Questions tagged [enriched-category-theory]
The enriched-category-theory tag has no usage guidance.
147
questions
10
votes
1
answer
245
views
Structural properties of $\mathcal{V}$-$\mathsf{Cat}$
In this question $(\mathcal{V}, \otimes, e)$ is a (bi)complete symmetric monoidal category.
We have an adjunction $$\mathscr{l}: \mathsf{Cat} \leftrightarrows \mathcal{V}\text{-}\mathsf{Cat} :(-)_0,$$ ...
- 8,475
11
votes
3
answers
846
views
Relation between Ind-completion and "additive"-ind-completion
Suppose that $\mathcal{C}$ is a skeletally small additive category.
To enlarge $\mathcal{C}$ and produce a bigger category whose "small" objects can be identified with those in $\mathcal{C}$,...
- 425
3
votes
0
answers
109
views
Density with respect to a family of diagrams, versus a class of weights
In Theorem 5.19 of Kelly's Basic Concepts of Enriched Category Theory, it is proven that a fully faithful functor $K \colon \mathcal A \to \mathcal C$ is dense if and only if $\mathcal C$ is the ...
- 9,521
12
votes
1
answer
700
views
Yoneda Lemma for monoidal functors
Let $(\mathcal V,\otimes,I)$ be a closed symmetric monoidal category, and let $\mathcal C$ be a $\mathcal V$-enriched category. The (weak) enriched Yoneda Lemma gives us a nice description of the set $...
- 1,790
6
votes
1
answer
768
views
Examples of (co)ends
I am reading texts about (co)ends, and everywhere I see a lack of examples. I am not an expert in this area, and without examples it is difficult for me to use my intuition to grasp the idea. MacLane ...
- 7,366
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
5
votes
0
answers
213
views
Is there a nice way to define discrete enriched categories?
In the setting of classical category theory, one defines the discrete category associated to a set $X$ as the category $X_\mathsf{disc}$ having the elements of $X$ as its objects and only identities ...
- 11.5k
4
votes
1
answer
747
views
How to compute Homotopy Pullback
What on Earth is a homotopy pullback of
$$A \rightarrow B \leftarrow C \ \ \ \ \ ???$$
Here $A,B,C$ are elements of a category ${\mathcal V}$ enriched in topological spaces (any convenient category ...
- 12.2k
1
vote
0
answers
151
views
Reference request: a class of matrices leading to interesting metric geometry
For $0 \le A \in GL(n,\mathbb{R})$, let $Aw = \Delta(A)$, where $\Delta$ denotes the map taking a matrix to a vector of its diagonal entries and/or forming a diagonal matrix from a vector, according ...
- 15.3k
18
votes
2
answers
914
views
Relationship between enriched, internal, and fibered categories
In this question, let $(\mathcal{V}, \otimes, [-,-], e)$ be a nice enough symmetric monoidal closed bicomplete category.
The usual set-based Category theory has been generalized in many directions, ...
- 8,475
2
votes
1
answer
110
views
"Lie theory" for anchored bundles and reflexive graphs
Perhaps Lie theory is not the correct term, but I'm thinking of the intermediate result in the Lie groupoid to Lie algebroid correspondence. Given a Lie groupoid $G$ over $M$, we may construct the Lie ...
- 1,253
1
vote
0
answers
134
views
Degree shift of multilinear maps
Let $V$ be a graded vector space over $\mathbb{k}$ and $V[1]$ its odd degree shift.
Given $k$, $l\in \mathbb{N}_0$, is there a natural way to define the following map,
$$
\psi: \hom_{\mathbb{k}}(V^{\...
- 456
7
votes
2
answers
438
views
Enrichment as extra structure on a category
We will suppose, for the sake of simplicity, that everything is happening within a fixed 'metacategory' $\textbf{SET}$ of sets and functions. So, from now on, a 'category' just means a category object ...
- 333
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