# 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 ...
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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 ...
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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 ...
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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 $\mathsf{Map}:\mathscr{C}^{\mathrm{op}} \times \mathscr{C} \to \mathsf{An}$ ...
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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. ...
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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 ...
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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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### 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 ...
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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 ...
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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:...
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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 ...
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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,...
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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: $\require{AMScd}$ \...
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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-...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $s_1$. "...
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### 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” $\mathcal{D}_1$,...
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### 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 ...
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### 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 ...