# Questions tagged [enriched-category-theory]

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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### “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 ...
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### Why is the category of all small $\mathbf{S}$-enriched categories locally presentable?

In Lurie's Higher Topos Theory Proposition A.3.2.4, the author used Proposition A.2.6.15 to prove that for any combinatorial monoidal model category $\mathbf{S}$ with all objects cofibrant and weak ...
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### Generating an enriched multicategory

Let $C$ be an $(M,\otimes,1)$-enriched category. I am looking for a reference for a notion of “generating the morphisms of $C$” (for ordinary categories, but also for multicategories, see below). My ...
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### Isomorphisms in enriched categories

Let $(M,\otimes,1)$ be closed monoidal category and $C$ an $M$-enriched category. Assume we have $C$-objects $X$ and $X'$ and a morphism $f:1\to C(X,X')$ in $M$. We call $f$ an isomorphism if there is ...
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### 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 ...
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### Monoidal functors $\mathcal C \to [\mathcal D,\mathcal V]$ are monoidal functors $\mathcal C \otimes \mathcal D \to \mathcal V$?

It is well known (e.g., Reference for "lax monoidal functors" = "monoids under Day convolution" ) that if $\mathcal C$ is a monoidal $\mathcal V$-enriched category, then a monoid ...