Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
1,197
questions
7
votes
1
answer
148
views
Stably-framed cobordism $(\infty,n)$-category
In Lurie's treatment of the cobordism hypothesis, the domain is $\mathsf{Bord}^{fr}_n$, the symmetric monoidal $(\infty,n)$-category of $k$-bordisms with $n$-framing for $0\leq k\leq n$.
If I ...
- 521
4
votes
1
answer
119
views
Fibre sequence of module spectra induces a fibre sequence of $K$-theory spectra?
Let $A$ be an $\mathbb{E}_\infty$-ring spectrum, and let $R_1$, $R_2$ and $R_3$ be $\mathbb{E}_\infty$-$A$-algebras.
We assume there is a homotopy fibre sequence
$$
R_1\to R_2 \to R_3
$$
in the stable ...
- 449
3
votes
1
answer
147
views
Gluing isomorphism in derived categories along filtered colimit
Let $X$ be a locally finite type algebraic stack $X$ (but feel free to pretend it's a scheme) with a presentation as the filtered colimit of finite type open substacks $U_i$. By descent, at the level ...
- 53
7
votes
1
answer
177
views
2-completeness of stacks
I am looking for a reference which discusses the 2-categorical properties of the 2-category $St(C,J)$ of stacks and the stackification $\dashv$ inclusion of presheaves 2-adjunction.
My stacks are ...
- 735
1
vote
0
answers
35
views
Finitely continuous fibrant replacement functor for localization of simplicial presheaves with projective model structure
Let $C$ be a model category given by generators and relations in the sense of Dugger (that is, $C$ is a left Bousfield localization of a global projective model model structure on simplicial ...
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7
votes
1
answer
260
views
Why does the 2-category of groups have (some, strict) coinserters but not (strict) inserters?
Let me begin by mumbling some abstract nonsense, and then attempt to be concrete.
The category of groups inherits the structure of a strict 2-category from the 2-category of small categories.
...
- 1,741
10
votes
1
answer
329
views
Big list: barycentric subdivision of simplicial sets
I'm preparing a seminar on the barycentric subdivision of simplicial sets and I'm looking for some examples of this construction appearing in the literature. Since it's a useful technique (at least in ...
4
votes
0
answers
83
views
State-sum for 4d TQFT from fusion 2-categories and invariants of Morita equiavalence classes beyond Drinfeld center
If $\mathcal{C}$ is a fusion 1-category, the Turaev-Viro state-sum produces a 3d TQFT whose modular tensor category is the Drinfeld center of $\mathcal{C}$. In particular this means that the Turaev-...
- 453
2
votes
1
answer
271
views
Why do we need enriched model categories?
As far as I understand, model categories mainly provide tools for studying the "homotopy theories" (that is, $\infty$-categories) that are ubiquitous in mathematics. From this point of view, ...
- 5,260
2
votes
1
answer
167
views
Homotopy coherent nerve for algebraic model categories
Is a homotopy coherent nerve defined for algebraic model category that returns algebraic quasi-categories as Urs Schreiber wrote about? Or do we not know how to determine it / does it seem impossible?
...
- 5,260
8
votes
1
answer
270
views
Does the Zariski spectrum of a ring arise formally from the inclusion of the big Zariski topos into the classifying topos for rings?
Let $\iota_\ast : \mathcal A \to \mathcal B$ be a geometric morphism. I'm looking for some functor
$$F_{\mathcal A \to \mathcal B} : \mathrm{Topos}_{//\mathcal B} \to \mathrm{Topos}_{//\mathcal A}$$
...
- 55.4k
3
votes
1
answer
99
views
Does the forgetful functor from a pointed $\left(\infty, 1\right)$-category only create weakly contractible colimits?
Consider an $\left(\infty, 1\right)$-category $\mathcal{C}$ with a terminal object $1$. (I'm particularly interested in the case where $\mathcal{C}$ is a topos.) It is known that the forgetful functor ...
- 41
3
votes
0
answers
378
views
Infinite dimensional dg-manifolds
In Def 2.5.1 in " Derived Quot schemes" by Ciocan-Fontanine and Kapranov, we can find the notion of dg-manifolds.
In detail, let $X$ be a dg-scheme over $k$ ($k$ : an algebraic closed field ...
- 137
4
votes
0
answers
82
views
Truncations of orientals
Let $n,m $ be naturals such that $n >m$ and $(-)_{\leq m}$ the left adjoint of the embedding of
m-categories into n-categories, which we call m-truncation.
Is there a canonical presentation of the ...
- 331
4
votes
0
answers
64
views
The internalization hierarchy
For a complete category $\mathcal{C}$, we can consider the strict $2$-category ${\sf Cat}(\mathcal{C}$) of internal categories in $\mathcal{C}$. Similarly, for any continuous functor $F:\mathcal{C}\to\...
- 8,294
5
votes
0
answers
48
views
Is the functor $O$ from the simplex category to the category of orientals cofinal
Let $\Delta$ be the full subcategory of the category of small categories spanned by the non-empty totally ordered sets of the form $[n]$ for $n \geq 0$. Let $\mathfrak{O}$ be the full subcategory of ...
- 331
4
votes
1
answer
162
views
Final and strongly final objects in Higher Topos Theory
Lurie introduced in subchapter 1.2.12 of his Higher Topos Theory
the notion of final and strongly final objects:
Definition 1.2.12.1. let $\mathcal{C}$ be a topological category (e.g. simplicial cats,
...
- 393
6
votes
1
answer
198
views
$K_1$ of Categories for intuition
Maybe there is no good answer to this, but I'm trying to get a feel for what the $K$-theory of a (permutative or symmetric monoidal $\infty$-)category computes.
In algebraic $K$-theory, we have ...
- 3,984
8
votes
0
answers
141
views
Symmetric monoidal structures on the functor taking presheaves
Let $\mathrm{Cat}_\infty$ be the $\infty$-category of small $\infty$-categories, $\mathrm{Pr}^\mathrm{L}$ be the $\infty$-category of presentable $\infty$-categories, and $\mathcal P(\mathcal C)$ be ...
- 435
4
votes
0
answers
70
views
Lifting adjunctions along a localisation of 2-categories
Let $\mathcal S$ be a base category and let $Fib_\mathcal S^{split}$ be the 2-category consisting of split fibrations, fibered functors which strictly preserve the splitting and vertical ...
- 735
3
votes
0
answers
69
views
Are dagger-categories / categories with duality related to unoriented field theories?
Recall that a dagger-category is a category $C$ equipped with an identity-on-objects, involutive anti-equivalence $\dagger: C^{op} \to C$. I believe that the correct $\infty$-categorical ...
- 55.4k
2
votes
3
answers
266
views
Categorifying the definition of a principal $G$ bundle
For a Lie group $G$, we can define a principal $G$ bundle as a submersion of manifolds $\pi:P \to X$ equipped with a free right $G$-action on $P$ that is transitive on the fibres over $X$.
What goes ...
- 303
6
votes
1
answer
187
views
Checking 2-dualizability
Let $(\mathcal C, \otimes, I)$ be a symmetric monoidal 2-category, and let $X \in \mathcal C$ be a dualizable object, with dual $X^\vee$, unit $coev: I \to X \otimes X^\vee$, and counit $ev : X^\vee \...
- 55.4k
6
votes
0
answers
236
views
Goodwillie-entire functors
Let $F : \mathsf{Spaces} \to \mathsf{Spaces}$ be a functor, and for $X \in \mathsf{Spaces}$, let $F_X : \mathsf{Spaces}_{/X} \to \mathsf{Spaces}_{/F(X)}$ denote the induced functor.
Recall that in ...
- 55.4k
6
votes
0
answers
227
views
Is there any reason not to use Hofmann-Streicher universes?
Let $\mathcal C$ be a small category, and consider the topos $Psh(\mathcal C)$ of $Set$-valued presheaves on $\mathcal C$. For simplicity, assume that there exists a proper class of inaccessible ...
- 55.4k
3
votes
0
answers
127
views
Higher theory o $C^{\ast}$-algebras and the $C^{\ast}$-algebra of a $\infty$-groupoid
Has someone already worked out what would be the infinity categorical analogue of the category of $C^{\ast}$-algebras? Given a groupoid $G$ we can associate a $C^{\ast}$-algebra $C^* (G)$, can we do ...
- 1,019
6
votes
0
answers
106
views
Homotopy fibers in the Joyal model structure and the Kan–Quillen model structure
While playing around with $\infty$-categories, I ran into the following problem:
Let $p:\mathcal{C}\to\mathcal{D}$ be a functor of $\infty$-categories. Does one of the following condition imply the ...
- 1,040
2
votes
0
answers
100
views
Gluing data for $\infty$-sheaves?
Let $\mathcal{F}$ and $\mathcal{G}$ be two $\infty$-sheaves on $X$ resp. $Y$. I want to understand exactly when we can "glue" $\mathcal{F}$ and $\mathcal{G}$ to give a $\infty$-sheaf on $X\...
- 21
9
votes
1
answer
370
views
Do compactly generated spaces have a more direct definition?
Is there an elementary way to define Haussdorf-compactly generated weakly Hausdorff topological spaces in a way that does not need defining topological space first?
Weakly Hausdorff sequential spaces ...
- 1,733
6
votes
0
answers
109
views
Mapping space between $n$-groupoids is an $n$-groupoid
Consider two simplicial sets $K$ and $L$. Their mapping space (or mapping complex) is the internal hom of simplicial sets, i.e. $\underline{\mathrm{Hom}}(K,L)$, where
$$
\underline{\mathrm{Hom}}(K,L)...
- 61
6
votes
0
answers
110
views
Presenting geometric morphisms by geometric morphisms
It's known that any $\infty$-topos $\mathcal{E}$ can be presented by a Quillen model category $\mathbf{E}$ that is itself a 1-topos. For instance, if $\mathcal{E}$ is a left exact localization of a ...
- 62.7k
17
votes
0
answers
561
views
What is the status of the cobordism hypothesis?
Let $\mathscr{C}$ be a symmetric monoidal (weak) $n$-category. A framed extended TQFT of dimension $n$ with values in $\mathscr{C}$ is a symmetric monoidal functor from the framed bordism $n$-category ...
- 171
5
votes
1
answer
236
views
Base change isomorphism for left Kan extensions
Let $\mathscr{C}$, $\mathscr{D}$, and $\mathscr{E}$ be (infinity) categories, and assume we're given a Cartesian diagram: $\require{AMScd}$
\begin{CD}
\mathscr{C} \times_{\mathscr{E}} \mathscr{D} @>...
- 2,889
6
votes
0
answers
144
views
(Co)cartesian fibrations and left Kan extensions
Let $p: \mathscr{C}\to\mathscr{D}$ be a functor of (small) $\infty$-categories. Let $\mathscr{E}$ be a cocomplete $\infty$-category. Assume that $\mathscr{C}, \mathscr{D}, \mathscr{E}$ admit finite ...
- 1,724
2
votes
1
answer
96
views
Morphisms in category of left fibrations
I am trying to better understand the straightening-unstraightening equivalence of Lurie in the $\infty$-categorical setting. In the case that I am interested in, this equivalence states that
$$
\...
- 221
6
votes
0
answers
308
views
Higher Algebra, Section 2.2.2
I am reading Section 2.2.2 of Higher Algebra by Jacob Lurie. There are proofs which I cannot understand, so I need someone's help.
First, I cannot understand the proof of Lemma 2.2.7. In the proof, ...
3
votes
0
answers
139
views
Hochschild homology of stable categories as topological chiral homology
Sorry for repost from Math Stack Exchange:
Let $\mathscr{C}_0$ be a small idempotent complete stable category tensored over some symmetric monoidal category $\mathcal{E}$.
Its Ind-completion $\mathscr{...
- 515
4
votes
1
answer
245
views
HTT, Remark 4.2.4.5
In his book Higher Topos Theory, Lurie proves that in some favorable cases, functor categories of $\infty$-categories admits a rigid model. More precisely, he proves in Proposition 4.2.4.4 the ...
- 1,040
5
votes
0
answers
278
views
What is an $\infty\text{-}E_{\infty}$ morphism?
My question is essentially what the title says, but here is some background that I have gathered from skimming through the literature. Please feel free to correct me if my understanding is wrong at ...
- 151
2
votes
0
answers
67
views
Hom-spaces of Segal spaces versus their realization in $\mathbf{Cat}$
Consider the $\infty$-category of simplicial spaces $s\mathcal{S} = \mathbf{P}(\Delta)$. The
inclusion $\Delta \to \mathbf{Cat}$ induces a left adjoint
$i_! : s\mathcal{S} \to \mathbf{Cat}$. It is ...
- 373
6
votes
0
answers
301
views
$(\infty,1)$-topoi generated by $(n,1)$-categories
A (1,1)-topos (i.e. an ordinary Grothendieck topos) is called localic if the following two equivalent conditions hold:
It is the category of sheaves on a (0,1)-site with finite limits$^*$ (i.e. a ...
- 62.7k
5
votes
0
answers
110
views
$(\infty,2)$-categories as colimits of orientals
Let $\mathcal{C}$ be an $\infty$-category represented by a fibrant simplicial set in the Joyal model structure. It is well known that $\mathcal{C}$ can be expressed as the (homotopy) colimit over its ...
- 223
10
votes
0
answers
146
views
Rectifying diagrams of $\infty$-categories
If $C$ is a 1-category and $D$ is a locally presentable $(\infty,1)$-category presented by a combinatorial model (1-)category $M$, then any $(\infty,1)$-functor $C\to D$ can be represented by a strict ...
- 62.7k
5
votes
1
answer
179
views
Fibrations of sites for $\infty$-topoi
For any geometric morphism $f:\mathcal{F} \to \mathcal{E}$ of Grothendieck 1-topoi, there exists a functor of small categories $\ell :D\to C$ and left exact localizations $\mathcal{F} \hookrightarrow \...
- 62.7k
7
votes
1
answer
110
views
Which direction does a lax dinatural transformation go?
In 2-category theory we often encounter notions with noninvertible coherence 2-cells, in which case there is a choice about which way the 2-cell should point. Generally, one of these directions is ...
- 62.7k
3
votes
1
answer
155
views
Can there be a cospan of symmetric monoidal $\infty$-categories whose maps are lax symmetric monoidal but the pullback is not symmetric monoidal?
Given symmetric monoidal $\infty$-categories $A, B, C$ and lax symmetric monoidal maps $F:A\to C$, $G:B\to C$, I am curious if the pullback (when I say pullback here I will really mean homotopy ...
- 327
9
votes
0
answers
376
views
Using higher topos theory to study Cech cohomology
It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...
- 821
5
votes
1
answer
345
views
On the definition of infinity-category
On 8:38 of Session 9: Masterclass in Condensed Mathematics an $\infty$-category is defined as a simplicial set $\mathcal{S}$ (i.e a functor $\Delta^{op}\rightarrow Sets$) such that for every horn $\...
2
votes
0
answers
105
views
Cat as a bicategory of monads over another category
Let's assume infinitely many Grothendieck universes exist. Let's call $\kappa$-Cat the bicategory of $\kappa$-small categories with anafunctors and anatural transformations. Now for any $\lambda$ and ...
- 399
4
votes
1
answer
340
views
Is the concept of an $H$ object still interesting, when we have the $\infty$-version of it?
Recently I got acquainted with $\infty$-algebraic theories. I expect $\infty$-algebraic objects (of ordinary Lawvere theories) in $\infty\text{-}\mathrm{Groupoid}$ to behave much better than algebraic ...
- 5,260