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Results tagged with infinity-categories
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user 156537
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
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Counterexamples concerning $\infty$-topoi with infinite homotopy dimension
In "Higher Topos Theory", Lurie introduces three different notions of dimension for an $\infty$-topos $\mathcal{X}$, namely:
Homotopy dimension (henceforth h.dim.), which is $\leq n$ if $n$-connectiv …
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1
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Accepted
Counterexamples concerning $\infty$-topoi with infinite homotopy dimension
To close this thread off, I will try to expand Lurie's helpful comment into an answer:
First of all concerning examples of $\infty$-topoi that are locally, but not globally, of finite homotopical dime …
9
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1
answer
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Coherent objects in a hypercomplete $\infty$-topos
In Lurie's "Spectral Algebraic Geometry", Proposition A.6.6.1 (2) shows that for $\mathcal{X}$ an $\infty$-topos that is both locally coherent and hypercomplete, the full subcategory $\mathcal{X}^{coh …
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2
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1
answer
172
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(Local) Homotopy dimension of $\infty$-topoi on paracompact spaces
I have a question concerning the proof of Corollary 7.3.6.5 in Luries "Higher Topos Theory" (the same issue also occurs in the proof of 7.3.6.10, but it is clearer here). Given is a continuous map $p: …
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4
votes
1
answer
219
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Gluing a manifold along its boundary, via chain complexes
Given closed oriented $n$-manifolds $M, M', M''$ and bordisms $W, W'$ with $\partial W = M \sqcup - M'$ and $\partial W' = M' \sqcup - M''$, we can collar-glue them to obtain a bordism from $M$ to $M' …
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Gluing a manifold along its boundary, via chain complexes
I might have a idea how to prove my claim, that also generalizes to any stable $\infty$-category with duality functor:
Let $C$ be a chain complex, remember that there are natural diagonal and codiagon …
- 898
1
vote
1
answer
210
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Proof in Higher Algebra that $\mathcal{C}at(\mathcal{K})$ is presentable
In Higher Algebra Lemma 4.8.4.2, Lurie shows that for $\mathcal{K}$ a small set of simplicial sets, the $\infty$-category $\mathcal{C}at(\mathcal{K})$ of small $\infty$-categories with $K$-shaped coli …
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Accepted
Proof in Higher Algebra that $\mathcal{C}at(\mathcal{K})$ is presentable
Note that by HTT 5.4.1.2 since $\tau$ is an uncountable regular cardinal, an $\infty$-category is $\tau$-compact iff it is $\tau$-small. Our first step is to show that the inclusion $\mathcal{C}at(\ma …
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