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V. Hinich

Publications and source records attributed to V. Hinich.

13 recordsLinked to original sources
arXiv2021

Yoneda lemma for enriched infinity categories

We continue the study of enriched infinity categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched infinity categories are associative monoids in an especially designed monoidal category of enriched quivers. We prove that, in case the monoidal structure in the basic category M comes from direct product, our definition is essentially equivalent to the approach via Segal objects. Furthermore, we compare our notion with the notion of category left-tensored over M, and prove a version of Yoneda lemma in this context. Version 2: An error in 2.6.2 corrected. Version 3: a few minor corrections. Version 4: Section 8 added, describing correspondences of enriched categories. In case the basic monoidal category M is a prototopos with a cartesian structure, we prove that the category of correspondences is equivalent to the category of enriched categories over [1]. Version 5: terminology changed (former bicartesian fibrations became bifibrations), a few misprints corrected. Version 6: Section 2.11 added, dealing with operadic sieves. A number of corrections and clarifications made per referee's request. Version 7: final version, accepted to Advances in Math. Version 8: a minor correction of 2.8.9-2.8.10.

math.CT↗
arXiv2020

So, what is a derived functor?

In the context of infinity categories, we rethink the notion of derived functor in terms of correspondences. This is especially convenient for the description of a passage from an adjoint pair (F,G) of functors to a derived adjoint pair (LF,RG). In particular, canonicity of this passage becomes obvious. 2nd version: added comparison to Deligne's definition (SGA4) and a discussion of diagrams of derived functors. Introduction rewritten and references added. 3rd version: description of Kan extensions in terms of correspondences more detailed. 4th version: the final version accepted to HHA.

math.CT↗
arXiv2016

Enriched Yoneda Lemma

We present a version of enriched Yoneda lemma for conventional (not infinity-) categories. We require the base monoidal category to have colimits, but do not require it to be closed or symmetric monoidal.

math.CT↗
arXiv2015

Dwyer-Kan localization revisited

A version of Dwyer-Kan localization in the context of infinity-categories and simplicial categories is presented. Some results of the classical papers by Dwyer and Kan on simplicial localization are reproven and generalized. It is proven that a Quillen pair of model categories gives rise to an adjoint pair of the DK localizations. Also a result on localization of a family of infinity-categories is proven. This, in particular, is applied to localization of symmetric monoidal infinity-categories, where some (partial) results are obtained.

math.QA↗
arXiv2015

Rectification of algebras and modules

Let O be a topological (colored) operad. The Lurie infinity-category of O-algebras with values in (infinity-category of) complexes is compared to the infinity-category underlying the model category of (classical) dg O-algebras. This can be interpreted as a "rectification" result for Lurie operad algebras. A similar result is obtained for modules over operad algebras, as well as for algebras over topological PROPs.

math.QA↗
arXiv2015

Formality theorem and bialgebra deformations

In this paper we prove formality of the exterior algebra on V+V* endowed with the big bracket considered as a graded Poisson algebra. We also discuss connection of this result to bialgebra deformations of the symmetric algebra of V considered as bialgebra.

math.QA↗
arXiv2015

Homotopy coherent nerve in Deformation theory

In this note we explain that homotopy coherent simplicial nerve has to used intead of the standard definition in the author's papers on formal deformation theory. A convenient version of the notion of fibered category is presented which is useful once one works with simplicial categories.

math.QA↗
arXiv2005

Drinfeld double for orbifolds

We prove that the Drinfeld double of the category of sheaves on an orbifold is equivalent to the category of sheves on the corresponding inertia orbifold.

math.QA↗
arXiv2003

Erratum to "Homological algebra of homotopy algebras"

Theorem 6.1.1 of [H.A.H.A.] on the existence of a model structure on the category of operads is not valid in the generality claimed. We present here a counter-example (due to B. Fresse) and a corrected version of the theorem.

math.QA↗
arXiv1999

Deformations of homotopy algebras

Let $k$ be a field of characteristic zero, $\CO$ be a dg operad over $k$ and let $A$ be an $\CO$-algebra. In this note we define formal deformations of $A$, construct the deformation functor $$\Def_A:\dgar(k)\to\simpl$$ from the category of artinian local dg algebras to the category of simplicial sets. In the case $\CO$ and $A$ are non-positively graded, we prove that $\Def_A$ is governed by the tangent Lie algebra $T_A$ (defined as derivations of a cofibrant resolution of $A$). A very easy example shows that the result does not hold without this condition.

math.AG↗
arXiv1998

DG coalgebras as formal stacks

The category of unital (unbounded) dg cocommutative coalgebras over a field of characteristic zero is provided with a structure of simplicial closed model category. This generalizes the model structure defined by Quillen in 1969 for 2-reduced coalgebras. In our case, the notion of weak equivalence is structly stronger than that of quasi-isomorphism. A pair of adjoint functors connecting the category of coalgebras with the category of dg Lie algebras, induces an equivalence of the corresponding homotopy categories. The model category structure allows one to consider dg coalgebras as very general formal stacks. The corresponding Lie algebra is then interpreted as a tangent Lie algebra which defines the formal stack uniquely up to a weak equivalence. An example of the coalgebra of formal deformaions of a principal $G$-bundle on a scheme $X$ is calculated.

math.AG↗