Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Grothendieck Construction and Gradings for Enriched Categories   The Grothendieck construction is a process to form a single category from a
diagram of small categories. In this paper, we extend the definition of the
Grothendieck construction to diagrams of small categories enriched over a
symmetric monoidal category satisfying certain conditions. Symmetric monoidal
categories satisfying the conditions in this paper include the category of
$k$-modules over a commutative ring $k$, the category of chain complexes, the
category of simplicial sets, the category of topological spaces, and the
category of modern spectra. In particular, we obtain a generalization of the
orbit category construction in [math/0312214]. We also extend the notion of
graded categories and show that the Grothendieck construction takes values in
the category of graded categories. Our definition of graded category does not
require any coproduct decompositions and generalizes $k$-linear graded
categories indexed by small categories defined by Lowen.
  There are two popular ways to construct functors from the category of graded
categories to the category of oplax functors. One of them is the smash product
construction defined and studied in [math/0312214,0807.4706,0905.3884] for
$k$-linear categories and the other one is the fiber functor. We construct
extensions of these functors for enriched categories and show that they are
``right adjoint'' to the Grothendieck construction in suitable senses.
  As a byproduct, we obtain a new short description of small enriched
categories.

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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: physics
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-lat
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an