Questions tagged [higher-category-theory]
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
1,339
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Localizations that are endofunctors
Usually when $F: C \rightarrow D$ is a localization functor, the categories $C$ and $D$ are not equivalent. My question is when is it possible for $C, D$ to be abstractly equivalent but $F$ is not an ...
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Self-enrichment for a closed monoidal bicategory
First, there are two possible generalization of the notion of closed category, vertical and horizontal.
I'm interested in the vertical one, something saying, I guess, that a monoidal bicategory $\...
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Soft question: Deep learning and higher categories
Recently, I have stumbled upon certain articles and lecture videos that use category theory to explain certain aspects of machine learning or deep learning (e.g. Cats for AI and the paper An enriched ...
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Why do we say IndCoh(X) is analogous to the set of distributions on X?
$\DeclareMathOperator\IndCoh{IndCoh}\DeclareMathOperator\QCoh{QCoh}$I've seen it written (for example, in Gaitsgory–Rozenblyum) that for a scheme $X$, the category $\IndCoh(X)$ is to be thought of as ...
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Completeness of comma $\infty$-categories
Let $\mathsf{A},\mathsf{B},$ and $\mathsf{C}$ be (ordinary) categories and $F : \mathsf{A}\to\mathsf{C}$ and $G : \mathsf{B}\to\mathsf{C}$ be functors such that
$\mathsf{A}$ and $\mathsf{B}$ are ...
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Does the Gray tensor product exhibit Gray as a monoidal Gray-category?
Crans constructs a lax Gray tensor product for strict omega-categories that defines a monoidal structure on the category of strict omega-categories $\omega Cat$. We write Gray for this monoidal ...
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Can you deduce the correspondence between 2D oriented TQFTs and commutative Frobenius algebras from the (framed) Cobordism Hypothesis?
Background
I am currently writing an MSc dissertation on TQFTs (and Khovanov homology, but that is unrelated to this question).
After having read most of Kock's book on the equivalence between 2D ...
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Piecewise construction of a functor from an $(\infty,1)$-category with an orthogonal factorization system
For the simpler case of 1-categories, consider a 1-category $C$ and an orthogonal factorization system $(L,R)$ on $C$. Let $C_L$ and $C_R$ denote the wide subcategories of $C$ corresponding to the ...
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Finitely presentable objects in the categories of algebras of $\infty$-algebraic theories
By default, all terms are understood in the infinity sense (“category” means “$(\infty, 1)$-category”, etc.).
An object $A$ in a category is said to be finitely presentable (or compact) if the functor ...
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How to prove a 1-localization of a 1-category is already an $(\infty,1)$-localization?
I don't even know this fits in here or in Mathematics Stack Exchange, but let me ask. I'm new to simplicial stuff, so a good reference would be quite helpful.
Let's say $C$ is a certain category, and ...
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Colimits from van Kampen cocones
Let $\mathcal{C}$ be a category with pullbacks, $\mathcal{J}$ a small category, $F : \mathcal{J} \to \mathcal{C}$ a diagram and $\kappa : F \Rightarrow \Delta X$ a cocone in $\mathcal{C}$. Let $\...
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Does the 2-category of double categories and vertical transformations have flexible limits?
Consider the 2-category of pseudo-double categories (with the weak composition in the horizontal direction and the strict composition in the vertical direction), strong double functors, and vertical ...
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Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical colimit?
For ordinary category theory, we have the following fact.
A weighted colimit of a functor can always be equivalently expressed as a colimit of a different functor.
Specifically, the weighted colimit ...
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Natural ways to make a functor adjoint
Let $F: C \to D$ be a functor between two categories without a right adjoint. What are some natural ways to create a right adjoint for $F$?
Of course, this does not make sense on the nose. One needs ...
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Tangent $(\infty,1)$ topos
I am trying to understand the tangent $(\infty,1)$ category. It is the fiberwise stabilization of the codomain fibration (which is a functor from the arrow category to the category).
But, intuitively ...
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