Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Categories with families and first-order logic with dependent sorts   First-order logic with dependent sorts, such as Makkai's first-order logic
with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed
(intuitionistic) first-order logic (DFOL), may be regarded as logic enriched
dependent type theories. Categories with families (cwfs) is an established
semantical structure for dependent type theories, such as Martin-L\"of type
theory. We introduce in this article a notion of hyperdoctrine over a cwf, and
show how FOLDS and DFOL fit in this semantical framework. A soundness and
completeness theorem is proved for DFOL. The semantics is functorial in the
sense of Lawvere, and uses a dependent version of the Lindenbaum-Tarski algebra
for a DFOL theory. Agreement with standard first-order semantics is
established. Applications of DFOL to constructive mathematics and categorical
foundations are given. A key feature is a local propositions-as-types
principle.

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The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
24
The categories of this paper are: dg-ga
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph