Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
1,350
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How to prove the classical definition of addition map for additive ordinary categories is commutative and associative up to coherent homotpy?
Could somebody please help me with this?
We know if $C$ is an ordinary additive category, then the addition of maps $f,g:x \rightarrow y$ coincide with the composition
$x \xrightarrow{\bigtriangleup} ...
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Tamari lattice and bicategory coherence
Reference links:
Tamari lattice (Wikipedia): https://en.wikipedia.org/wiki/Tamari_lattice
Associahedra: https://en.wikipedia.org/wiki/Tamari_lattice#/media/File:Tamari_lattice.svg
The Tamari lattice ...
- 4,145
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2-cells in the double category of 2-functors
Mike Shulman has in the answer to my previous question argued that for 2-categories $C$ and $K$ there is a double category whose objects are 2-functors between them and morphisms are lax and colax ...
- 933
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$E_k$-operads and actions on objects inside $k$-tuply monoidal $n$-category
I understood more or less the claim that $k$-tuply monoidal $n$-categories can be seen as $n$-categories equipped with an action of the $E_k$-operad.
For $k=2$, we have a homotopy equivalence $E_2(r) \...
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Generalized $\infty$-operads are an analog of ??? in $\infty$-category theory
In Section 2.3.2 of Higher Algebra, Lurie introduces the notion of generalized $\infty$-operads. This is a functor $p:\mathcal{O}^\otimes \to \mathcal{F}\mathrm{in}_\ast$ of $\infty$-categories, where ...
- 2,154
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Equivalence of two definitions of relative limits
This is a question on seemingly equivalent definitions of relative limits, formulated in the language of quasi-categories. I will use notations from Higher Topos Theory.
Let $p:\mathcal{C}\to\mathcal{...
- 2,154
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Is the adjunction between spaces and chain complexes monadic?
Consider the adjunction of $\infty$-categories $\mathbb{Z}[-]: \mathcal{Spaces} \rightleftarrows \text{Ch}_{\ge 0}(\mathbb{Z}): |-|$ where the left adjoint takes a space to its singular chain complex ...
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Reference for the tricategory of elements associated to a trifunctor
The theory of bicategorical fibrations has been relatively well studied, e.g. by Baković and by Buckley. In particular, given a trifunctor $F : \mathcal K \to \mathbf{Bicat}$ from a bicategory $\...
- 9,521
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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{...
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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 ...
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Universal property of category of categories
As discussed here, Using the universal property of spaces, the $(\infty,1)$-category of spaces has a universal property: it is the free $\infty$-categorical cocompletion of the terminal category $*$. ...
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The analogy between dualizable categories and compact Hausdorff spaces
Efimov has in his recent preprint K-theory and localizing invariants of large categories, Appendix F, a long table of analogies between the categories $\text{Cat}^\text{dual}_\text{st}$ and $\text{...
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The infinity category of dg-categories is bicomplete
We can define the $\infty$-category of dg-categories $dgCat_\infty$ as the definition of the $\infty$-category of $\infty$-categories which given gy the section.3 of J.Lurie "Higher Topos Theory&...
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Notion of $\kappa$-sifted categories?
Let $\kappa$ be a regular cardinal. It seems reasonable to introduce the following definition:
Definition. A simplicial set $K$ is $\kappa$-sifted if, for every set $E$ with $\lvert E\rvert<\kappa$...
- 2,446
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Double category of monads and pseudo monad-morphisms
We can construct bicategories of monads in a bicategory $B$, $Mnd_l(B)$/$Mnd_c(B)$ with lax and colax monad-morphisms respectively.
I am failing to find a good notion of pseudo monad-morphisms.
Is ...
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