# Questions tagged [higher-category-theory]

For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.

1,197
questions

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### 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 ...

4
votes

1
answer

119
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### 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 ...

3
votes

1
answer

147
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### 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 ...

7
votes

1
answer

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### 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 ...

1
vote

0
answers

35
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### 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 ...

7
votes

1
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### 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.
...

10
votes

1
answer

329
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### 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
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### 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-...

2
votes

1
answer

271
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### 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, ...

2
votes

1
answer

167
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### 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?
...

8
votes

1
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### 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
$$F_{\mathcal A \to \mathcal B} : \mathrm{Topos}_{//\mathcal B} \to \mathrm{Topos}_{//\mathcal A}$$
...

3
votes

1
answer

99
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### 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 ...

3
votes

0
answers

378
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### 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 ...

4
votes

0
answers

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### 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 ...

4
votes

0
answers

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### 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\...

5
votes

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### 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 ...

4
votes

1
answer

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### 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,
...

6
votes

1
answer

198
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### $K_1$ 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 ...

8
votes

0
answers

141
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### 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 ...

4
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### 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 ...

3
votes

0
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### 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 ...

2
votes

3
answers

266
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### 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 ...

6
votes

1
answer

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### 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 \...

6
votes

0
answers

236
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### 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 ...

6
votes

0
answers

227
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### 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 ...

3
votes

0
answers

127
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### 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 ...

6
votes

0
answers

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### 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 ...

2
votes

0
answers

100
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### 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\...

9
votes

1
answer

370
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### 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 ...

6
votes

0
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109
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### 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)...

6
votes

0
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### 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 ...

17
votes

0
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561
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### 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 ...

5
votes

1
answer

236
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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} @>...

6
votes

0
answers

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### (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 ...

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
$$
\...

6
votes

0
answers

308
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### 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
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### 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{...

4
votes

1
answer

245
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### 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 ...

5
votes

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### 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 ...

2
votes

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### 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 ...

6
votes

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answers

301
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### $(\infty,1)$-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 ...

5
votes

0
answers

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### $(\infty,2)$-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 ...

10
votes

0
answers

146
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### 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 ...

5
votes

1
answer

179
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### 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 \...

7
votes

1
answer

110
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### 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 ...

3
votes

1
answer

155
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### 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 ...

9
votes

0
answers

376
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### 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 ...

5
votes

1
answer

345
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### 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
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### 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 ...

4
votes

1
answer

340
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### 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 ...