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3
votes
1answer
455 views

Opposite Symmetric Monoidal Structure on an Infinity Category

Given an $\infty$-category (in the sense of Lurie) $C$, and a symmetric monoidal structure on $C$ associated to a coCartesian fibration $p:C^\otimes\to N(Fin_\ast)$, Lurie says in Remark 2.4.2.7 of ...
7
votes
1answer
158 views

On the coherence theorem for bicategories

The coherence theorem for bicategories, as usually stated, reads Any bicategory $B$ is biequivalent to a (strict) 2-category. It is possible to give an explicit construction of the ...
4
votes
2answers
643 views

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 ...
6
votes
2answers
507 views

Connections on principal bundles via stacks?

Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to [G]$, where $[G]$ is the differentiable stack ...
2
votes
2answers
251 views

When do zero-simplices of a simplicial diagram determine its homotopy colimit?

Suppose that I have a diagram of simplicial sets $X_\bullet:\mathscr{C} \to Set^{\Delta^{op}},$ with $\mathscr{C}$ a small category such that for each $C \in \mathscr{C},$ $X_\bullet(C)$ is a Kan ...
2
votes
1answer
149 views

strictifying tricategories

Every tricategory is equivalent to a Gray-categories. However any Gray-category is not equivalent to a 3-category. As far as I know, this is similar to the fact that braided monoidal categories are ...
33
votes
4answers
6k views

Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...
2
votes
0answers
119 views

A Cartesian model structure (and straightening for) on $n$-trivial simplicial sets

A pair $(X,tX)$, with $X$ a simplicial set and $tX$ a collection of simplices of $X$, is said to be stratified if no $0$-simplex is in $X$ and all degenerate simplices of $X$ are in $tX$. Recall a ...
-2
votes
1answer
304 views

$E_n$ structures on Symmetric Monoidal Stable infinity-categories

Let $C$ be a stable $∞$-category with a monoidal structure on it, hence a monoidal stable $∞$-category; if this monoidal structure is "maximally symmetric" (in Wikipedia's sense) $C$ is a symmetric ...
0
votes
1answer
346 views

What is the delooping of a looping?

What is $\mathbf{B}\Omega A$, where $A$ is a pointed object of an $(\infty,1)$ category with point $*\to A$, $\Omega A$ is the loop space of $A$, and $\mathbf{B}X$ is the delooping of $X$? The ...
5
votes
2answers
152 views

For a quasicategory $C$, why is $\mathrm{Fun}(\Lambda^2_0,C) \to \mathrm{Fun}(\Delta^{\{2\}},C) \cong C$ a cocartesian fibration?

More generally, I expect that the following is true: Let $D$ be a diagram quasicategory, let $d \in D$ be a vertex, and use this to define $D' = D \amalg_{\Delta^{\{0\}}} \Delta^1$. Then ...
5
votes
1answer
459 views

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 ...
1
vote
0answers
77 views

Function complex and simplicial presheaves

Let $\mathcal{C}$ be a small catgeory, $\mathcal{E}$ be a model category and $A\: : \: \mathcal{C}\to \mathcal{E}$ be a functor. Let $\tilde{A}$ be an objectwise cosimplicial frame on A. Consider the ...
8
votes
1answer
356 views

Non-unique splittings of homotopy idempotents

By a homotopy idempotent I mean a map $f:X\to X$, where $X$ is a space, equipped with a homotopy $f\circ f \sim f$. In contrast to the situation in stable homotopy theory (where $X$ would be a ...
7
votes
2answers
320 views

Constructing a “geometric” model structure on Cat by localizing the “categorical” model structure

Let $\text{Cat}$ be the category of (small) categories and functors. There is a "categorical" (also called "canonical" or "folk") model structure on $\text{Cat}$ in which the weak equivalences are the ...
7
votes
0answers
207 views

A model category for E-infty algebras in a non-monoidal model category?

Given a suitable nice symmetric monoidal category $C$, symmetric monoidally enriched, tensored, and cotensored over a symmetric monoidal category $S$, and an operad $\mathcal{O}$ in $S$, we can ...
1
vote
1answer
271 views

(Homotopy) limits and colimits in a dg-category

It is known that differential graded categories (or, also, $A_\infty$-categories) are 'incarnations', in some sense, of (stable) $(\infty,1)$-categories. I'm not used to the theory of ...
7
votes
1answer
267 views

Can any object in a presentable category be written as a colimit of generators?

Let $\mathcal{C}$ be a presentable category, and let $S$ be a set of objects such that $S$ generates $\mathcal{C}$ under colimits, i.e., such that the smallest cocomplete subcategory of $\mathcal{C}$ ...
6
votes
1answer
673 views

$(\infty, 1)$-Yoneda embedding via the Grothendieck construction

Let $C$ be a quasi-category. Then there is an imbedding $$ C^{op} \times C \to \mathrm{Kan}$$ where $\mathrm{Kan}$ is the quasi-category of Kan complexes. This is essentially constructed in Lurie's ...
12
votes
1answer
695 views

Lower Algebra: Modules over the monoidal category of abelian groups

Proposition 6.3.2.18 of Higher Algebra identifies $Mod_{Sp}(Pr^L)$, the symmetric monoidal category of right modules over the monoidal category $Sp$ of spectra in $Pr^L$ the category of presentable ...
6
votes
2answers
298 views

Relationship between Hochschild cohomology and Drinfeld centers

Let $HH_*(A,N)$ (or $HH^*(A,N)$) be the Hochschild homology (or cohomology) of an associative algebra $A$ with coefficients in an $A$-bimodule $N$. I was reading nlab's entry on Hochschild cohomology ...
13
votes
2answers
435 views

Infinite dimensional 2-Hilbert spaces

Is there a definition of an infinite dimensional 2-Hilbert space? Finite dimensional 2-Hilbert spaces have been discussed by Baez in http://arxiv.org/abs/q-alg/9609018 In the more recent paper by ...
8
votes
4answers
823 views

Is there a sense in which the homotopy theory of simplicial sets is the “paradigmatic” one?

I could not come up with a better title for my question. What I am asking is this (preemptive excuses to all experts in homotopy theory for naivetes of all kinds you may find herein): the ...
5
votes
1answer
205 views

Mayer-Vietoris sequence for twisted R-homology

In this paper Ando, Blumberg, Gepner, Hopkins and Rezk define the twisted $R$-Homology of a ring spectrum $R$ together with a map $f \colon X \to R$-$Line$ to be $$ R^f_n(X) = \pi_0(map_R(\Sigma^nR, ...
4
votes
2answers
421 views

2-category theory

I know that we can do a lot of 2-category theory, seeing 2-categories as Cat-enriched categories. Yet, I know that there are some limitations of this approach. I also know that there are many articles ...
4
votes
0answers
200 views

Reference Request for TQFTs

I had originally asked this question on Math StackExchange but have not obtained any answers, so I decided to post this here. (I have flagged the MSE post to be moved to MathOverflow, for the ...
8
votes
2answers
288 views

Why does a tetracategory with one object, one 1-morphism and one 2-morphism give a symmetric monoidal category

According to the periodic table of k-tuply monoidal n-categories, it should be the case that a tetracategory (= weak 4-category) with one object, one 1-morphism and one 2-morphism is effectively ...
14
votes
1answer
433 views

Lurie's approach to the bar-cobar adjunction

I've been trying to read Jacob Lurie's approach to the bar-cobar constructions (Higher algebra §5.2 in the 2014-09 version) but I don't recognize what I know about these constructions. I wonder if ...
11
votes
2answers
336 views

A bit of history of Verdier duality

I was wondering who originated the presentation of Verdier duality as an equivalence between categories of sheaves and cosheaves ? I learnt it reading Jacob Lurie's Higher Algebra and Justin Curry's ...
3
votes
1answer
145 views

Adjoint of simplicial left Kan extension

Let $\mathcal{C}$ be a small simplicial category and let $F\: : \:\mathcal{C}^{op}\to sSet$ be a simplicial functor, we denote with $$ \int_{\mathcal{C}}F $$ the category where objects are triples ...
6
votes
2answers
289 views

Obstructions for $E_n$-algebras

In Alan Robinson's paper, Classical Obstructions and S-algebras, he provides conditions for a ring spectrum to have an $A_n$ and $\mathbb{E}_\infty$-structure. Have the obstructions for an object ...
-3
votes
1answer
118 views

Relations between ordinary functor categories and higher categories [closed]

Definitions of ordinary functor categories and higher categories are considered with very similar algebraic and geometric methods such as graph structures and simplicial sets. I know the differences ...
-1
votes
1answer
221 views

Kan extensions and special cases

Kan extensions specify the adjoint structures between $\mathbf{Sets^{C^{op}}}$ and $\mathbf{Sets^{D^{op}}}$, where there exists a functor $f:\mathbf{C} \to \mathbf{D}$ and $\mathbf{C}$ and ...
4
votes
1answer
249 views

$\omega$-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 ...
6
votes
0answers
241 views

What is a stable $(n,1)$-category?

My question in its simplest form is whether we have any understanding of stable $(n,1)$-categories, by analogy with stable $(\infty,1)$-categories and abelian categories (the latter being like "stable ...
4
votes
0answers
91 views

Trying to understand straightening functor associated to a right fibration of simplicial sets

Let $p:X \to S$ be a right fibration of simplicial sets; one can roughly think of it as some sort of "functor" $S^{op} \to Set_\Delta$ (where $Set_\Delta$ denotes simplicial sets) sending $s \in S$ ...
4
votes
0answers
83 views

Limits in Span(Vec)

Let Vec be the category of real vector spaces and linear maps. Let Span(Vec) be the bicategory of correspondences between real vector spaces. I am trying to understand lax limits in Span(Vec). What ...
4
votes
0answers
176 views

$n$-Fold Framed Functions

Suppose that $M$ is a manifold. One can consider a suitably constructed space of generalized framed Morse functions on $M$, let's call it $\mathrm{Fun}^\mathrm{fr}(M)$. This space is known to be ...
5
votes
3answers
385 views

What is a Homotopy between $L_\infty$-algebra morphisms II

I would like to continue on question I asking, what is a homotopy between Lie infinity algebras, since I'm not satisfied in two directions: 1.) The naive approach to define a homotopy would be ...
9
votes
1answer
307 views

$k$-Disk algebras versus $E_k$ algebras

Background: The little $k$-cubes operad is the $(\infty,1)$-operad defined by embedding disjoint unions of $k$-dimensional open cubes rectilinearly into one another, that is using maps ...
0
votes
0answers
62 views

Complex of “homotopy coherent dg-natural transformations” between dg-functors

Let $\mathcal B$ be a dg-category. I define the dg-category of morphisms $\underline{\mathrm{Mor}}(\mathcal B)$ as follows: objects are triples $(A,B,f)$, where $A,B \in \mathcal B$ and $f \in ...
39
votes
5answers
4k views

Why higher category theory?

This is a soft question. I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...
11
votes
0answers
380 views

Chiral categories versus braided monoidal categories

Let $X$ be a curve over $\mathbf{C}$. As I understand from the 2008 Talbot notes, a chiral category on $X$ consists of a crystal of categories on the Ran space $\mathrm{Ran}(X)$ (see these notes of ...
8
votes
1answer
563 views

Higher vector spaces

As far as I know there are different ways to categorify the notion of vector space/module. These appear (for example) when trying to find extended TQFTs. There are at least two ways (presented at ...
9
votes
1answer
223 views

Extended TFT with coefficients in spans in any $\infty$-topos

In the TFT classification article by Jacob Lurie (arXiv:0905.0465) the (∞,n)-category of correspondences (there: $Fam_n$) plays a key role, whose $k$-morphisms are $k$-fold spans of ...
10
votes
0answers
238 views

“extended TQFT” versus “TQFT with defects”

There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related? According to the Atiyah-Segal axioms, a ...
1
vote
1answer
90 views

Is the extension of a quasi-functor again a quasi-functor?

Let $\mathcal A, \mathcal A', \mathcal B$ be dg-categories over a field $k$ (this assumption allows me not to derive the tensor product, I don't think it is really essential). Let $F : \mathcal A \to ...
8
votes
0answers
95 views

What suffices to check completeness in an n-fold Segal space?

Recall that a Segal space is a simplicial space $X : \Delta \to \mathrm{Spaces}$, $\bullet \mapsto X_\bullet$, which satisfies the Segal condition: For each $j$, the map $$ X_j \to ...
0
votes
0answers
165 views

Goodwillie tower of $\Omega^n$?

What are layers of the Goodwillie tower of the functor "n-th iterated loop space" from based spaces to based spaces? I know the answer for n=0.
7
votes
1answer
418 views

Why $( \infty , n)$-categories are useful for?

I know that mathematicians are trying to construct adequate models for $( \infty, n)$-categories. Although, it seems to be an interesting task, I would like to know some explicity examples where this ...