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      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 \...
      16
      votes
      2answers
      892 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 ...
      8
      votes
      0answers
      244 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 ...
      7
      votes
      1answer
      180 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 ...
      10
      votes
      1answer
      229 views

      The category theory of Span-enriched categories / 2-Segal spaces

      The category $\mathsf{Span}$ of spans of sets is symmetric monoidal closed under $\times$ (the cartesian product from $\mathsf{Set}$, which is not the categorical product in $\mathsf{Span}$), ...
      2
      votes
      0answers
      86 views

      Classification of unitary pointed monoidal category

      I wonder if the following classification results are true (and are there any references): Unitary pointed monoidal categories (the fusion rule of the objects is given by a finite group $G$) are ...
      2
      votes
      1answer
      331 views

      Colimits in the category of simplicial categories

      A simplicial category is a category enriched over the monoidal category of simplicial sets (morphism sets are now simplicial sets), and the collection of all such categories forms a category itself (...
      1
      vote
      0answers
      100 views

      Posets as (0,1)-categories

      I am reading on the nLab that a poset can be seen as a (0,1)-category. I was assuming all along that an ($n$,$r$)-category were a category where all morphisms of order larger than $n$ are trivial. ...
      14
      votes
      1answer
      380 views

      Why is every object cofibrant in an excellent model category?

      In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent ...
      21
      votes
      2answers
      1k views

      How to stop worrying about enriched categories?

      Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they ...
      4
      votes
      1answer
      654 views

      Properties of loop space functor from homotopy types to group objects in homotopy types

      I am trying to understand some properties of categories enriched in homotopy types, and the following question has become important: When we take the loop-space of a (connected) homotopy type, we get ...
      1
      vote
      2answers
      191 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
      3
      votes
      1answer
      308 views

      About a Double-pseudo-category generalization of the module bicategory construction

      To a category with finite limits $\mathscr{C}$, it is associated the bicategory of its spans $Span(\mathscr{C})$. Furthermore the bicategory of (bi)modules (and monoids) on $Span(\mathscr{C})$ is ...
      12
      votes
      2answers
      620 views

      What are the higher morphisms between enriched higher categories?

      This question is about $n$-categories, or perhaps $(\infty,n)$-categories, or ... My guess is that the answer will not depend sensitively on the model of higher categories, so rather than have me ...

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