Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
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Applications of spectral Artin representability?
The spectral Artin representability theorem says that a functor $X:CAlg^{cn}\rightarrow S$ from the $\infty$-category of connective $E_{\infty}$-rings to the $\infty$-category of small topological ...
user74900
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Internal van Kampen colimits
Let $C$ be a category with pullbacks. Recall that a colimit in $C$ is van Kampen if it is preserved by $C/(-): C^{op} \to CAT$, $c \mapsto C/c$.
We can $C$-internalize everything in sight:
Let $\...
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Cofree conilpotent (cocommutative) coalgebra for $\infty$-categories
Let $\mathrm{K}$ be a field.
Denote $ \mathrm{Vect}_{\mathrm{K}} $ the category of K-vector spaces, $\mathrm{Coalg }^{\mathrm{conil } } $ the category of
conilpotent, coaugmented, coassociative ...
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Cartesian liftings in double categories
The question: I wonder whether the following definition, or something similar, has appeared somewhere (see below for motivations). Any reference or pointer is welcome!
(In what follows, I denote ...
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Categorical formalism for higher non-abelian group cohomology / obstruction theory for gerbes?
I'm sure this is very well known but I haven't found any references for this searching the internet so hence the question:
What's the neat abstract framework for obstruction theory for non-abelian ...
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Dense (∞,1)-subsites
So if $C$ is a 1-site and $D$ is a subsite (with the induced coverage), there are some conditions that ensure that the pre-composition and right Kan extension functors yield an equivalence of ...
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Can representable presheaves be made injectively fibrant?
I suspect that the answer to my question is no, but let me give it a shot anyway.
If $\mathcal{A}$ is a small simplicially enriched category, then the category of simplicial presheaves $\mathsf{sSet}^...
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A definition of the homotopy colimit of a coherent diagram
Suppose I am given a homotopy coherent diagram of spaces of shape $I$ (This is a simplicial functor $F:\mathfrak{C}[I] \to Top$, where $\mathfrak{C}$ is the standard cofibrant replacement functor in ...
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How to realize the descent data of Qcoh as a (pseudo)-limit in Cat?
It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ ...
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Is hypercompletion functorial?
Given an infinity topos $\mathcal{E}$, one can pass to its hypercompletion $\widehat{\mathcal{E}}$, which is the full subcategory on those objects local with respect to $\infty$-connected morphisms (...
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Rigorous recursive definition of $m$-algebras
Naively, $m$-algebras over a field $\mathbb{k}$ are easily defined recursively: the category of 0-algebras is the symmetric monoidal category of $\mathbb{k}$-vector spaces, and for $m>0$ an $m$-...
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Mathematics needed for higher dimensional category theory? [closed]
I'm a undergrad(third year, Manchester uni and want to do a PhD) that is thinking of doing a PhD in this area or category theory in general.(Sorry for asking it here, Maths exchange stack didn't help ...
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Homotopy Pushouts via Model Structure in Top
As far as I know, one way to take a homotopy colimit in a model category is to replace (up to acyclic fibration) all arrows in the diagram with cofibrations, and take the strict colimit of the ...
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Bar construction in commutative algebras is calculated by pushout
$\DeclareMathOperator\colim{colim}$
Also asked in MathStackexchange here
This is a statement in Lurie's Higher Algebra 5.2.2.4.
Proposition 3.2.4.7 in HA said that the monoidal structure on $\text{...
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Examples of categories cofibered in groupoids
In Chapter 2 of Lurie's Higher Topos Theory, the first main theorem establishes a connection between categories cofibered in groupoids and left fibrations and asserts the importance of studying left ...
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The category of groupoids vs the category of sets
Hopefully this question makes sense. As we know that Kan complexes are the "$\infty$-version" of groupoids for $\infty$-categories as groupoids for categories. On the other hand, the $\infty$...
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Homotopy factorization of morphisms of chain complexes
This is a sort of follow up to this MO question.
Let $R$ be a ring (eventually with good properties) and $\mathrm{Chains}(R)$ be the category of chain complexes of $R$-modules (eventually bounded). ...
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What are natural transformations in 1-categories?
It's well-known that, for lots of concrete categories (but by no means all), we can think of the objects as themselves being small categories, and morphisms are the functors between these categories. ...
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Computation of Joins of Simplicial Sets
It turns out that joins of simplicial sets are fairly easy to define, but hard to manage. In lots of cases, we'd like to compute what a join is, does it look like a horn?, a boundary?, etc? and ...
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$\omega$-nerve versus $\Theta$-nerve
To which extent the adjunction $F\dashv N_\omega$ generated by the $\omega$-nerve described at $n$Lab - oriental (obtained as a particular instance of the nerve-realization paradigm) is linked to the ...
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Which E_∞-spaces are homotopy colimits of k-truncated E_∞-spaces?
This question is closely related to my previous question about modules over truncated sphere spectra, in particular, it has the same motivation.
Recall that every space (or ∞-groupoid) can be ...
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Motivation for the covariant model structure on SSet/S
I was reading HTT 2.1.4, and I just totally lost what was going on. Could someone provide some motivation for this section? Why do we want another model structure?
I'm sorry for not providing ...
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Does the 2-category of topoi have exponential objects?
Does the 2-category of Grothendieck topoi have exponential objects?
There are size issues: Since Grothendieck topoi are supposed to have a small set of generators, the collection of objects of a ...
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Is the "homotopy category" functor well-defined?
$$\def\Cat{\mathbf{Cat}} \def\qCat{\mathbf{qCat}} \def\Catinf{{\mathcal{C}at_\infty}} \def\Catone{{\mathcal{C}at_1}} \def\cC{\mathcal{C}}$$ Let $\Cat$ be the 1-category of small categories, $\qCat$ be ...
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What's so special about $1$-categories?
I have been pretty thoroughly convinced for some time now that, when thinking about mathematics, one really should be thinking 'categorically', that is, one should always be thinking of the morphisms ...
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How can I see that the slice of a presheaf category is equivalent to the presheaf category of the category of elements?
Let $\mathcal{C}$ be a small category and $P$ a presheaf on $\mathcal{C}$. Then there is an equivalence $$\widehat{\mathcal{C}} / P \cong \widehat{\int_{\mathcal{C}}} P$$ Now there are (at least) two ...
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Local Joyal-simplicial presheaves?
It is well known that left Bousfield localizations of the global functor model category $Func(C^{op}, SSet_{standard})$ of functors with values in simplicial sets equipped with the standard model ...
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Explicit description of the right adjoint
Let $C$ be a diagram. Consider a functor $F: C \to \mathbb{E}_{\infty}(Sp)$ from the diagram to the category of $\mathbb{E}_{\infty}$-rings in spectra. Let $R$ be the limit of this diagram.
Given the ...
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Remark 5.4.2.15 in HTT
In Remark 5.4.2.15 in Higher Topos Theory, Lurie explains in what sense an accessible functor $F:\mathcal{C}\rightarrow \mathcal{D}$ between accessible $\infty$-categories is "determined by small ...
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Example of a non-$\infty$-category whose homotopy category is a groupoid
What is an example of a simplicial set $S$ such that its homotopy category $hS$ is a groupoid, but such that $S$ is not an $\infty$-category?
I know that if $S$ is an $\infty$-category, then $S$ is a ...
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Is there a definition of reduced $E_\infty$ ring?
[Edit: I have completely changed the question in response to the replies given]
I am curious if there is well defined notion of reduced $E_\infty$-ring.
Let $CAlg$ denote the $\infty$-category of $E_\...
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Uniqueness of $\infty$-adjoints
Adjoints in a 2-category are essentially unique, in the following strong sense. If $\mathbf{2}$ denotes the "walking arrow" category $(\cdot \to \cdot)$, then there is a 2-category $\mathrm{Adj}_1$ ...
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2 and 3 pullbacks
If $F:A\to C$ and $G:B\to C$ are morphisms in $Cat$, then their pseudo-pullback (I hope it's the right notion) can be calculated as the strict limit $A\times_C C^I \times_C B$, where $I$ is the ...
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Models for, and motivation for, (oo,n)-categories for general n
First: Is there a precise meaning to the term "model for (oo,n)-categories"? A related question, which might actually be the same question, is: what exactly do we want to get out of (oo,n)-category ...
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Where does the "easy" definition of a weak n-category fail?
Okay, I'm going to ask a naiive question that surely has an interesting answer. So, a first approximation of defining a (small) weak n-category probably goes something like this. Take a pre-n-category ...
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Are there universal homological functors?
There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable?
That is, for each small abelian ...
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Pushforwards of stacks of algebras?
This is a refined/sheafified version of this previos question of mine.
Let $(X,\mathcal{O}_X)$ be a ringed space or more in general a ringed stack, where the structure sheaf $\mathcal{O}_X$ is a ...
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3-functoriality of the lax Gray tensor product
In Formal category theory: adjointness for 2-categories, Gray defines a tensor product of 2-categories, now more commonly known as the lax Gray tensor product, which I will denote by $\otimes_l$. For ...
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limits and products stable $\infty$-category
In an abelian category $\mathcal{A}$, for a system $\{F_i,\phi_{ij}\}$ we have an exact sequence
$0\to \lim F_i\to \prod F_i \to \prod F_i$
where the second map is given by $id-\prod\phi_{ij}$. Is ...
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Defining (infinity,1)-categories in HoTT using only an interval type
In this article, Emily Riehl and Michael Shulman describe a type theory in which one can do $\infty$-category theory synthetically. Their framework allows them to define simplices $\Delta^n$, and a ...
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Diagonal of a diagram of codescent objects
Given the following diagram in a $2$-category, in which squares of the same "type" commute, where each column and each row is a strong codescent diagram (Edit: it should be reflexive as well), is ...
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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 ...
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Remark 2.4.1.4 Higher Topos Theory
In HTT, given a inner fibration $p : X \rightarrow S$ of simplicial, an edge $f : x \rightarrow y$ of the simplicial set $X$ is said to be a $p$-Cartesian if the induced map
$$ X_{/f} \rightarrow ...
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Universal property of sheaf category
Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...
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The conceptual difference in notations of Cat
There have been some places in which a conceptual difference (that of criteria of identification, etc) is avoided in the categorical notation, namely
Let $Cat$ be at the same time the 2-category of ...
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Higher and lower analogues of Yoneda's lemma
Here's a statement of Yoneda's lemma for n-category.
Let C be a n-category and $C^{\wedge}=[C^o,n-1Cat]$ be the n-category of presheaves on C.
$C^o$ is the opposite n-category of C and $n-1Cat$ is ...
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Small objects in categories
I would like to pick out small objects from a category. I would like to find such a notion which
Dream 1. Picks out the schemes of finite type over $k$ from the category of $k$-schemes. Or at least ...
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How does one Segal-subdivide a 2-category?
Let $\mathcal{C}$ be a small category. Then, its Segal subdivision $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to ...
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references for models of stable infinity categories
There's a fair amount of literature comparing different models for the homotopy theory of homotopy theories, or the homotopy theory of $(\infty,1)$-categories. Julie Bergner has a survey of this ...
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Base change isomorphism for left Kan extensions
Let $\mathscr{C}$, $\mathscr{D}$, and $\mathscr{E}$ be (infinity) categories, and assume we're given a Cartesian diagram: $\require{AMScd}$
\begin{CD}
\mathscr{C} \times_{\mathscr{E}} \mathscr{D} @>...
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