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Results tagged with infinity-categories
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user 76636
Part of higher category theory that for instance in Algebraic Topology enables us to capture finer homotopic distinctions. As in say Eilenberg-Maclane spaces.
3
votes
Accepted
Algebras over the trivial $\infty$-operad
One can avoid relative Kan extensions, but since this calculation hinges on the fact that $\mathcal{C}^\otimes$ is right Kan extended from $\{\langle 1 \rangle\} \to \mathrm{Triv}^\otimes$, I suspect …
- 1,337
8
votes
Accepted
Functoriality of infinite suspension spectrum functor on infinity groupoids!
I'm not sure if this counts as a direct way, but a more practical way to show functoriality at the level of $\infty$-categories is to not construct $\Sigma^\infty$ as a functor directly, but instead s …
- 1,337
5
votes
Accepted
How to prove $\text{Map}_C(X,Y)$ is a grouplike commutative monoid object of the $\infty$-ca...
A reference is Corollary II.17 of the lecture notes on algebraic and hermitian K theory by Fabian Hebestreit (typeset copy by Ferdinand Wagner available here). The argument is relatively short, so I h …
- 1,337
2
votes
Accepted
Connectedness of truncated version of cosimplicial indexing category
I don't believe the first claim is true, but I can give a somewhat formal argument for the connectivity of $\Delta_{\le n} \times_{\Delta} \Delta_{/m}$.
Let us write $u$ for the inclusion $\Delta_{\le …
- 1,337
4
votes
Accepted
Limits of infinity categories and mapping spaces
Yes. To see this, let us make the preliminary observation that it suffices to prove that this holds for products and pullbacks since we can decompose a general limit into these two special cases.
Let …
- 1,337
2
votes
Accepted
Cofinal maps from posets (HTT, 4.2.3.16)
[Rephrasing my comment as an answer]
While I cannot speak for the actual proof, two points are worth noting. First, a similar construction appears in Kerodon (https://kerodon.net/tag/02QA) but there ( …
- 1,337
3
votes
Monomorphisms of diagrams in an $\infty$-category
(Assuming $\mathcal{C}$ is finitely complete for convenience)
A partial answer at least:
For the first, $\eta$ is an monomorphism iff its diagonal map $\delta : f \to f \times_g f$ is an isomorphism. …
- 1,337