# Tagged Questions

Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

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### Are there any detailed references for the enriched yoneda lemma?

I am just starting out learning enriched category theory, and I am looking for a reference proof of the Yoneda lemma for categories enriched in a monoidal category. Thank you for your help.
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### What is the “quaternionic” super Brauer group?

In addition to the two reasonably well-known categories $\mathrm{SuperVect}_{\mathbb R}$ and $\mathrm{SuperVect}_{\mathbb C}$ of real and complex super vector spaces, each of which is monoidally ...
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### Does every bicategory have a “delaxing object”?

If I'm not mistaken, there is a bicategory $\mathsf{Monad}$ given as follows: Start with the associative operad. Deloop it to obtain a multicategory. Adjoin objects and morphisms as necessary to ...
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### Directed homotopy in the Cayley graph of a monoid

There is a the notion of the Cayley graph $C(G)$ of a group $G$ (which depends on a given presentation $G \cong \mathcal F(S) / \sigma$ where $\mathcal F$ is the free group functor and $\sigma$ some ...
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### reference request: category of crystals on a scheme is locally noetherian

I'm looking for a reference to the fact that the category of crystals on any scheme is locally noetherian. This is stated in Gaitsgory's paper "Crystals and D-modules" but he doesn't provide a ...
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### What is an Elementary “Homotopy, Model” Topos?

Context: Def (Rezk): A (Grothendieck) homotopy topos is a homotopy left exact Bousfield localization of the model category of simplicial presheaves sPsh(C) on a small simplicial category C. Thm (...
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### Test categories applied to Dold-Kan correspondence?

Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to ...
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### Is the modularisation of a unitary fusion category always unitary?

Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières ...
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### Reference request: Morita bicategory

I have two closely related questions: Who first recognized that Morita equivalence was equivalence in the bicategory of Rings, Bimodules, and Intertwiners? I've heard this bicategory called the "...
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### What kinds of limits does localization of commutative rings reflect?

Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of ...
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### Category enriched over a monoidal 2-category

Consider a monoidal 2-category (or bicategory) B. For example, B could by the 2-category (finite sets, finite correspondences, isomorphisms of correspondences) with monoidal structure given by ...
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### “Why” are monadicity and descent related?

This question is probably too vague for experts, but I really don't know how to avoid it. I've read in several places that under mild conditions, a morphism is an effective descent morphism iff the ...
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### Topology of categories, very basic facts surrounding Quillen's Higher Algebraic K-Theory I

In his paper Higher Algebraic K-Theory I (see [here][1]), Quillen introduces a topological space $BC$, called the classifying space of $C$, and tries to relate its topology to the categorical ...
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### Faithful exact functors to tensor categories

Let $P$ be a "nice" $k$-linear abelian tensor category (e.g. A tannakian category or a fusion category over a field $k$) and $F: M\to P$ an additive $k$-linear exact and faithful functor. I want to ...
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### Many-sorted nominal sets as sheaves

The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on $\mathbb{I}^\mathsf{op}$ under the atomic topology, where $\mathbb{I}$ is the category of finite ...
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### Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an ...
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### Defining Inertia Stack

Let $X$ be a topos and $F: \zeta \rightarrow X$ a stack on $X$. Now in the paper http://arxiv.org/pdf/math/0411337v2.pdf ( Definitions 2.1.1.5 and 2.1.1.1) of Lieblich, he describes the inertia stack ...
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Fix a presheaf topos $\mathcal{E}\triangleq \mathbf{Set}^{\mathcal{C}^\mathsf{op}}$. Then, for any object/presheaf $X:\mathcal{E}$, I can define the presheaf of relations on $X$ as $\wp(X)\triangleq\... 2answers 307 views ### In a fibration, where does the generic object live? In Bart Jacobs. Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, 1999. isbn: 9780444508539, the author writes, p. 326: Sometimes, for ... 0answers 188 views ### Operadic Lift of Lurie's Relative Tensor Product In Section 4.4 of his book Higher Algebra, Lurie introduces, for a monoid object$A$of a monoidal quasicategory$C$, and right and left$A$-modules$M,N$, the relative tensor product$M\otimes_AN$. ... 0answers 98 views ### Finitely presented algebras with isomorphic semilattices of congruences Let$\mathbb{T}$be a finitary algebraic theory. For each$\mathbb{T}$-algebra$A$, let$Q (A)$be the join semilattice of finitely generated congruences on$A$. There is an evident pushforward ... 1answer 178 views ### Exterior derivative as only (up to multiple) natural operator$\Lambda ^kT^\ast \rightsquigarrow \Lambda ^{k+1}T^\ast$In Kolar, Michor, & Slovak's book Natural Operations in Differential Geometry, it is proved the exterior derivative is universal in the following sense. Proposition 25.4. For$k>0$all natural ... 2answers 335 views ### Natural operators in differential geometry - why are they natural? I'm reading bits and pieces of Kolar, Michor, & Slovak's Natural Operations in differential Geometry, and I'm having "doubt" about some of the definitions. All I'm trying to do is sheafify some of ... 1answer 249 views ### Model independent proof of colimit formula for left Kan extensions I am interested in finding a proof of the colimit formula for left Kan extensions$(\infty,1)$-categories which does not rely on a chosen model. (By a model inpendent proof I mean a proof which uses ... 2answers 178 views ### Equivalence of natural transformations Let$\mathcal{C}$be a small category and$\mathrm{Cat}$be the 2-category of small categories. Let$F,G : \mathcal{C} \to \mathrm{Cat}$be two functors and$\theta : F \to G$be a natural ... 1answer 173 views ### Definition of Left Operadic Kan Extension for$\infty$-operads In Lurie's book Higher Algebra, he makes the following definition: Definition 3.1.2.2: Let$M^\otimes\to N(Fin_\ast)\times\Delta^1$be a correspondence from an$\infty$-operad$A^\otimes$to another$...
The simpler question is to study the 2-groupoid $\mathrm{Aut}(\mathsf{Cat})$ of all autoequivalences of the category of (small) categories (i.e. the groupoid of equivalences of categories \$\mathsf{Cat}...