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Results tagged with infinity-categories
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user 51164
Part of higher category theory that for instance in Algebraic Topology enables us to capture finer homotopic distinctions. As in say Eilenberg-Maclane spaces.
2
votes
What is the functoriality of the $\infty$-categorical slice construction?
Here is something which seems like a plausible description of ??. Let $\mathrm{RFib} \subseteq \mathrm{Ar}(\mathrm{Cat}_{\infty})$ be the full subcategory of the arrow category of $\mathrm{Cat}_{\inft …
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2
votes
Accepted
On equivalences of cartesian fibrations
Yes. Since $X^{\natural} \to S$ and $Y^{\natural} \to S$ are both cartesian fibrations they are fibrant and cofibrant objects in the cartesian model structure over $S$, which is a simplicial model str …
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4
votes
Accepted
Compact objects in the $\infty$-category presented by a simplicial model category
If $X$ is such that $X \times \Delta^n$ is compact for every $n$ then yes. This happens, for example, if the cotensor functor $(-)^{\Delta^n}$ preserves filtered colimits, a condition which is quite c …
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11
votes
Accepted
Homotopy fibers of infinity functors
In general the homotopy pullback of the diagram given by $i:\{y\} \to \mathcal{D}$ and $f:\mathcal{C} \to \mathcal{D}$ is given by first replacing $i$ and $f$ by fibrations between fibrant objects (so …
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11
votes
Accepted
How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?
One way to construct the duality functor ${\cal C} \to {\cal C^{\rm op}}$ is through the notion of a pairing of $\infty$-categories (see HA, Definition 5.2.1.5). In particular, in this case we're talk …
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6
votes
Accepted
Compatibility of Grothendieck construction with pullback
Yes, though it is usually written as the commutativity of unstraightening with pullback (on the $\infty$-categorical level it doesn't matter, since straightening and unstraightening are inverse equiva …
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1
vote
Construction for algebras over little cubes operad
As you point out, it indeed seems that in order to get a well-behaved answer one should work in a suitable homotopical setting, for example, that of $\infty$-categories and $\infty$-operads. Using thi …
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4
votes
Accepted
Homotopy theoretic description of homotopy fixed points (and obstructions) for an action of ...
I'm not sure exactly what kind of answer you're looking for, but I can try to give some context which may make things sound more reasonable. Let us think of of groupoids as $1$-truncated $\infty$-grou …
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5
votes
Gray product on $(\infty,2)$-categories
For question (2), there is actually a left Quillen bifunctor
$$ \times_{\mathrm{gr}}: \mathrm{Set}_\Delta^{\mathrm{sc}} \times \mathrm{Set}_\Delta^{\mathrm{sc}} \to \mathrm{Set}_\Delta^{\mathrm{sc}} $ …
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15
votes
Why Grothendieck's Homotopy Hypothesis is so difficult?
First, let me remark that your question does not seem to be about the homotopy hypothesis, but about rectification. More specifically, the homotopy hypothesis concerns the question of whether (some ve …
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5
votes
Accepted
real and complex vector spaces as topological categories
I think the answer is no. Suppose there exists an enrichment satisfying your requirements, and let $U: Vect_{\mathbb{C}} \to Vect_{\mathbb{R}}$ be the forgetful functor. Let $C \subseteq Map(\mathbb{R …
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