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Questions tagged [higher-category-theory]

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. (Def: http://en.m.wikipedia.org/wiki/Higher_category_theory)

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the infinite category constructed from a model category has all limits and colimits

I am reading a survey on derived algebraic geometry. On the page 28, I read such a paragraph: Let $C$ be a model category, and $I$ be a small category, let $C^{I}$ be the model category of diagrams in ...
Yang's user avatar
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Adjoint triplet induced by exact functor of stable categories

For $\mathscr{A}$ a small stable $\infty$-category, we can consider the following diagram: where $\mathcal{Y}_\mathscr{A}$ denotes the ordinary Yoneda embedding $\mathscr{A} \to \mathcal{P}(\mathscr{...
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Why the non-abelian 4-cocycle condition?

In a monoidal category it holds by definition (together with the identity coherence) an associativity coherence axiom, stating commutativity of the pentagon Now, if we categorify vertically, we can ...
Nikio's user avatar
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derived version of Picard functor as a derived stack

The notation originates from the paper virtual Cartier divisor and derived blow up I read recently, in its proof of proposition 3.2.6, there is a derived stack: $\\$$\underline{Pic} ^{\simeq}$: $(...
Yang's user avatar
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Cosimplicial resolution associated to a monad

Let $\mathcal{C}$ be a category and $\mathbf{T}$ a monad on $\mathcal{C}$ with functor part $T : \mathcal{C} \to \mathcal{C}$ (I would actually like to consider the case where $\mathcal{C}$ is an $(\...
Brendan Murphy's user avatar
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Is there a notion of infinity multicategories?

I know there are infinity operads, but I could not find anything about infinity multicategories. Is there any such notion or is it just useless to even consider such an object?
toby flenderson's user avatar
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Reference for k-invertible cells in an n-category being (k+1)-invertible

A while ago, I read that if a cell in a strict n-category is invertible with respect to composition along its k-boundary, it is also invertible with respect to composition along its (k+1)-boundary. ...
Wilf Offord's user avatar
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On the induced morphism between colimits in an $\infty$-category

Let $\mathcal{C}$ be an $\infty$-category (quasicategory), $F\colon K\rightarrow\mathcal{C}$ a diagram in $\mathcal{C}$ and $y$ a colimit of $F$. To be precise, there is a colimit cone $\overline{F}\...
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Simplicial presheaves present $\infty$-presheaves related question

I am trying to work out the details that every $\infty$-topos is presented by a model topos. By presented I mean it is the image under the homotopy coherent nerve. A model topos is a model category ...
Secher Nbiw's user avatar
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Naturality for the Homotopy Fiber Sequence of Mapping Spaces

For a cartesian fibration $p\colon\mathcal{E}\rightarrow\mathcal{C}$ of $\infty$-categories (quasicategories) and objects $x,y$ of $\mathcal{E}$, the induced map $\mathrm{map}_{\mathcal{E}}(x,y)\...
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Geometric Morphisms and Fibrations in Higher Category Theory

I have been exposed to the idea of geometric morphisms in an ordinary categorical sense. There is a paper titled 'Fibred categories a la Jean Benabou' this somewhat links geometric morphisms and ...
Siyabonga Mthimkulu's user avatar
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the infinite category of pullback squares in an infinite stable category is also stable

I am currently reading Lurie's paper infinite stable category, in the proof of proposition 4.4, to show that every pushout square in an infinite stable category $C$ is also a pullback, he considers ...
Yang's user avatar
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$\Sigma_n$ action on algebras of symmetric monoidal $\infty$-categories

Let $\mathcal{C}^\otimes$ be a symmetric monoidal ($\infty$)-category. In $1$-category theory, given $X^{\otimes n} \in \mathcal{C}$, there is an action on the permutation group $\Sigma_n \...
Frusciante's user avatar
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Multiplication map of algebras of monoidal ($\infty$)-categories

Lurie defines (DAG-II, Def. 1.1.2) a monoidal $\infty$-category $\mathcal{C}$ as a coCartesian fibration $\mathcal{C}^\otimes \to N(\Delta)^\text{op}$ satisfying the Segal condition, i.e. such that ...
Frusciante's user avatar
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Infinite category structure on SCRing and 'space of commutative squares' in SCRing

When I read this paper 'virtual cartier divisors and blow ups', I often meet with such phrase like 'mapping space of infinite category $SCRing_{A}$'. See lemma 2.3.5 in the above paper: $Map_{SCRing_{...
Yang's user avatar
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The core of an $\infty$-category and pointwise invertible maps $\Delta^1 \times \partial \Delta^n \cup \{1\} \times \Delta^n \to \underline \hom(B,X)$

I'm currently working through Theorem 3.5.11 of Cisinski's Higher Categories and Homotopical Algebra. The section in which the theorem is inscribed concers the core of an $\infty$-category. Trying to ...
qualcuno's user avatar
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Higher Coherences and Maps from Colimits

In higher category theory there is the important mantra that commutativities are additional data and not just a property as in $1$-category theory. So the following question I'm proposing will ...
Qi Zhu's user avatar
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1 answer
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Natural transformation picking out the map from the initial object

As so often, I'm failing to construct a map in $\infty$-category theory that is easily constructed in $1$-category theory. Let $\mathscr{C}$ be an $\infty$-category and let $F: \mathscr{C} \to \mathsf{...
Qi Zhu's user avatar
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Stable categories are tensored over spectra

It's a folklore result by Lurie that for a (presentable) stable $\infty$-category $\mathscr{C}$ the mapping space functor $\mathsf{Map}:\mathscr{C}^{\mathrm{op}} \times \mathscr{C} \to \mathsf{An}$ ...
Qi Zhu's user avatar
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Forgetful functor from derived category $D(\mathbb{Z})$ to Spectra

How is concretely defined the (canonical?) forgetful functor from $D(\mathbb{Z})$, the derived category of the ring of integers, to the catetegory of spectra $\text{Sp}$? (here is refered to such map. ...
user267839's user avatar
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Internal Category Theory for Double Fibrations

Are there books (besides Handbook of Categorical Algebra Vol 1) that introduce the theory of internal category theory? My purpose for this is to have an intuition for double fibrations, at least for ...
Makhulukhulu's user avatar
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Terminal object in $\infty$-category

Let $\mathcal{C}$ be an $\infty$-category. How to show that the following definitions agree? An object $X\in\mathcal{C}$ is terminal if the space $\mathrm{Map}_{\mathcal{C}}(Z,X)$ is contractible for ...
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$\infty$-groupoid with initial (or terminal) object is contractible.

I am currently learning about $\infty$-categories and we were asked to prove the following: Let $\mathcal{C}$ be an $\infty$-groupoid (i.e. an Kan complex) with an initial object. Show that $\mathcal{...
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Quillen adjunction between Kan and Joyal model structures on sSet

I have read "The thoery of quasi-categories and its application" by A.Joyal. Theorem.6.22 shows that Kan and Joyal model structures on sSet are Quillen adjoint. ($k_! : sSet_{Kan} \to sSet_{...
Keima's user avatar
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Confused about definition of 2-category of categories over $\mathcal{C}$

Let $\mathcal{C}$ be a category. Here the following definition is given for the 2-category of categories over $\mathcal{C}$. The 2-category of categories over $\mathcal{C}$ is the 2-category defined ...
Grujo's user avatar
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1 answer
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Show that $\mathcal{A}n$ is generated under colimits by its terminal object

In a course on $\infty$-categories I am attending we were asked to show that $\mathcal{A}n$ (by which I mean the $\infty$-category of anima which we defined as the simplicial nerve of the subcategory ...
Womm's user avatar
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Does geometric realization commute with finite limits?

I am trying to find out if geometric realizations i.e. the functor $|-|\colon\text{sSet}\to \text{Top}$ commutes with finite limits. In the following post the user claim that this is well known: https:...
Womm's user avatar
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The geometric realization of a simplicial set does not determine it

I read in Gallauer's notes on infinity categories, in "Warning 1.9." without proof, that the geometric realization of a simplicial set does not determine it and wanted to make sure I ...
kindasorta's user avatar
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4 votes
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Stable and Pointed Infinity-Operads

I wanted to understand the construction of the maps $\mathsf{An}^{\times} \xrightarrow{(-)_+} \mathsf{An}_{*/}^{\wedge} \xrightarrow{\Sigma^{\infty}} \mathsf{Sp}^{\otimes}$ of commutative algebras in $...
Qi Zhu's user avatar
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1 answer
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Are Diagrams with equal source equivalent if their limits are?

Let $d_{1}:I\to Cat_{\infty}$ and $d_{2}:I\to Cat_{\infty}$ be two diagrams with the same source and with equivalent limits: there exists an equivalence of $\infty$-categories $lim d_{1}\simeq lim d_{...
Matteo Doni's user avatar
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0-connective objects in an $\infty$-topos

I am trying to understand Marc Hoyois's proof of the Van Kampen theorem in an $\infty$-topos given as Lemma 6 in : https://ncatlab.org/nlab/files/Hoyois_PlusConstruction.pdf In the course of the proof,...
dicemaster666's user avatar
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Are conservative functors stable under pullback?

Let $A, B$ and $C$ be three small categories and let $f: A\to B$ and $g: C\to B$ two functors such that $g$ is conservative. We can take the pullback as in the following picture: $\require{AMScd}$ \...
Matteo Doni's user avatar
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Associativity of a spectrum with relation to complex bordism spectrum

I am reading Lurie's notes about chromatic homotopy theory. In Lecture 22, he considers the p-local complex bordism spectrum $\operatorname{MU}^p$ with $\pi_*(\operatorname{MU}^p)=\mathbb{Z}_p [t_1,...
Runner's user avatar
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2 votes
1 answer
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Relation between the category of categories with finite limits and category of categories with finite colimits.

Let $Cat^{rex}$ be the category whose objects are (small) categories admitting finite colimits and let $Cat^{lex}$ be the category whose objects are (small) categories admitting finite limits. My ...
Matteo Doni's user avatar
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0 answers
39 views

Bar construction for cocartesian monoidal structure is calculated by pushout

$\DeclareMathOperator\colim{colim}$ 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 $\text{CAlg}(\mathcal{C})$ is cocartesian. I ...
Xiong Jiangnan's user avatar
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What should $n$-transfors be?

I'm having trouble understanding why modifications are the right notions of $3$-morphisms. For natural transformation, we get the definition by requiring $1\text{-}Cat$ to be cartesian closed (nlab). ...
Fernando Chu's user avatar
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1 vote
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differential graded nerve of a category whose mapping complexes are non-positive

Let $C$ be a differential graded (abbreviated to dg) category, i.e. a category enriched by the category of chain complexes. Then $N_{dg}(C)$, the differential graded nerve of $C$ is a simplicial set ...
Kanae Shinjo's user avatar
3 votes
1 answer
69 views

Elements of $\infty$-cats Corollary 4.1.3

I am currently studying the book Elements of $\infty$-Cats and stumbled across the following Corollary (this is Corollary 4.1.3 on page 88 in the book): I am a bit confused on how to get the induced ...
h3fr43nd's user avatar
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Internal Homs of (Higher) Operads and $(\infty, 2)$-Categories

While $(\infty, 1)$-categories continue to scare me (but also bring me joy!), it is almost frightening how naturally $(\infty, 2)$-categories seem to pop up if you're interested in $(\infty, 1)$-...
Qi Zhu's user avatar
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9 votes
2 answers
100 views

Category of isomorphisms is equivalent to underlying $\infty$-category

Let $\mathscr{C}$ be an $\infty$-category (for example taking quasicategories as a model). Recall that the arrow category is $\mathsf{Ar}(\mathscr{C}) = \mathsf{Fun}([1], \mathscr{C})$. We denote by $\...
Qi Zhu's user avatar
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5 votes
1 answer
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Sanity Check: Monoids as Algebras over an Operad

I should probably study the classical viewpoint first. I haven't yet and I will eventually but let us stick to $\infty$-operads for this question. I'm following the lecture notes from Hebestreit-...
Qi Zhu's user avatar
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0 votes
1 answer
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Homotopy orbits

Let $\mathcal{C}$ be an $\infty$-category, and let $G$ be a group. Denote by $\mathcal{C}^{\text{BG}}$ the functor $\infty$-category $\text{Fun}(\text{BG}\longrightarrow \mathcal{C})$. The homotopy ...
kindasorta's user avatar
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4 votes
2 answers
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Cobordism Hypothesis in dimension 1

I am trying to understand consequences of the Cobordism Hypothesis in dimension 1, following section 4.2 of Lurie's "On the Classification of Topological Field Theories". Especially, I want ...
Hyunbok Wi's user avatar
2 votes
1 answer
52 views

Is the category of monoids $\textsf{Mon}(\mathcal{C})$ in a monoidal category $\mathcal{C}$ itself monoidal?

I have read about the forgetful functor $U$ from $\textsf{Mon}(\mathcal{C})$ to $\mathcal{C}$ (see e.g. this nLab page). I think I have read somewhere that this functor is monoidal, but I cannot find ...
user11718766's user avatar
2 votes
1 answer
98 views

What's the Lurie's tensor product of dg categories?

I learn dg categories from Toën's lecture notes Lectures on dg-categories, where the tensor product $T\otimes T'$ of two dg-cateogries is defined to be such that it has objects of pairs $(x,x')$ and ...
Yining Chen's user avatar
2 votes
1 answer
110 views

Waldhausen's $S.$ construction as an adjoint

I am studying K-theory with Weibel's K-book and have just read the definition of the $S.$ construction for Waldhausen categories. I also recently watched a series of talks by Thomas Nikolaus which ...
DevVorb's user avatar
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0 votes
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single set category

I'm trying to understand the single-set category in Categories for the Working Mathematician , specifically the 2-category version, and I'm confused about the commutation of $s_0$ and $s_1$. "...
noCrayCray's user avatar
2 votes
1 answer
33 views

Are (commutative) squares in some sense universal among edge-symmetric double categories?

Definitions: Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$,...
hasManyStupidQuestions's user avatar
1 vote
1 answer
71 views

Morphism $\infty$-categories

Let $\mathcal{C}$ be an $\infty$-category, and let $X, Y$ be a pair of its objects. It is said that $\text{Mor}_{\mathcal{C}}(X,Y)$ has again the structure of an $\infty$-category. What is this ...
kindasorta's user avatar
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0 votes
1 answer
39 views

Objects in an $\infty$-category

I tried reading about the definition of a pre-additive $\infty$-category, there it is said that a pointed $\infty$-category is pre-additive if all finite products and co-products exist, and the ...
kindasorta's user avatar
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