All Questions
Tagged with enriched-category-theory higher-category-theory
25
questions
3
votes
0
answers
58
views
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 $\...
- 351
7
votes
1
answer
232
views
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 ...
- 161
3
votes
1
answer
186
views
Enriched cofibrant replacement in spectrally enriched categories
If $\mathcal{V}$ is a monoidal model category with all objects cofibrant, Theorem 13.5.2. of Categorical Homotopy Theory will guarantee that the functorial cofibrant replacement of a $\mathcal{V}$-...
- 5,511
2
votes
1
answer
408
views
Why do we need enriched model categories?
As far as I understand, model categories mainly provide tools for studying the "homotopy theories" (that is, $\infty$-categories) that are ubiquitous in mathematics. From this point of view, ...
- 6,684
2
votes
1
answer
198
views
Strictification of $\mathcal{V}$-pseudofunctors
Let $\mathcal{B}$ be a bicategory. Section 4.10 of Gordon, Power and Street's paper "Coherence for Tricategories" states that there is a bicategory $\textbf{st}\mathcal{B}$ and a ...
- 23
5
votes
0
answers
92
views
Constructing lax limits from lax limits
Let $K$ be a 2-category. It's well-known that if $K$ has all PIE limits, then $K$ also has all lax limits. But I don't know a general "limit-decomposition" result which works "...
- 62.6k
4
votes
1
answer
271
views
Explicit description of a pullback of $(2,1)$-categories
In the 1-category of 2-categories, with objects being categories enriched over Cat, and morphisms being 2-functors, is there an explicit way to describe a pullback of two functors $G:E\to D$ and $F:C\...
- 85
3
votes
1
answer
199
views
Morphisms of $\infty$-groupoids
As far as I understand, there are several ways of defining $\infty$-categories. One of them is to think of $\infty$-cateogries as $top$-enriched categories. Hence we can think of $\infty$-groupoids as ...
- 4,286
10
votes
1
answer
421
views
Theory of weak enrichment in higher categories
Has there been work towards a general theory of weak enrichment in higher categories? To be more pointed, has there been any work towards trying to make sense of statements such as
There is a (weak) $...
- 607
8
votes
2
answers
443
views
Classification of absolute 2-limits?
Let $\mathcal V$ be a good enriching category. Recall that an enriched limit weight $\phi: D \to \mathcal V$ is called absolute if $\phi$-weighted limits are preserved by any $\mathcal V$-enriched ...
- 62.6k
9
votes
1
answer
356
views
Are (complete) 2-Segal spaces the same as Span-enriched infinity categories?
The question is basically in the title. More generally, I would like to know if this, or any reasonable variant of it, is true. Or perhaps, to understand better the gap between 2-Segal spaces and Span-...
- 2,280
3
votes
0
answers
73
views
On cofibrations of simplicially enriched categories
Let $\mathbb{C}$ be an strict 2-category and denote by $C$ is underlying 1-category viewed as as a 2-category only having identity 2-cells.
We have a canonical inclusion functor ,
$$i: C \...
- 457
18
votes
3
answers
1k
views
A multicategory is a ... with one object?
We all know that
A monoidal category is a bicategory with one object.
How do we fill in the blank in the following sentence?
A multicategory is a ... with one object.
The answer is fairly ...
- 907
9
votes
0
answers
522
views
In what context can enriched category theory be done?
There are many possible situations one can do enriched category theory. See https://ncatlab.org/nlab/show/category+of+V-enriched+categories#possible_contexts for a list.
My question is what ...
- 753
11
votes
1
answer
494
views
Can an enriched functor be expressed as a colimit of representable functors?
Suppose that $\mathcal C$ is an ordinary category and $F:\mathcal C^{op}\longrightarrow Set$ a functor. Then, we can form the category $\mathcal C/F$ as follows : each object is a morphism of functors ...
- 111