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Results tagged with higher-category-theory
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user 11546
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
25
votes
2
answers
787
views
Explicit left adjoint to forgetful functor from Cartesian to symmetric monoidal categories
There is a forgetful functor from the category of (small) Cartesian monoidal categories (a symmetric monoidal category in which the tensor product is given by the categorical product) to the category …
14
votes
2
answers
960
views
Are n-truncated quasicategories a model for n-categories?
In Higher Topos Theory, Lurie gives a description of $n$-categories (section 2.3.4) in terms of quasicategories. In other words, he gives a definition (2.3.4.1) of an $\infty$-categorical notion of $n …
11
votes
1
answer
833
views
What does the homotopy coherent nerve do to spaces of enriched functors?
Suppose I have two fibrant simplicially enriched categories $C$ and $D$. Then I can consider the collection of simplicial functors $sFun(C,D)$ which, if I recall correctly, has the structure of a simp …
10
votes
2
answers
541
views
Simplicial nerve functor commutes with opposites
There are two "opposite" functors:
$$ op_\Delta\colon sSet\to sSet$$
and
$$op_s\colon sCat\to sCat.$$
The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite …
10
votes
0
answers
329
views
When is Fun(X,C) comonadic over C with respect to the colimit functor?
Because I'm primarily interested in this question from the point of view of $\infty$-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular …
9
votes
1
answer
779
views
Lurie's Endomorphism Space vs. Endomorphisms
In Jacob Lurie's book Higher Algebra, for an object $M$ of a monoidal $\infty$-category $\mathcal{C}$, he constructs a category $\mathcal{C}[M]$ which can be thought of as "maps in $\mathcal{C}$ of th …
9
votes
2
answers
391
views
Monoidal structures on modules over derived coalgebras
Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can f …
9
votes
1
answer
733
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 …
8
votes
3
answers
461
views
Cofiber of the inclusion of an $E_0$-algebra $M$ into the free $E_k$-algebra generated by it
Let $\mathcal{C}$ be the $E_k$-monoidal $\infty$-category of left modules over a fixed connective $E_{k+1}$-ring spectrum $A$. Suppose that $M$ is an object of $\mathcal{C}$ which is an $E_0$-algebra, …
8
votes
1
answer
696
views
Interaction of Grothendieck Construction with Coherent Nerve
There are a number of Grothendieck constructions: one for discrete categories, one for enriched categories (see Tamaki's paper here) and one for quasicategories (see the Unstraightening and Straighten …
7
votes
0
answers
173
views
Simplicial right Kan extensions and Cartesian transformations
I will write the concrete question first, in case the answer is clear independently of the context:
Question: Given an $\infty$-topos $\mathfrak{X}$ and a diagram $F\colon\Delta^1\times\Delta_+^{op}\ …
5
votes
How does homotopy theory simplify topology but allow for complexity in higher category theory?
I think one way to look at this is to say that, from a certain point of view, topological spaces are far more complicated than categories and homotopy types are somewhere in the middle. The only wrink …
4
votes
Accepted
Interaction of Grothendieck Construction with Coherent Nerve
Answering this question took me some time. First of all, Liang Ze Wong and I had to write down a version of the enriched Grothendieck construction that worked for these purposes, as Tamaki's construct …
4
votes
1
answer
948
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 Hi …
4
votes
1
answer
152
views
Are monomorphisms in an $\infty$-topos preserved by $0$-truncation?
Let $\mathfrak{X}$ be an $\infty$-topos and let $f\colon X\to Y$ be a morphism of $\mathfrak{X}$. We say that $f$ is a monomorphism if it is $(-1)$-truncated which means that for every $Z\in\mathfrak{ …