All Questions
Tagged with enriched-category-theory 2-categories
11
questions
5
votes
0
answers
75
views
Free cocompletion of a 2-category under pseudo colimits, lax colimits, and colax colimits
Let $\mathscr K$ be a small 2-category. It follows from $\mathrm{Cat}$-enriched category theory that the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors is given by the 2-...
- 9,521
7
votes
1
answer
144
views
Preservation of lax limits in categories of functors and lax natural transformations
Let $f:\mathbb{C} \to \mathbb{D}$ be a functor of 2-categories and let $\operatorname{Fun}(\mathbb{C},\operatorname{Cat})^{\operatorname{lax}}$ denote the 2-category of functors and lax natural ...
- 295
3
votes
0
answers
75
views
Conditions for natural transformations of weights to induce adjunctions of weighted limits
Suppose we have:
-) A $2$-category $\mathsf{J}$
-) A natural transformation of $\alpha : M \Longrightarrow W : \mathsf{J} \longrightarrow \mathsf{Cat}$
-) A functor $X:\mathsf{J} \longrightarrow \...
- 151
3
votes
1
answer
54
views
Reference request for facts about bi(co)descent objects
I know the following facts are true, but I struggle to find adequate references for them:
Let $T$ be a pseudo-monad on a bicategory $\mathcal{C}$, and let $A$, $B$ be pseudo-algebras for $T$. Then, ...
- 1,001
1
vote
0
answers
130
views
Universal property of the V-Mat construction
Internal categories and enriched categories can both be realised as monads in certain bicategories. If $\mathcal E$ is a category with pullbacks, then a monad in $\mathbf{Span}(\mathcal E)$ is a ...
- 9,521
4
votes
1
answer
356
views
Weak enrichment and bicategories
I'm trying to find examples where the following perspective on bicategories is developed.
We can define a 2-category as being enriched in Cat, where Cat is treated as a monoidal category using the ...
- 1,253
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
7
votes
1
answer
354
views
Are sifted (2,1)-colimits of fully faithful functors again fully faithful? (And a de-categorified variant)
1) Suppose that I have a sifted diagram of categories $\mathcal{C}_i$, another of the same shape $\mathcal{D}_i$, and that I have a system $F_i:\mathcal{C}_i\to\mathcal{D}_i$ commuting with the ...
- 6,071
9
votes
0
answers
320
views
To what kind of generalized Lawvere theory does the "free cartesian closed category" 2-monad on $\mbox{Cat}_g$ correspond?
Thinking of Cat as a mere 1-category, there is a 1-monad $\Lambda$ for the free cartesian closed category on a category. To every category X it assigns the category $\Lambda(X)$ whose objects are ...
- 1,532
2
votes
1
answer
471
views
Comma category as weighted limit
Let $F : C \to E$ and $G : D \to E$ be functors. Consider the comma category $(F \downarrow G)$ with its projections $\pi_1 : (F \downarrow G) \to C$ and $\pi_2 : (F \downarrow G) \to D$. Using the ...
- 476
1
vote
2
answers
216
views
Completion under weighted limits/colimits
Is there any further reference besides "Basic Concepts of Enriched Categories" (Kelly) for completion under T-(weighted) limits/colimits?
(in which T is a set of weights)
Thank you in advance
- 875