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2
votes
1answer
117 views

Kan extensions in the $2$-category of monoidal categories

Kan extensions make sense in any $2$-category. But so far I have only really seen them in the case of the $2$-category of categories, functors, natural transformations and the $2$-category of ...
0
votes
1answer
384 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 ...
5
votes
0answers
115 views

Is there a classification of 2d extended TQFTs with defects?

Chris Schommer-Pries has classified 2d extended TQFTs (topological quantum field theories) in his PhD thesis. The result is a (not necessarily abelian) separable symmetric Frobenius algebra (possibly ...
2
votes
2answers
310 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 ...
45
votes
7answers
14k views

Is Mac Lane still the best place to learn category theory?

For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ... Is Mac Lane still ...
2
votes
1answer
107 views

Are lax functor categories into a cartesian closed 2-category cartesian closed?

Suppose that $C$ is a complete closed monoidal category and $I$ is any small category. Then the functor category $Fun(I,C)$ is again a closed monoidal category with the pointwise tensor product $F ...
8
votes
1answer
298 views

Separation condition for higher Deligne-Mumford stacks

Let $X$ be a stack of $n$-groupoids on the site of affine schemes over a fixed base, with the etale topology. If $n=1$ then for $X$ to be Deligne-Mumford, aside from having an etale atlas from an ...
0
votes
0answers
45 views

About cartesian closure of lax.functors categories

Let $\mathscr{A}$ a category and $F, G, H: \mathscr{A}^{op}\to CAT$ lax.functors. I wish find a possible "natural correspondence" between categories: $[F\times G, H]_O \leftrightarrow [F, H^G]_O$ ...
5
votes
1answer
171 views

Difference between coherent nerve of simplical model category and simplicial category

Suppose I have a simplicial model category $M$. Then I can take the homotopy coherent nerve of $M$ to obtain a quasicategory. This, however, only depends on the fact that $M$ is a category enriched in ...
10
votes
1answer
460 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 ...
2
votes
3answers
323 views

Integral transform on noncommutative spaces

In their paper "Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry" the authors show that for perfect stacks $X$ and $Y$ over $k$, and their $k$-linear $\infty$-categories of ...
3
votes
1answer
203 views

What is the applications of the dg-enhancements of derived categories of sheaves

Let $X$ be a scheme and let $D^b_{\text{coh}}(X)$ be the derived category of complexes of sheaves with bounded, coherent cohomologies. We know that the category $D^b_{\text{coh}}(X)$ has some ...
5
votes
0answers
167 views

Is there a higher, “orientalish” version of geometric realisation?

Geometric realisation of simplicial sets can be roughly thought of like this: In some category $\mathcal{C}$, we choose an object for every abstract $n$-simplex. In topological spaces, we would ...
5
votes
1answer
113 views

What homomorphisms $G \to BrPic(\mathcal{C})$ correspond to group-theoretical $G$-extensions of $\mathcal{C}$?

For a fusion category $\mathcal{C}$ the Brauer-Picard group $\text{BrPic}(\mathcal{C})$ is the group of all invertible $\mathcal{C}$-bimodule categories under multplication $\boxtimes_\mathcal{C}$. ...
7
votes
1answer
521 views

What are $( \infty , n)$-categories 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 ...
2
votes
0answers
94 views

References for the bicategory of ring-bimodule pairs

One of the standard examples of a bicategory is the bicategory of rings (with bimodules as 1-morphisms), which is sometimes denoted $\operatorname{Bim}$ and in other sources $\operatorname{Ring}$ (or ...
17
votes
6answers
2k views

Simplicial homotopy book suggestion for HTT computations

I'm struggling through Lurie's Higher Topos Theory, since it appears that someone reading through the book is expected to be somewhat comfortable with simplicial homotopy theory. The main trouble ...
3
votes
1answer
487 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
186 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
666 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
520 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
1answer
167 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 ...
34
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
129 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 ...
-1
votes
1answer
325 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
355 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
164 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
465 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
80 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 ...
7
votes
2answers
338 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
223 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 ...
7
votes
1answer
360 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
704 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
706 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
349 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
452 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
824 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
213 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
484 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
214 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
300 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
476 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
345 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
159 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
310 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
122 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
227 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
253 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
253 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
98 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 $s \in S$ ...