feat: 풀스택 할일관리 앱 구현 (통합 모달 + 간트차트)

- Backend: FastAPI + MongoDB + Redis (카테고리, 할일 CRUD, 파일 첨부, 검색, 대시보드)
- Frontend: Next.js 15 + Tailwind + React Query + Zustand
- 통합 TodoModal: 생성/수정 모달 통합, 탭 구조 (기본/태그와 첨부)
- 간트차트: 카테고리별 할일 타임라인 시각화
- TodoCard: 제목/카테고리/우선순위/태그/첨부 한줄 표시
- Docker Compose 배포 (Frontend:3010, Backend:8010, MongoDB:27021, Redis:6391)

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

backend/app/database.py

@@ -0,0 +1,98 @@
+from motor.motor_asyncio import AsyncIOMotorClient, AsyncIOMotorDatabase
+import redis.asyncio as aioredis
+from app.config import get_settings
+
+
+class Database:
+    client: AsyncIOMotorClient | None = None
+    db: AsyncIOMotorDatabase | None = None
+    redis: aioredis.Redis | None = None
+
+
+db = Database()
+
+
+def get_database() -> AsyncIOMotorDatabase:
+    """MongoDB 데이터베이스 인스턴스를 반환한다.
+    주의: `if not db.db:` 사용 금지 (pymongo 4.9.x NotImplementedError)
+    """
+    if db.db is None:
+        raise RuntimeError("Database not initialized")
+    return db.db
+
+
+def get_redis() -> aioredis.Redis:
+    """Redis 클라이언트 인스턴스를 반환한다."""
+    if db.redis is None:
+        raise RuntimeError("Redis not initialized")
+    return db.redis
+
+
+async def connect_db():
+    """MongoDB + Redis 연결 초기화"""
+    settings = get_settings()
+
+    # MongoDB
+    db.client = AsyncIOMotorClient(settings.mongodb_url)
+    db.db = db.client[settings.mongodb_database]
+
+    # 인덱스 생성
+    await create_indexes(db.db)
+
+    # Redis
+    db.redis = aioredis.from_url(
+        settings.redis_url,
+        encoding="utf-8",
+        decode_responses=True,
+    )
+
+
+async def disconnect_db():
+    """연결 종료"""
+    if db.client is not None:
+        db.client.close()
+    if db.redis is not None:
+        await db.redis.close()
+
+
+async def create_indexes(database: AsyncIOMotorDatabase):
+    """컬렉션 인덱스 생성"""
+    todos = database["todos"]
+    categories = database["categories"]
+
+    # todos 텍스트 검색 인덱스
+    await todos.create_index(
+        [("title", "text"), ("content", "text"), ("tags", "text")],
+        name="text_search_index",
+        weights={"title": 10, "content": 5, "tags": 3},
+    )
+
+    # todos 복합 인덱스
+    await todos.create_index(
+        [("category_id", 1), ("created_at", -1)],
+        name="category_created",
+    )
+    await todos.create_index(
+        [("completed", 1), ("created_at", -1)],
+        name="completed_created",
+    )
+    await todos.create_index(
+        [("priority", 1), ("created_at", -1)],
+        name="priority_created",
+    )
+    await todos.create_index(
+        [("tags", 1)],
+        name="tags",
+    )
+    await todos.create_index(
+        [("due_date", 1)],
+        name="due_date",
+    )
+    await todos.create_index(
+        [("completed", 1), ("due_date", 1)],
+        name="completed_due_date",
+    )
+
+    # categories 인덱스
+    await categories.create_index("name", unique=True, name="category_name_unique")
+    await categories.create_index("order", name="category_order")