ...

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### Simplicial localisation and infinity categories

If $(\mathcal{C},W)$ is a category with weak equivalences then we may naturally form its Dwyer-Kan simplicial localisation $L(\mathcal{C}, W)$. This is a simplicial category which naturally gives a ...

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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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### Internal categories in simplicial sets

Is there a model structure (or more generally a homotopy theory) on the category of internal categories in simplicial sets, which presents the theory of $(\infty,1)$-categories?
Note that this ...

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### Stable infinity categories vs dg-categories

What is the relation between dg-categories and stable $\infty$-categories?
Given a dg-category one can form its dg-nerve and get a $\infty$-category
(which will be stable if the dg-category is?).
...

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### Right Notion of Localizing Subcategory in Quasicategory

Given a stable quasicategory $C$, what are the conditions required of a subcategory for it to descend to a localizing subcategory in $Ho(C)$? Is it enough for it to be reflective? Perhaps exact and ...

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### generalisations of the Seifert-van Kampen Theorem?

I have been reading Jacob Lurie's book "Higher Algebra", version May 8, 2011. One is grateful to him for covering such a lot of ground and for making it all so readily available.
My attention was ...

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### Adjoint functor theorem for infinity categories

In HTT, a version of the adjoint functor theorem for (locally) presentable infinity categories is proven (Corollary 5.5.2.9). Is there a more refined version of this somewhere, which more closely ...

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### On small generators of an infinity category

Suppose that $\mathcal{C}$ is an $\infty$-category with pullbacks and small coproducts. Suppose that $\mathcal{D}$ is a small subcategory for every object $C,$ the canonical map ...

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### Limitations on model-categorical presentations

In higher category theory, it is common that a weak structure cannot be strictified in all directions simultaneously. For instance, a monoidal category is not (in general) equivalent to one that is ...

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### $(\infty, 1)$-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 ...

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### how to make the category of chain complexes into an $\infty$-category

I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes.
Has anyone ever written down ...

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### $\infty$-topos and localic $\infty$-groupoids?

It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales).
For the record, this is proved by, starting ...

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### 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 ...