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Results tagged with infinity-categories
Search options questions only
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user 2362
Part of higher category theory that for instance in Algebraic Topology enables us to capture finer homotopic distinctions. As in say Eilenberg-Maclane spaces.
7
votes
0
answers
155
views
Is strictness decidable?
Let $\mathcal C$ be an $\infty$-category. We can ask:
Q: Is $\mathcal C$ a 1-category?
That is, are the hom-spaces of $\mathcal C$ essentially discrete?
Roughly, my question is:
Proto-Question: Is Q …
5
votes
1
answer
204
views
Are $\infty$-categories functorially colimits of their simplices?
Let $\mathcal C$ be an $\infty$-category. If $C$ is a quasicategory modeling $\mathcal C$, then we have a coend decomposition
$$\mathcal C = \int^{[n] \in \Delta} \Delta[n] \times C_n.$$
This allows u …
10
votes
2
answers
377
views
When is the homotopy category of an accessible $\infty$-category accessible?
Let $\mathcal C$ be an accessible $\infty$-category, and let $ho(\mathcal C)$ be its homotopy category. I can think of two "trivial" reasons for $ho(\mathcal C)$ to be accessible:
$ho(\mathcal C) = …
5
votes
1
answer
262
views
What is the correct statement of Theorem 4.2 in Street's "Parity Complexes"?
Ross Street's 1991 paper Parity Complexes (apologies; I don't know how to find DOI links for Cahiers papers) develops some very useful tools for working with free strict $\omega$-categories. There is …
7
votes
2
answers
424
views
Which free strict $\omega$-categories are also free as weak $(\infty,\infty)$-categories?
There are a number of formalisms available for presenting free strict $\omega$-categories -- Street's parity complexes, Steiner's directed complexes, computads, polygraphs,... Typically one has a cert …
11
votes
1
answer
445
views
Proof of existence of Joyal model structure via Cisinski theory?
I'm looking for a proof of the existence of the Joyal model structure -- with its usual description -- which uses Cisinski theory directly. The closest thing I know of is Theorem 5.26 of Ara's Higher …
9
votes
3
answers
873
views
Decomposing a (co)limit by decomposing the indexing diagram
Let $D: I \to \mathcal C$ be a diagram, and suppose we have a colimit decomposition $I = \varinjlim_{j \in J} I_j$ in $Cat$. Then under certain conditions, we can decompose the colimit of $D$ as $\var …
6
votes
1
answer
211
views
If $\mathcal C$ is a $\kappa$-accessible $\infty$-category, then is $Mor \mathcal C$ $\kappa...
If $\mathcal C$ is a $\kappa$-accessible 1-category, then the category of morphisms $Mor \mathcal C$ is a $\kappa$-accessible 1-category, with the $\kappa$-presentable objects being those morphisms wh …
4
votes
0
answers
163
views
Interlocking (weak) factorization systems
I'm interested in instances of the following data:
$C$ is a (possibly higher) category;
$(L,M)$ is a weak factorization system (wfs) on $C$;
$(M,R)$ is a unique factorization system (fs) on $C$.
…
7
votes
1
answer
319
views
Is there a model-independent characterization of the gaunt strict $n$-categories amongst the...
Recall that a strict $n$-category $C$ is called gaunt if every $k$-morphism in $C$ with a weak inverse is an identity, for all $k$; let $Gaunt_n$ denote the strict 1-category of gaunt $n$-categories. …
5
votes
0
answers
260
views
Surprising examples of functors which preserve cofiltered limits but not all limits?
Question: What are some "surprising" examples of functors (resp. $\infty$-functors) $F$ which preserve cofiltered limits?
I'm not quite sure what "surprising" means, but I think that
"Surprising" sho …
6
votes
2
answers
403
views
Can conservativity depend on the universe?
Question 1: Let $F: C \to D$ be a conservative, $\kappa$-cocontinuous functor between small, $\kappa$-cocomplete categories. Is the induced functor $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ als …
6
votes
1
answer
186
views
Can a locally presentable category have a proper class of accessible localizations?
Question: What is an example of a locally presentable category $\mathcal C$ such that there exists a proper class of accessible localizations $(\mathcal C \to \mathcal D_i)_{i < ORD}$?
In other words …
13
votes
0
answers
243
views
Categorification of "Every domain embeds into a field"?
In the category of commutative rings, every domain embeds into a field. Is this true in the category of presentably symmetric monoidal stable $\infty$-categories? Here's what I mean by that.
Let $Pr …
6
votes
1
answer
216
views
Is the inclusion of its 2-skeleton into the walking idempotent homotopy cofinal?
Let $Idem = Idem^{(\infty)}$ be the walking idempotent [1], and let $Idem^{(n)}$ be its n-skeleton. Note that $Idem$ has one nondegenerate simplex in each dimension. Let $\iota_n^m: Idem^{(n)} \to Ide …